An upper bound on Reidemeister moves more

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Topology, Geometry And Topology, Mathematics, Geometric Topology, Complexity Theory, Knots and Surfaces, and Knot Theory

AN UPPER BOUND ON REIDEMEISTER MOVES ALEXANDER COWARD AND MARC LACKENBY Abstract. We provide an explicit upper bound on the number of Reidemeister moves required to pass between two diagrams of the same link. This leads to a conceptually simple solution to the equivalence problem for links. 1. Introduction It is one of the most fundamental theorems in low-dimensional topology that any two diagrams of a knot or link in R3 diﬀer by a sequence of Reidemeister moves, illustrated in Figure 1. Since Reidemeister’s seminal paper [22] in 1926, it has been speculated as to whether there is an explicit upper bound for the number of moves that are needed, as a function of the number of crossings in the initial and terminal diagrams. See [23] or page 15 of [1] for example. In a celebrated paper [5], Hass and Lagarias provided such a bound when the link in question is the unknot. In this paper we answer the general question with the following theorem, which applies to all knots and links. Theorem 1.1. Let D1 and D2 be connected diagrams for some knot or link in R3 , and let n be the sum of their crossing numbers. Then D2 may be obtained from D1 by n a sequence of at most exp(c ) (n) Reidemeister moves, where c = 101,000,000 . Figure 1. Reidemeister moves Here, a link diagram is a 4-valent graph embedded in R2 with each vertex decorated with ‘over’ and ‘under’ crossing information. If a knot or link is oriented, then its diagrams have directed edges that agree with this orientation. Theorem 1.1 applies to link diagrams with or without orientation. We view two diagrams as the same if their decorated (oriented) graphs are ambient isotopic in R2 . The function exp(x) is the exponential function 2x , and exp(r) (x) means iterate this function r times. Thus, n exp(c ) (n) is shorthand for a tower of 2s with an n at the top, the height of the tower being cn . This upper bound is very large indeed, but it is explicit and computable. It demonstrates for the ﬁrst time that the number of Reidemeister moves required is bounded by a primitive recursive function of the number of crossings in the two diagrams. It furthermore leads to a conceptually very simple algorithm for solving the equivalence problem for links. Given two links L1 and L2 with connected diagrams D1 and D2 , one can decide whether they are the same link as follows. Let n be the sum of their crossing numbers. Apply all possible sequences of Reidemeister moves to D1 of MSC (2010): 57M25, 57N10 1 2 n ALEXANDER COWARD AND MARC LACKENBY length at most exp(c ) (n). If L1 and L2 are equivalent links, one of these diagrams will be ambient isotopic to D2 , and this can readily be determined. On the other hand, if L1 and L2 are inequivalent, then none of these diagrams will be isotopic to D2 . It is trivial that there is some function F : N × N → N such that any two connected diagrams D1 and D2 of a link with n1 and n2 crossings diﬀer by a sequence of at most F (n1 , n2 ) Reidemeister moves. This follows from the fact that there are only ﬁnitely many connected diagrams with a given number of crossings, and Reidemeister’s theorem. However, the existence of a computable function F is a much deeper result, as the following simple theorem demonstrates. Theorem 1.2. The following are equivalent: (1) There is a computable function F : N × N → N such that for any two connected diagrams D1 and D2 of a link with n1 and n2 crossings, there is a sequence of at most F (n1 , n2 ) Reidemeister moves that takes D1 to D2 . (2) There is an algorithm to solve the equivalence problem for links. In other words, there is an algorithm that takes as input two link diagrams and determines whether or not they represent equivalent links. The proof of (1) ⇒ (2) is described above. For (2) ⇒ (1), we need to produce an algorithm that, given natural numbers n1 and n2 , computes a natural number F (n1 , n2 ) with the required properties. To do this, the computer enumerates all connected link diagrams with at most max{n1 , n2 } crossings. Then using the hypothesised algorithm, the computer arranges these into groups according to their link type. Then, the computer searches for Reidemeister moves relating all diagrams of each type. Such sequences of moves exist by Reidemeister’s theorem. Hence, eventually, an upper bound on the number of moves will be computed. We call a function F : N × N → N a Reidemeister move function for a link L (which may be oriented or unoriented) if for any two connected diagrams D1 and D2 of L with n1 and n2 crossings, there is a sequence of at most F (n1 , n2 ) Reidemeister moves that takes D1 to D2 . Thus, Theorem 1.1 gives a Reidemeister move function that applies to all links. The equivalence problem for links was solved by Haken [4] and Hemion [9] in the 1960s and 1970s. In fact, an alternative solution to the homeomorphism problem for hyperbolic link complements, and hence the equivalence problem for hyperbolic knots, was given by Dahmani and Groves [3], based on work of Sela [24]. Theorem 1.2 thus implies the existence a computable Reidemeister move function. The existence of an explicit, primitive recursive Reidemeister move function, as exhibited in Theorem 1.1, is a much stronger statement. Theorem 1.1 is proved using triangulations and Pachner moves. Much as any two diagrams of a link are related by a sequence of Reidemeister moves, any two triangulations of a PL manifold are related by a sequence of Pachner moves [20, 21]. The key theorem we use is an adaptation of a result of Mijatovi´ [17], who provides an c explicit upper bound on the number of Pachner moves required to pass between two triangulations of a knot exterior. However, the translation from a bound on Pachner moves to a bound on Reidemeister moves is not a straightforward one. AN UPPER BOUND ON REIDEMEISTER MOVES 3 Starting with two connected diagrams D1 and D2 for a link, we pick embeddings L1 and L2 of the link in R3 which project to these diagrams and which both lie in some convex 3-ball. We use D1 and D2 to build two triangulations for the link exterior in this 3-ball. Using an adaptation of Mijatovi´’s theorem, these are related by a bounded c number of Pachner moves. These induce an explicit PL homeomorphism between the triangulated link exteriors. This extends to a homeomorphism of the 3-ball sending L1 to L2 . The bound on the number of Pachner moves provides some control on this homeomorphism. We then apply Alexander’s trick to specify an ambient isotopy sending L1 to L2 , and this ambient isotopy is again of controlled complexity in a certain sense. The bound on the complexity of this ambient isotopy is ultimately what provides us with our upper bound for the number of Reidemeister moves required to pass between D1 and D2 . A signiﬁcant complication arises from the fact that Mijatovi´’s theorem is not quite c suﬃcient for our purposes. His result provides an explicit upper bound on the number of Pachner moves required to pass between two triangulations of a link exterior, up to a homeomorphism that is isotopic to the identity in the hyperbolic pieces of the link’s JSJ decomposition. This is insuﬃcient, for two reasons. Firstly, many link exteriors have Seifert ﬁbred pieces in their JSJ decomposition, and so the resulting homeomorphism of the link exterior may act non-trivially on the boundary, in which case it may not extend to a homeomorphism of the 3-ball. But Mijatovi´’s theorem c is not suﬃcient for our purposes even in the case of hyperbolic knots, because it only provides a bound on the number of Pachner moves up to ambient isotopy, and yet it is exactly an explicit ambient isotopy that we are aiming to construct. It is therefore necessary for us to prove a strengthened version of Mijatovi´’s theorem. c This takes some eﬀort, because it requires us to go through his proofs, originally exposed in [14], [15], [16] and [17], and adapt them. We have thus endeavoured to give an accessible outline of Mijatovi´’s work, highlighting the similarities and diﬀerences c between his methods and ours. The following is our result. Theorem 1.3. Let M be a compact orientable irreducible 3-manifold with boundary a non-empty collection of tori. Suppose that the closure of each component of the complement of the characteristic submanifold of M satisﬁes at least one of the following conditions: • it does not ﬁbre over the circle; or • it is not a surface semi-bundle; or • it has at least two boundary components. ′ Let T∂M be any triangulation of ∂M. Then there is a triangulation Tcan of M with the following properties. Its restriction to ∂M equals T∂M . Further, if T is any triangulation of M with t tetrahedra, such that the restriction of T to ∂M also equals T∂M , t then there is a sequence of at most exp(a ) (t) interior Pachner moves, followed by a ′ homeomorphism of M that is the identity on ∂M, taking T to Tcan . This homeomorphism is isotopic to one equal to the identity on the complement of the characteristic submanifold of M. Here, a = 2162 . 4 ALEXANDER COWARD AND MARC LACKENBY By an interior Pachner move, we mean a Pachner move that does not aﬀect the triangulation of the boundary of M. See Section 2 for a precise deﬁnition. A surface semi-bundle is a compact orientable 3-manifold obtained from two I-bundles over nonorientable surfaces by identifying their horizontal boundaries via a homeomorphism. ′ We use the phrase ‘canonical triangulation’ for Tcan . The word ‘canonical’ needs to be used with some caution, because it depends on the given triangulation T∂M of ∂M, and arbitrary choices are made in its construction. It is built out of a collection of surfaces in M, which is very nearly a hierarchy. The above theorem can of course be used to bound the number of interior Pachner moves required to pass between two triangulations of M that are equal on ∂M. We call a function F : N × N → N a Pachner move function for a compact 3-manifold M if for any two triangulations T1 and T2 for M, equal on ∂M and with at most n1 and n2 tetrahedra, there is a sequence of at most F (n1 , n2 ) interior Pachner moves, followed by a homeomorphism of M that is the identity on ∂M, that takes T1 to T2 . Thus Theorem 1.3 implies that if M is a 3-manifold satisfying the hypotheses of the n n theorem, then (n1 , n2 ) → exp(a 1 ) (n1 ) + exp(a 2 ) (n2 ) is a Pachner move function for M where a = 2162 . It is in this way that we will apply Theorem 1.3. The bound on Pachner moves in Theorem 1.3 gives the bound on Reidemeister moves in Theorem 1.1, via the following result. Theorem 1.4. Let L be an oriented non-split link in S 3 . Suppose that PL : N×N → N is a Pachner move function for the exterior of L in S 3 . Then RL : N × N → N given by RL (n1 , n2 ) = exp(2) (400(PL (214 (n1 + n2 ), 214 (n1 + n2 )) + 216 (n1 + n2 ))) is a Reidemeister move function for L. An outline of this paper is as follows. In Section 2, we introduce some of the basic PL machinery that we require. In Section 3, we explain how to work with triangulations of the 3-ball rather than the 3-sphere. In Section 4, we explain how to construct, starting with a connected diagram of a link L, an explicit triangulation T of a 3-ball that contains L as a subcomplex and also possesses some other properties that we will require. In Section 5, we review Alexander’s trick and use it to show how one may pass from a homeomorphism of the 3-ball, of known complexity, sending L1 to L2 , to an ambient isotopy sending L1 to L2 also of known complexity. This ambient isotopy induces a 1-parameter family of link projections interpolating between D1 and D2 . However these link projections may not be diagrams in the usual sense. Thus in Section 6, we show how one may pass from a 1-parameter family of link projections to a sequence of diagrams each related to the next by a single Reidemeister move. In Section 7, Theorem 1.4 is proved, leading quickly to Theorem 1.1. In Sections 8 and 9, we state Mijatovi´’s result, compare it with Theorem 1.3, and summarise his proof c in the case of simple and Seifert ﬁbred 3-manifolds. In Sections 10, 11, 12 and 13, we give our proof of Theorem 1.3. AN UPPER BOUND ON REIDEMEISTER MOVES 5 2. Piecewise linear theory Our main theorem is proved using piecewise linear techniques. We therefore start by giving precise deﬁnitions from this theory. For n ∈ N, let ∆n be the standard n-simplex. An abstract ∆-complex K is: • an indexing set A(n), for each natural number n; • a copy ∆n of the standard n-simplex for each α ∈ A(n); α • for each ∆n , where n > 0, an aﬃne homeomorphism between each (n − 1)α n−1 dimensional face of ∆n and some ∆β for β ∈ A(n − 1). α The term ∆-complex was also used by Hatcher in [7]. However, his use of the term was slightly diﬀerent. He ordered the vertices of each simplex ∆n and required that α n−1 each aﬃne homeomorphism from a face of ∆n to ∆β preserved the ordering of the α vertices. We will not make that requirement. The basic example of an abstract ∆-complex is a simplicial complex. One forms the underlying space |K| of an abstract ∆-complex K by starting with the disjoint union of the simplices, and identifying each face of each simplex ∆n with α n−1 the corresponding simplex ∆β . We shall use the term ∆-complex to denote either an abstract ∆-complex or its underlying space. A combinatorial isomorphism between abstract ∆-complexes K and K ′ is, for each natural number n, a bijection between their indexing sets A(n) and A′ (n), together with a collection of aﬃne homeomorphisms ∆n → ∆n′ for each α ∈ A(n) associated α α with α′ ∈ A′ (n), such that for each (n − 1)-dimensional face F of ∆n and F ′ of ∆n′ α α that correspond, the following diagram commutes: F ∼ = ∨ n−1 ∆β > F′ ∼ = ∨ > ∆n−1 β′ Here, the vertical maps are the aﬃne homeomorphisms in the deﬁnition of a ∆complex. The top horizontal map is the restriction of the aﬃne homeomorphism ∆n → ∆n′ . The bottom horizontal map is the aﬃne homeomorphism arising from the α α bijection between A(n − 1) and A′ (n − 1). Put simply, a combinatorial isomorphism is a bijection that preserves all the structure of the complex. The combinatorial isomorphism between K and K ′ determines a homeomorphism |K| → |K ′ | such that the interior of each simplex of |K| is sent to the interior of a simplex of |K ′ | via an aﬃne homeomorphism. We shall also term this homeomorphism a combinatorial isomorphism. A subdivision of a ∆-complex K is a ∆-complex K ′ together with a homeomorphism h : |K ′ | → |K| such that each simplex σ of |K ′ | is mapped into a simplex of |K| and the restriction of h to σ is aﬃne. A triangulation T of a space M is a ∆-complex K together with a homeomorphism M → |K|. (When M already has a PL structure, we insist that this homeomorphism is PL.) We let |T | denote |K|. When K is actually a simplicial complex, we term this a genuine triangulation. (In traditional PL theory, what we term a ‘genuine 6 ALEXANDER COWARD AND MARC LACKENBY triangulation’ is just called a ‘triangulation’, and only these are considered. However, it has become standard practice, particularly in low-dimensional topology, to work with ∆-complexes rather than restrict to simplicial complexes.) When K ′ is a subdivision of K, there is an induced homeomorphism M → |K ′ | which is the subdivided triangulation. Two triangulations h : M → |K| and h′ : M → |K ′ | of a space M are equal if there is a combinatorial isomorphism c : |K| → |K ′ | such that the following diagram commutes: M h h′ > < |K| c > |K ′ | This is a rather restrictive condition. If we are only given a combinatorial isomorphism c : |K| → |K ′ |, there is an associated homeomorphism h′−1 ch : M → M which realises this combinatorial isomorphism. There is no guarantee that this is the identity. We say that the two triangulations are isotopy-equivalent if this homeomorphism is isotopic to the identity on M. We also say that two triangulations are homeomorphismequivalent if their ∆-complexes are combinatorially isomorphic, but with no restriction on the commutativity of the above diagram. Thus, homeomorphism-equivalence of triangulations is essentially the same concept as combinatorial isomorphism, but it is useful to have two diﬀerent terms. It is helpful to consider an example. Suppose that M is a space with inﬁnite mapping class group. Then it is immediate that if it admits a triangulation with N simplices, it admits inﬁnitely many such triangulations that are homeomorphism-equivalent but isotopy-inequivalent, as follows. Starting with one such triangulation h : M → |K|, one may apply non-isotopic homeomorphisms φi : M → M (i ∈ N) to obtain new triangulations hφi . Since K is ﬁnite, there are only ﬁnitely many combinatorial isomorphisms |K| → |K|, and these realise only ﬁnitely many isotopy classes of homeomorphisms M → M. Hence, inﬁnitely many of these triangulations of M are distinct, even up to isotopy-equivalence. This lack of ﬁniteness is an important phenomenon that occurs often in the situations we will examine. For example, the torus has inﬁnite mapping class group, as does the exterior of every satellite knot. We will be concerned with triangulations of manifolds. In this case, there is a well known way to vary a triangulation, via Pachner moves, which are deﬁned as follows. Let T be a triangulation of an n-manifold M. Let ∆ be a standard (n + 1)-simplex. Let D be a non-empty subset of ∂∆, consisting of i < n + 2 faces of dimension n. Suppose that some subcomplex of T has interior that is combinatorially isomorphic to the interior of D. (The typical case is when the subcomplex itself is combinatorially isomorphic to D, but we wish to permit the possibility that it may be obtained from a copy of D by identifying simplices on the boundary.) Then the operation of removing from T the copy of the interior of D and inserting the interior of cl(∂∆ − D) is an interior Pachner move. This is known as an (i, n + 2 − i) move, since it replaces i n-simplices with n + 2 − i ones. See Figure 2. AN UPPER BOUND ON REIDEMEISTER MOVES 7 (1, 4) (4, 1) (2, 3) (3, 2) Figure 2. Three-dimensional Pachner moves If triangulations T and T ′ are related by an interior Pachner move, then they have a common subdivision. More precisely, one can also consider the triangulation T ′′ that is obtained by replacing the interior of D by the cone on ∂D. It is possible to realize this cone as a subdivision of both D and cl(∂∆ − D), in a way that is canonical up to isotopy equivalence. We therefore obtain homeomorphisms h : |T ′′| → |T | and h′ : |T ′′ | → |T ′ |, and hence a homeomorphism h′ ◦ h−1 : |T | → |T ′ |. See Figure 3. T ′′ h h′ h′ ◦ h−1 T T′ Figure 3. T ′′ is a common subdivision of both T and T ′ If the n-manifold M has non-empty boundary, Pachner moves do not aﬀect the triangulation of ∂M. Thus, it is necessary to introduce a related move, which we call a boundary Pachner move. Here, one starts with a standard n-simplex ∆ and a non-empty subset D of ∂∆ consisting of i < n + 1 faces of dimension n − 1. One 8 ALEXANDER COWARD AND MARC LACKENBY ﬁnds a subset of ∂M whose interior is combinatorially isomorphic to the interior of D, and one attaches ∆ to it along D. Alternatively, one performs the reverse of this procedure. This operation has the eﬀect of changing the triangulation on ∂M by an interior Pachner move. A Pachner move refers to an interior or a boundary Pachner move. The following key result about Pachner moves is due to Pachner [20, 21]. See also [12]. Theorem 2.1. Let T and T ′ be two triangulations of a closed PL manifold M. Then, up to homeomorphism-equivalence, there is a ﬁnite sequence of Pachner moves that takes T to T ′ . One might then wish to determine a bound on this number of moves, solely in terms of the number t and t′ of top-dimensional simplices in T and T ′ . It trivially holds that there is a function F : N × N → N such that any two triangulations T and T ′ diﬀer by a sequence of at most F (t, t′ ) Pachner moves, up to homeomorphismequivalence. However, any useful bound is impossible in general, as the following theorem demonstrates. This is proved in much the same way as Theorem 1.2. Theorem 2.2. Let M be a compact orientable PL m-manifold, where m ≤ 4. Then the following are equivalent: (1) There is a computable function F : N × N → N such that for any two triangulations T1 and T2 of M, with t1 and t2 m-simplices respectively, there is a sequence of at most F (t1 , t2 ) Pachner moves that takes T1 to a triangulation homeomorphism-equivalent to T2 . (2) There is an algorithm to recognize M among all triangulated compact PL mmanifolds. In other words, there is an algorithm that takes as input the triangulation of some compact m-manifold and determines whether or not this manifold is PL homeomorphic to M. Note that, by a theorem of Novikov, there are closed m-dimensional manifolds which cannot be recognized among the set of all triangulated PL m-manifolds, provided m ≥ 4. Indeed, the m-sphere falls in this class when m ≥ 5. (See the appendix to [18] for example.) An explicit example of such a manifold when m = 4 is the connected sum of 16 copies of S 2 × S 2 [25]. One might wonder why it is necessary to assume that m ≤ 4 in the above result. This is required in the proof of (2) ⇒ (1). Here, one starts with a recognition algorithm for M and from this, one constructs an algorithm to compute F (t1 , t2 ) for any positive integers t1 and t2 . To do this, one constructs all spaces obtained from max{t1 , t2 } m-simplices by identifying their (m − 1)-dimensional faces in pairs. But one must then discard all spaces that do not form an m-manifold, and to do this, a recognition algorithm for the (m−1)-sphere is required. This is only known to exist for m ≤ 4 [26]. Then, once one has constructed this collection of triangulated m-manifolds, one then applies the recognition algorithm for M to discard the triangulations of manifolds not PL homeomorphic to M. The resulting triangulations of M are all related by sequences of Pachner moves, which one can eventually construct. One deﬁnes F (t1 , t2 ) to be the maximum length of each such sequence of Pachner moves. AN UPPER BOUND ON REIDEMEISTER MOVES 9 The solution to the recognition problem for Haken 3-manifolds was established by Haken [4] and Hemion [9]. Hence, for 3-manifolds M in this class, there are computable functions F as above. Moreover, Mijatovi´ in [17] provided an explicit and easily c computed function F for a large class of Haken 3-manifolds that includes the exteriors of all non-split links in the 3-sphere. It is Mijatovi´’s explicit upper bound, and the c technology behind it, that is the key to this paper. 3. Pachner moves on punctured 3-manifolds In this section, we will deal with 3-manifolds which may have a 2-sphere boundary component. Our goal is to prove the following theorem. Theorem 3.1. Let T1 and T2 be triangulations of a 3-ball B. Suppose that in a collar neighbourhood of ∂B, these triangulations are equal, and are the same standard triangulation of ∂B × I. Let t1 and t2 be the number of 3-simplices in T1 and T2 respectively. Suppose that links L1 and L2 are subcomplexes of T1 and T2 respectively, and that they have triangulated neighbourhoods N(L1 ) and N(L2 ) that are combinatorially isomorphic. Suppose that the homeomorphism from N(L1 ) to N(L2 ) that realises this combinatorial isomorphism preserves the longitudal slope of each boundary component. Suppose also that L1 and L2 are ambient isotopic, and that P : N×N → N is a Pachner move function for the exterior of this ambient isotopy class of link in S 3 . Then there exists a sequence of at most 100P (t1 , t2 ) + 100(t1 + t2 ) Pachner moves, followed by a combinatorial isomorphism, that takes T1 to T2 , and N(L1 ) to N(L2 ). Furthermore, none of the Pachner moves aﬀect N(L1 ) or ∂B, and the combinatorial isomorphism restricts to the identity on ∂B. We say that the standard triangulation of the 2-sphere S is the boundary of a 3simplex. Let TS×I be the following triangulation of S × I. Give each component of S × ∂I the same standard triangulation. For each edge e of this triangulation, insert e × I into S × I. Place a vertex in the interior of this rectangle, and cone oﬀ the rectangle from this vertex. These rectangles divide S × I into a collection of balls. Insert a vertex in the interior of each such ball and cone oﬀ the ball from this vertex. The resulting triangulation is TS×I , which we term the standard triangulation of S × I. Theorem 3.1 will be a consequence of the following result. Theorem 3.2. Let M be a compact orientable 3-manifold, and let S be a 2-sphere boundary component of M. Suppose that the 3-manifold that results from attaching a 3-ball to S has P : N × N → N as a Pachner move function. Let T1 and T2 be triangulations of M with at most t1 and t2 tetrahedra respectively. Suppose that T1 and T2 are equal on ∂M, and that in a collar neighbourhood of S, T1 and T2 are equal and combinatorially isomorphic to TS×I . Then there is a sequence of at most 100P (t1, t2 ) + 100(t1 + t2 ). interior Pachner moves, followed by a homeomorphism that is the identity on ∂M, taking T1 to T2 . We shall need some terminology before embarking on the proof of Theorem 3.2. 10 ALEXANDER COWARD AND MARC LACKENBY ˆ Deﬁnition 3.3. Let M be a compact orientable 3-manifold, let T be a triangulation ˆ → |T | of M , and let x be a point in M that is disjoint from the image under φ−1 ˆ ˆ φ: M of the 2-skeleton. We deﬁne the following puncturing operation. Remove the interior ˆ of the 3-simplex containing φ(x), and insert TS×I . Let P (M, T, x) be the resulting ˆ triangulation of the manifold M − int(N(x)). ˆ Let T1 and T2 be triangulations of the same 3-manifold M . Suppose that they diﬀer by a sequence of N interior Pachner moves T1 = T 0 T1 ··· T N = T2 . For each relevant integer i, let K i be the ∆-complex associated with T i , and let hi : |K i−1 | → |K i | be the homeomorphism resulting from the Pachner move. Let x be ˆ ˆ a point in M disjoint from the inverse image in M of the 2-skeleton of |K 0 |, |K 1 |, . . . N and |K |. ˆ ˆ Lemma 3.4. The triangulations P (M, T 0 , x) and P (M, T N , x) diﬀer, up to ambient ˆ isotopy ﬁxed on ∂ M ∪ ∂N(x), by a sequence of at most 100N interior Pachner moves. ˆ ˆ Proof. It clearly suﬃces to prove that for each i, P (M, T i , x) and P (M, T i+1 , x) diﬀer by a sequence of at most 100 Pachner moves. An example is shown in Figure 4, where T i and T i+1 diﬀer by a (1, 4)-move. (Note that, for clarity, not all simplices in these triangulations are drawn.) Since there are only 4 types of move, it is clear that there ˆ ˆ is a universal constant k such that P (M, T i , x) and P (M, T i+1 , x) diﬀer by at most k interior Pachner moves. An elementary calculation, which is omitted, proves that k = 100 suﬃces. ˆ P (M, T i , x) Figure 4 ˆ P (M, T i+1 , x) ˆ Lemma 3.5. Let T be a triangulation of a compact orientable 3-manifold M with ′ ˆ that are in the complement of the 2t tetrahedra. Let x and x be points in M ˆ ˆ ˆ skeleton. Then, P (M , T, x) and P (M, T, x′ ) are triangulations of M − int(N(x)) and AN UPPER BOUND ON REIDEMEISTER MOVES 11 ˆ M − int(N(x′ )) respectively, from which ∂N(x) and ∂N(x′ ) inherit triangulations. Let ˆ ˆ h : ∂N(x) → ∂N(x′ ) be a simplicial isomorphism. Then, P (M, T, x) and P (M , T, x′ ) diﬀer by a sequence of at most 100t interior Pachner moves, followed by a homeomorˆ ˆ phism M − int(N(x)) → M − int(N(x′ )). We may arrange that this homeomorphism ˆ equals h on ∂N(x), equals the identity on ∂ M , and extends to a homeomorphism ˆ ˆ M → M that is isotopic to the identity. Proof. Suppose that x and x′ lie in distinct 3-simplices ∆ and ∆′ that share a face. Let ∆ ∪ ∆′ denote the 3-ball obtained by gluing ∆ and ∆′ along this face. There may be other identiﬁcations on the boundary of this ball, but we do not yet make these. The triangulations P (∆, ∆, x) ∪ ∆′ and ∆ ∪ P (∆′ , ∆′ , x′ ) are triangulations of (∆ ∪ ∆′ ) − int(N(x)) and (∆ ∪ ∆′ ) − int(N(x′ )) respectively. Pick a homeomorphism φ : (∆ ∪ ∆′ ) − int(N(x)) → (∆ ∪ ∆′ ) − int(N(x′ )) that restricts to h on ∂N(x), equals the identity on ∂(∆ ∪ ∆′ ) and extends to a homeomorphism (∆ ∪ ∆′ ) → (∆ ∪ ∆′ ) that is isotopic, relative to its boundary, to the identity. Then, using φ to pull back the triangulation of (∆ ∪ ∆′ ) − int(N(x′ )), we obtain another triangulation of (∆ ∪ ∆′ ) − int(N(x)). It is clear that there is some number k of interior Pachner moves, followed by an ambient isotopy that equals the identity on ∂(∆ ∪ ∆′ ) ∪ ∂N(x) taking one to the other. One may easily check that k = 100 suﬃces. Since only interior Pachner moves were used, any further identiﬁcations on the boundary of ∆ ∪ ∆′ do not aﬀect this argument. ˆ Now suppose that x and x′ are arbitrary points in M disjoint from the 2-skeleton. ′ There is a sequence of points x = x0 , . . . , xr = x of points disjoint from the 2-skeleton, where r ≤ t, and for each i, xi and xi+1 lie in distinct 3-simplices that share a face. Thus, at most kr ≤ 100t Pachner moves, followed by a homeomorphism as described ˆ ˆ in the lemma, suﬃce to take P (M, T, x) to P (M, T, x′ ). ˆ Proof of Theorem 3.2. Let M be the result of attaching a 3-ball to M along S. Let ˆ1 and T2 be the result of removing the copy of TS×I from T1 and T2 , and inserting ˆ T ˆ ˆ ˆ ˆ a 3-simplex, giving triangulations φ1 : M → |T1 | and φ2 : M → |T2 |. Thus, T1 = ˆ ˆ ˆ ˆ P (M, T1 , y) and T2 = P (M, T2 , y), where y is a point in the interior of the newly attached 3-ball. By assumption, there is a sequence of N ≤ P (t1 , t2 ) interior Pachner moves ˆ T1 = T 0 T1 ··· TN, ˆ ˆ ˆ where T N is homeomorphism-equivalent to T2 , via a homeomorphism h : M → M that ˆ is the identity on ∂ M . For each relevant integer i, let K i be the ∆-complex associated i with T , and let hi : |K i−1 | → |K i | be the homeomorphism resulting from the Pachner ˆ move. Let x be a point in M disjoint from the inverse image of the 2-skeletons of ˆ ˆ ˆ every |K i |. By Lemma 3.4, the triangulations P (M, T1 , x) and P (M, T N , x) diﬀer by a sequence of at most 100N interior Pachner moves, up to ambient isotopy ﬁxed on ˆ ˆ ˆ ∂ M ∪ ∂N(x). By Lemma 3.5, P (M , T N , x) and P (M, T N , h−1 (y)) diﬀer by a sequence of at most 100t2 Pachner moves, followed by a simplicial isomorphism that acts as ˆ ˆ ˆ ˆ the identity on ∂ M . Now, P (M, T N , h−1 (y)) and P (M, T2 , y) are homeomorphismˆ equivalent, via a homeomorphism that is the identity on ∂ M . The composition of the 12 ALEXANDER COWARD AND MARC LACKENBY above homeomorphisms restricts to a simplicial isomorphism φ : ∂N(x) → ∂N(y). By ˆ ˆ ˆ ˆ Lemma 3.5, T1 = P (M, T1 , y) and P (M, T1 , x) diﬀer by a sequence of at most 100t1 interior Pachner moves, followed by a simplicial isomorphism that acts as the identity ˆ on ∂ M . Moreover, we may ensure that the induced map ∂N(y) → ∂N(x) equals φ−1 . Thus, we have related T1 and T2 by a sequence of at most 100P (t1, t2 ) + 100(t1 + t2 ) interior Pachner moves, followed by a homemorphism that acts as the identity on the boundary of M. Proof of Theorem 3.1. We are given triangulations T1 and T2 of the 3-ball, B. The links L1 and L2 and their regular neighbourhoods N(L1 ) and N(L2 ) are subcomplexes. ′ Let M = B − int(N(L1 )). Then T1 restricts to a triangulation T1 for M. Now, L1 and L2 are assumed to be equivalent, and so there is a homeomorphism h of B taking N(L1 ) to N(L2 ). After an isotopy, we may assume that the restriction of h to N(L1 ) ′ realises the given combinatorial isomorphism between N(L1 ) and N(L2 ). Let T2 be the −1 triangulation of M obtained by transferring the restriction of T2 to M via h . Thus, ′ ′ we have two triangulations of M, T1 and T2 , and these restrict to equal triangulations of ′ ′ ∂M. Note that the number of tetrahedra in T1 and T2 is at most t1 and t2 respectively. ˆ Let M be the result of attaching a 3-ball to the 2-sphere boundary component of M. ˆ By assumption, P : N × N → N is a Pachner move function for M. Applying Theorem 3.2, we obtain a sequence of at most 100P (t1 , t2 ) + 100(t1 + t2 ) interior Pachner moves, ′ ′ followed by a homeomorphism that is the identity on ∂M, taking T1 to T2 . This induces a sequence of at most this many interior Pachner moves, followed by a combinatorial isomorphism that is the identity on ∂B, taking T1 to T2 and N(L1 ) to N(L2 ). This completes the proof of Theorem 3.1. 4. Constructing a triangulation from a link diagram In this section, we show how a link diagram can be used to construct a triangulation of the link’s exterior. Deﬁnition 4.1. The standard triangulation of a cube is obtained as follows. Start by inserting a vertex into each face, and coning oﬀ each face from this vertex. This gives a triangulation of the boundary of the cube. Now place a vertex at the centre of the cube, and cone oﬀ the triangulation of the boundary. See Figure 5. Deﬁnition 4.2. For any positive integer n, the standard triangulation of the solid torus with length n is obtained by gluing together n cubes in a circular fashion, with their standard triangulations, where the intersection of a cube with its neighbours is precisely a pair of opposite faces of the cube and where the other faces patch together to form four annuli. Theorem 4.3. Let L be a link in S 3 with components L1 , . . . , Lr . Let D be a connected diagram of L with c(D) crossings. For 1 ≤ i ≤ r, let ci be the number of crossings in which at least one strand is part of Li . Let n1 , . . . , nr be integers satisfying ni ≥ 10ci for each i, and let n = n1 + · · · + nr . Then there is a triangulation T of a convex 3-ball in R3 with the following properties: • it has at most 212 c(D) + 120(n + 11) tetrahedra; AN UPPER BOUND ON REIDEMEISTER MOVES 13 Figure 5 • it contains L as a subcomplex, and also a neighbourhood N(L) = N(L1 ) ∪ · · · ∪ N(Lr ) of L; • the vertical projection of this copy of L onto the horizontal plane is the diagram D; • for each i, N(Li ) has the standard triangulation of the solid torus with length ni ; • each simplex of T is straight in the aﬃne structure on R3 ; • a collar neighbourhood of the boundary of the ball is triangulated as TS×I . Proof. Step 1. First suppose D is not the trivial unknot diagram. Let G be the underlying 4-valent planar graph of D. We start by embedding G in a square Q in R2 so that each edge of G is a union of at most 3 straight arcs. To ﬁnd such an embedding, we ﬁrst collapse parallel edges of G to a single edge and remove edge loops, forming a graph G. Using F´ry’s Theorem we may ﬁnd an embedding of G in the plane in which a every edge is straight. Now reinstate the parallel edges of G with 2 straight arcs each and the edge loops of G with 3 straight arcs. If D is the trivial unknot diagram then let G be a triangle formed of three straight edges in the interior of Q. Step 2. Replace each edge of G by 4 parallel edges. Replace each 2-valent vertex of G by 4 vertices joined by 3 straight edges. Replace each 4-valent vertex of G by 9 parallelograms. See Figure 6. Call the resulting graph G+ . Step 3. We will use G+ to triangulate Q as follows. Into each complementary region of G+ coming from the complementary regions of G we add straight edges until the region is triangulated. Denote by E the union of these edges together with G+ and ∂Q. An elementary Euler characteristic argument shows that E decomposes Q into at most max(10c(D) + 2, 8) ≤ 10c(D) + 8 triangles and at most max(27c(D), 9) ≤ 27c(D) + 9 convex quadrilaterals. Into the quadrilateral shaped regions add a vertex and cone from this vertex. The result is a triangulation of the square Q which we denote by TG . Step 4. Insert 4 copies of Q with its triangulation TG into the cube Q × I, one being the top face, one the bottom face, and two parallel copies between them. Insert a copy of E × I, lying vertically in the cube, running from top to bottom. The union of these copies of Q with E × I decomposes the cube into a collection of convex balls. 14 ALEXANDER COWARD AND MARC LACKENBY G G+ Figure 6 Triangulate each of the vertical faces by inserting a vertex into the centre of the face, and coning oﬀ. Insert a vertex into each ball and cone oﬀ. The result is a triangulation of the cube. It has at most 2364c(D) + 984 tetrahedra. However, it does not yet have all the required properties. Step 5. We have triangulated the cube Q × I, but we actually require a triangulation of a ball so that a collar neighbourhood of its boundary has the standard triangulation TS×I . Place a large copy of TS×I around Q × I. We need to triangulate the space between them. Do this by adding a cone over each face of Q × I, the cone point being a vertex of TS×I . Then triangulate the remaining space. This certainly adds at most 300 + 236c(D) tetrahedra to the triangulation. Step 6. Near each 4-valent vertex of G, there are 9 parallelograms which are complementary regions of G+ . A copy of these 9 parallelograms lies in each of the 4 copies of Q, and between these lies a parallelepiped made out of 27 smaller parallelepipeds which together look like a Rubik’s cube. We remove the entirety of this parallelepiped, and replace it with a ﬁxed triangulation of the parallelepiped with the same boundary. This triangulation contains two pairs of parallelepipeds, glued end-to-end, which together form two thickened arcs that realise the crossing of L. We call these parallelepipeds crossing parallelepipeds. See Figure 7. We triangulate each of these crossing parallelepipeds using the standard triangulation of a cube. We triangulate the rest of the large parallelepipeds using less than 300 tetrahedra each, with their boundaries remaining unchanged from the start of this step. Step 7. This new triangulation certainly has at most 212 c(D) + 1284 tetrahedra. It has all the required properties, with one exception. Each component N(Li ) of N(L) is a union of cubes, but the number of cubes is n′i , say, where n′i ≤ 10ci . Thus, we remove each cube of N(Li ), and replace it with the following triangulation. Four of its outer faces have the same triangulation as in the standard case. However, the faces F that are attached to other cubes of N(Li ) are given the following triangulation. It has a central square, coned oﬀ. There is an edge running from each vertex of AN UPPER BOUND ON REIDEMEISTER MOVES 15 Figure 7. Crossing parallelepipeds this square to the corresponding vertex of F . This creates 4 trapeziums. A vertex is inserted into each, and then we cone oﬀ. This speciﬁes the boundary of each of these new cubes. See Figure 8. A portion of the boundary triangulation Figure 8 The interior cubes Between opposite faces containing the new smaller squares, we insert ⌊ni /n′i ⌋ or ⌈ni /n′i ⌉ standard cubes. The union of these cubes will form the new N(L). Between the boundary of the new N(L) and the old N(L), we insert 2-simplices and 3-simplices so that the resulting triangulation has all the required properties. The number of tetrahedra lying within the old N(L) is at most 120n. 16 ALEXANDER COWARD AND MARC LACKENBY 5. Alexander’s trick and bounded isotopies In this section we prove the following result. Theorem 5.1. Let B be a convex 3-ball in R3 . Let h : B → B be a homeomorphism that ﬁxes ∂B pointwise and sends each straight arc in B to a concatenation of at most m straight arcs. Then there is an ambient isotopy ht : B → B for t ∈ [0, 1] such that h0 is the identity map on B, h1 = h and ht sends each straight arc to a concatenation of at most m + 2 straight arcs in R3 for all t ∈ [0, 1]. Proof. We will use Alexander’s trick. Let p be a point in the interior of B. For each r ∈ [0, 1], let Br denote the result of linearly scaling B with centre p by a factor of r, so that B1 = B and B0 = {p}. More precisely, for r > 0 let Br be the image of B under the map sr : x → r(x − p) + p. Let gr : Br → Br be given by gr (x) = sr ◦ h ◦ s−1 (x) r for r > 0. The map gr essentially applies the homeomorphism h to the dilated copy of B, namely Br , as shown in Figure 9. h B B gr Br Figure 9. The map gr Alexander’s trick may now be described as follows. Start with the identity map on B and set this to be h0 . As t increases ht acts by applying gt to Bt and leaving the rest of B unchanged, as shown in Figure 10. When t reaches 1, h1 is the same as h because B1 = B and g1 = h. Formally, ht is given by gt (x) x ∈ Bt . x x ∈ Bt / Now, a straight arc in B will be sent under ht to a single straight arc if it lies entirely outside Bt . It will be sent to at most m straight arcs if it lies entirely within Bt . If it ht (x) = Br AN UPPER BOUND ON REIDEMEISTER MOVES 17 ht B Figure 10. The map ht has one or both endpoints outside of Bt but part of its interior within Bt then it will be sent to at most m + 1 or m + 2 straight arcs respectively. 6. Continuous families of link projections In order to prove Theorem 1.1, we will want to ﬁnd a sequence of diagrams interpolating between two given diagrams for a link. However, in the course of the proof, we will obtain not a sequence of diagrams, but a continuous family of link projections. In this section, we will show how such a family can be used to produce a sequence of diagrams. We say that a piecewise linear map from a disjoint union of circles C to the plane is a link projection if it can be factorised as C → R3 → R2 , where C → R3 is an embedding and R3 → R2 is the standard vertical projection onto the ﬁrst two co-ordinates. In a link projection, one keeps track not just of the map C → R2 , also one records, for any two points in C with the same image in R2 , their relative heights in R3 . Link projections need not form diagrams in the usual sense. For example, more than two points in C may map to the same point in R2 ; indeed uncountably many points may have the same image. However, a link projection induces a diagram if only ﬁnitely many points in the plane have more than one inverse image point in the circles; each such point has precisely two inverse image points; and near each such point in the plane, the image of the link consists of two arcs intersecting transversely. Deﬁnition 6.1. Let D be a link diagram. Suppose that D ′ is obtained from D by adding a small unknot summand at a point in the interior of an edge of D. Then we say that D ′ is obtained from D by adding an unknot summand. We say that D is obtained from D ′ by removing an unknot summand. Proposition 6.2. Let H : B × [0, 1] → B be a piecewise linear isotopy of a convex polyhedral 3-ball B in R3 . For each t ∈ [0, 1], let ht : B → B be H(·, t). Suppose that h0 is the identity. Let L be a piecewise linear link in the interior of B. Suppose that, for each t ∈ [0, 1], ht (L) consists of at most n straight arcs. Suppose also that the projections of h0 (L) and h1 (L) are diagrams. Then, there is a sequence of diagrams relating the projections of h0 (L) and h1 (L) with the following properties: • successive diagrams are related by either a single Reidemeister move or the addition or removal of an unknot summand; • each diagram in the sequence has at most n2 crossings. B 18 ALEXANDER COWARD AND MARC LACKENBY Proving Proposition 6.2 is a fairly routine exercise in general position. Because of this, we only give an outline. Since H is piecewise linear, B × [0, 1] has a triangulation, T , such that H restricts to a linear map on every simplex of T . Now, L × [0, 1] intersects every simplex of T in a collection of aﬃne pieces which we collectively call A. These aﬃne pieces form a cell structure on the annulus L × [0, 1]. Let t1 , t2 . . . be the t-coordinates of the 0-cells of A, arranged in increasing order. For every i, if t ∈ (ti , ti+1 ) then L × {t}, consists of a ﬁxed number of straight pieces, this number depending only on i. Let Dt be the projection of the link ht (L) to the plane. By the discussion above, on each interval (ti , ti+1 ), the projections Dt vary in a fashion determined by the motion of the vertices of H(L × {t}), with the edges in between remaining aﬃne throughout. When t = ti , the projections Dt change in a way that is more complicated. Edges of Dt can shrink to length zero as t increases to ti , and new edges and vertices can appear where none existed before when t increases from ti . Thus it is at these values of t that we may need to remove and then add unknot summands. In order to complete the proof of Proposition 6.2 we need to arrange that Dt is a diagram for every t apart from ﬁnitely many values of t ∈ (0, 1), at each of which a single Reidemeister move takes place or an unknot summand is removed and then another added. This we achieve by perturbing H. First perturb H so that each 0-cell in the interior of A has a diﬀerent t-coordinate. Then 0 = t0 , t1 , . . . , tN = 1 are the t-coordinates of the 0-cells of A. Now introduce two new vertices to each 1-cell in the interior of A. If the endpoints of the 1-cell have times ti and tj , where ti < tj , then place the vertices at ti + ε and tj − ε, for some small ε > 0. From now on, all perturbations of H will be achieved by slightly moving the location in B of the images of the 0-cells of A. The remainder of the proof of Proposition 6.2 will be in several steps. The ﬁrst step will be to ensure that each Dti is a diagram. Next we will ensure that for each i, Dti ±ε is, for small enough ε, a diagram related to Dti by the addition or removal of an unknot summand. The third and ﬁnal step will be to arrange that the link projections between Dti +ε and Dti+1 −ε are diagrams for all but ﬁnitely many times, t, at which a single Reidemeister move takes place. Step 1 - Ensuring each Dti is a diagram: We need to perturb H to ensure the following: • any two edges of the projection Dti intersect in at most one point; • no three edges of Dti have a common point of intersection; • no vertex of the link has image that lies in a non-adjacent edge of Dti . This is a straightforward general position argument. Suppose that two non-adjacent edges intersect in more than one point. The images of their four endpoints in B lie in a 12-dimensional vector space. The subspace consisting of conﬁgurations where the two lines intersect in more than one point has dimension 10, for the following reason. The ﬁrst line is speciﬁed by 6 co-ordinates. The remaining line has projection that overlaps with the projection of the ﬁrst line, and so there are 4 co-ordinates that specify this line. Since 10 is less than 12, we may perturb hti to avoid this subspace. Perturb H to realize this perturbation of hti . We can make this perturbation suﬃciently small AN UPPER BOUND ON REIDEMEISTER MOVES 19 so that no other bad conﬁgurations arise. We can deal with the other cases similarly, and thereby ensure that each Dti is a link diagram. Note that the case where an edge of the link is projected to a single point is ruled out by the third condition. Step 2 - Perturbing each Dti ±ε : As t increases from ti ∈ (0, 1), the diagram Dti changes in the following way. All but one of the vertices of Dti move with constant velocity. From the remaining vertex, v say, a collection of vertices are formed and these all move away from v with constant velocity. The velocity of these new vertices are the projected velocities of new vertices in ht (L) moving away from the pre-image of v. Because Dti is a diagram, for ε small enough, Dti +ε is a diagram apart from possibly near v. To ensure that Dti +ε is a diagram perturb it in a similar fashion to Step 1. All the diagrams Dti +δ change, as δ varies in (0, ε], in a linear fashion with their vertices moving with constant velocity. Thus as t increases from ti we have arranged that near v we see a small unknot summand appear and grow in a linear fashion. A similar argument applies at times immediately prior to each ti ∈ (0, 1). At times t = 0 and t = 1 things are only slightly diﬀerent, with possibly several unknot summands appearing immediately after t = 0 and possibly several unknot summands disappearing as t reaches 1. Step 3 - Passing from Dti +ε to Dt(i+1) −ε with Reidemeister moves: So far we have ensured that we may pass from the projection of h0 (L) to the projection of h1 (L) by means of a sequence of diagrams where consecutive diagrams in the sequence are related by the addition or removal of a small unknot summand, or by moving their vertices with constant velocity, keeping the joining edges straight throughout. We ensure that the latter 1-parameter families of link projections may be perturbed to give rise to sequences of Reidemeister moves with the following proposition: Lemma 6.3. We may perturb each Dti ±ε slightly so that they remain diagrams, and so that Dt fails to be a diagram for only ﬁnitely many values of t in each [ti + ε, ti+1 − ε]. Moreover, at these values of t, a single Reidemeister move is performed. Proof. We ensure that, for t in [ti + ε, ti+1 − ε], the following conditions hold for every link projection Dt : • for any two non-adjacent edges of the link, their projections intersect in at most one point; • for any four edges of the link, their projections have no common point of intersection; • the link projection has at most one triple point; • there is at most one vertex of the link that projects to a point in the image of a non-incident edge; • no two vertices have a common projection. We will also ensure that the following possibilities arise for at most ﬁnitely many link projections: (1) the projection of some edge lies within the projection of an adjacent edge; (2) a vertex of the link has image lying within the projection of a non-incident edge; (3) there is a triple point. 20 ALEXANDER COWARD AND MARC LACKENBY Note that there will, in general, be moments when (1), (2) or (3) above do occur, and typically then a Reidemeister move will be performed, as illustrated in Figure 11. Move (1) Move (2) Figure 11 We start by arranging the ﬁrst of these conditions. We will perturb the image of L under the homeomorphisms hti +ε and hti+1 −ε , but keeping ti + ε and ti+1 − ε ﬁxed. Consider any two non-adjacent edges of the link. The images of their four endpoints under hti +ε and hti+1 −ε lie in a 24-dimensional vector space. We wish to consider the subset of this space consisting of conﬁgurations where the projections of the two edges intersect in more than one point. This subset lies in a subspace, the dimension of which can be computed as follows. The two edges have bad projections at some point in time. There is a one-dimensional family of possible times. One of the edges has arbitrary image at this time, and so its position is speciﬁed by 6 co-ordinates. The remaining edge must have projection lying in the same line as the ﬁrst edge. Thus, there are two parameters specifying the endpoints of its projection. Two further parameters give the height of these endpoints in R3 . As one moves away from this time, the endpoints of the edges vary linearly. Their derivatives are speciﬁed by 12 further parameters. Thus, the subspace that we need to avoid has at most 23 dimensions. Hence, by a small perturbation of hti +ε and hti+1 −ε , we can avoid this subspace. Thus, we can ensure that, throughout the isotopy, the projection of any two non-adjacent edges of the link intersect in at most one point. We next ensure that, throughout the isotopy, the projection of no four edges have a common point of intersection. This time the ambient space, giving the position of these four edges at times ti + ε and ti+1 − ε is 48-dimensional. The subspace containing the bad conﬁgurations has 47 dimensions: one giving the time of the bad conﬁguration, two parameters specifying the point of intersection in the projection, four parameters give the angles of the lines emanating from this point, eight further parameters give the lines themselves, eight parameters give the heights in R3 of their endpoints, and 24 co-ordinates give the derivatives of these endpoints at that time. Thus, we can ensure that only triple points arise. Using similar arguments to the ones above, we can ensure the remaining conditions. This proves Proposition 6.2. Of course, Proposition 6.2 is not suﬃcient for our purposes because the addition or removal of unknot summands do not obviously induce a sequence of Reidemeister moves. For this reason we will need to apply the following theorem, whose proof, which we omit, is an easy adaptation of the methods used by Hass and Lagarias to prove the main theorem of [5]. AN UPPER BOUND ON REIDEMEISTER MOVES 21 Theorem 6.4. Let D and D ′ be link diagrams and suppose that D ′ is obtained from D by the addition or removal of an unknot summand. Suppose that D and D ′ both have 11 at most n crossings. Then there is a sequence of at most 210 n Reidemeister moves relating D and D ′ . Thus, from a continuous family of link projections interpolating between D1 and D2 , we can ﬁnd a sequence of Reidemeister moves taking D1 to D2 . In order to eﬃciently bound the number of moves required we will need the following theorem. Theorem 6.5. Up to ambient isotopy, there are at most (24)n+1 connected, unoriented link diagrams with at most n crossings. Proof. This is an adaptation of an argument of Welsh [28]. We use a theorem of Tutte from [27], which counts ‘rooted bicubic maps’. A map is a cell structure on the 2-sphere, or equivalently an embedded connected planar graph. It is trivalent if the valence of each of its vertices is 3. It is bicubic if it is trivalent and bipartite. A map is rooted if one of its edges is chosen and oriented, and the two faces on either side of this edge are speciﬁed as lying on the left and right of the edge. The point of using rooted maps is that they have no symmetries, and so they are easier to count. It is easy to show that the faces of a bicubic map may be coloured using only three colours, and so that adjacent faces have distinct colours. This colouring is unique once the colours adjacent to some edge are chosen. If the bicubic map is rooted, then the root colour is the face colour not adjacent to the speciﬁed edge. Tutte proved in 4.3 of [27] that the number of rooted bicubic maps in which there are just n faces of the root colour is 2(2n)!3n . n!(n + 2)! Now, given a connected link diagram, there is a simple way of creating a bicubic map. One simply replaces each 4-valent vertex of the link projection with a square consisting of four vertices and four edges. One can root the bicubic map by picking one of the original edges of the diagram. The new squares are then assigned the root colour. Hence, we deduce that the number of connected rooted embedded 4-valent planar graphs with n vertices is also at most 2(2n)!3n . n!(n + 2)! There are 2n ways of assigning the crossing information and so there are at most 2(2n)!6n n!(n + 2)! connected link diagrams with n crossings. So, the number with at most n crossings is no more than n k=0 2(2k)!6k ≤ k!(k + 2)! n 2 k=0 2k k 6k ≤ (k + 1)2 n ∞ 24k k=0 k=1 1 k2 ≤ 24n+1, as required. Theorem 6.5 yields the following immediate corollary: 22 ALEXANDER COWARD AND MARC LACKENBY Corollary 6.6. Up to ambient isotopy, there are at most (48)n+1 connected, oriented link diagrams with at most n crossings. We may combine Theorem 6.4 and Corollary 6.6 to obtain the following corollary. Corollary 6.7. Suppose that there is a ﬁnite sequence of diagrams, starting with D1 and ending with D2 , so that successive diagrams are related by either a Reidemeister move or the addition or removal of an unknot summand. Suppose that each diagram in this sequence is connected and has at most n crossings. Then, there is a sequence 12 of at most 210 n Reidemeister moves joining D1 to D2 . Proof. Without loss of generality suppose that the hypothesized sequence of diagrams is chosen to be as short as possible. Then no two diagrams are isotopic and so, by Corollary 6.6, the sequence consists of at most (48)n+1 diagrams. By Theorem 6.4, 11 passing from one diagram to the next can be achieved with at most 210 n Reidemeister moves. The product of these two expressions is less than the desired bound. Combining Theorem 5.1, Proposition 6.2 and Corollary 6.7, we obtain the following. Corollary 6.8. Let h : B → B be a PL homeomorphism of a convex polyhedral 3-ball B, which ﬁxes ∂B pointwise, and which has the property that it sends each straight arc in B to a concatenation of at most m straight arcs. Let L be an non-split oriented link in B that is the concatenation of at most n straight arcs. Suppose that L and h(L) project to oriented diagrams D1 and D2 . Then D1 and D2 diﬀer by a sequence of at most 12 2 13 2 210 (n(m+2)) ≤ 210 (nm) Reidemeister moves. 7. Proof of the main theorem In this section, we will prove Theorem 1.4. This will quickly yield Theorem 1.1, assuming Theorem 1.3. All diagrams in this section will be oriented. Note that Theorem 1.1 for unoriented diagrams follows from the version for oriented diagrams. Let D1 and D2 be connected diagrams for some knot or non-split link L in the 3-sphere. Let n1 and n2 be their crossing numbers, and let n = n1 + n2 . We wish to ﬁnd a sequence of diagrams taking D1 to D2 , where successive diagrams are related by a Reidemeister move. We suppose that L is not the unknot, for in this case Theorem 1.4 follows from [5]. (Theorem 1.1 also follows immediately from [5] in this case.) Start by applying type 1 Reidemeister moves to D1 and D2 so the writhes of cor′ ′ responding components agree. Call the resulting diagrams D1 and D2 respectively. They have at most 2n1 and 2n2 crossings respectively. The number of Reidemeister moves required is at most n. Use Theorem 4.3 to create triangulations T1 and T2 of the 3-ball B with L1 and L2 ′ ′ as subcomplexes. The vertical projections of L1 and L2 are the diagrams D1 and D2 . Theorem 4.3 ensures that the triangulations T1 and T2 contain neighbourhoods N(L1 ) and N(L2 ) of L1 and L2 that are subcomplexes and that each have a standard triangulation of the solid torus. By choosing the integers ni in Theorem 4.3 appropriately, we AN UPPER BOUND ON REIDEMEISTER MOVES 23 may arrange that these triangulations of the solid tori have the same length, and hence ′ ′ are combinatorially isomorphic. Further, since D1 and D2 have the same writhes, we may also ensure that the homeomorphism N(L1 ) → N(L2 ) that realises this combinatorial isomorphism preserves longitudes. Since 213 n1 + 120(40(n1 + n2 ) + 11) < 214 n, the triangulations T1 and T2 may be taken to contain at most t = 214 n tetrahedra each. Apply Theorem 3.1 to give a sequence of at most N = 100PL (t, t) + 200t interior Pachner moves, followed by a combinatorial isomorphism, that takes T1 to T2 , and N(L1 ) to N(L2 ). Further, none of the Pachner moves aﬀect N(L1 ) or ∂B, and the combinatorial isomorphism restricts to the identity on ∂B. This sequence of Pachner moves gives a sequence of ∆-complexes joined by PL homeomorphisms 1 1 N |T1 | = |T 0 | −→ |T 1 | −→ · · · −→ |T N | = |T2 |. h h h Let h = φ−1 hN . . . h1 φ1 be the resulting homeomorphism B → B, where φ1 and φ2 are 2 the homeomorphisms associated with T1 and T2 . Note that h(L1 ) = L2 . We will need the following concept. Let T be a triangulation of a 3-manifold. Then an arc in |T | is said to be straight if it lies in a single simplex and is straight in the aﬃne structure on that simplex. We now apply the following straightforward lemma. Lemma 7.1. Suppose that T i and T i+1 are triangulations of a 3-manifold that are related by an interior Pachner move. Let hi+1 : |T i | → |T i+1 | be the resulting homeomorphism. Then, hi+1 sends each straight arc in |T i | to a concatenation of at most 4 straight arcs in |T i+1 |. Now, φ1 sends each straight arc in B to a concatenation of at most t straight arcs in |T1 |. By Lemma 7.1, this is sent to a concatenation of at most 4N t straight arcs in |T2 |. Each of these arcs is sent to a straight arc in B. Thus we obtain the following. Corollary 7.2. Each straight arc in B is sent, via h, to a concatenation of at most 4N t straight arcs. Now, since L1 lies in the 1-skeleton of T1 , it consists of at most 6t = 6 · 214 n straight arcs in B. Hence we may apply Corollary 6.8 to conclude that there is a sequence of at most 13 N 2 210 (6t·4 t) ≤ exp(1015 t4 24N ) Reidemeister moves taking D1 to D2 . This proves Theorem 1.4 since exp(1015 t4 24N ) = exp(1015 t4 24(100PL (t,t)+200t) ) ≤ exp(250+4t+4(100PL (t,t)+200t) ) ≤ exp(2400(PL (t,t)+4t) ) = exp(2) (400(PL (214 n, 214 n) + 216 n)). Now, Theorem 1.3 may be applied to non-split link exteriors because the 3-sphere cannot contain a 3-dimensional submanifold that has a single boundary component, ﬁbres over the circle, and is a surface semi-bundle. Indeed, no Z2 -homology 3-sphere 24 ALEXANDER COWARD AND MARC LACKENBY can contain such a submanifold. Hence Theorem 1.1 follows in the case where the link is not split, assuming Theorem 1.3, because t t t n exp(2) (400(2 exp(a ) (t) + 4t)) ≤ exp(2) (exp(a +10) (t)) = exp(a +12) (t) ≤ exp(c ) (n), where c = 2163·2 ≤ 101,000,000 . It remains to prove Theorem 1.1 in the case where L is split. In this case, we apply the following result. Proposition 7.3. Let L be a split link, and write L = L1 ∪ · · · ∪ Lr , where each Li is non-split, and when i = j, Li and Lj are separated by a sphere that lies in the complement of L. Let D be a diagram for L with n crossings, and let Di be the n restriction of this diagram to Li . Then there is a sequence of at most 2(r−1) exp(k ) (k n ) Reidemeister moves taking D to the distant union of D1 , . . . , Dr , where k = 2129 . Proof. It is a theorem of Hayashi [8] that if D is a diagram of a split link with n n crossings, then there is a sequence of at most exp(k ) (k n ) Reidemeister moves taking it to a disconnected diagram D ′ , where k = 2129 . (We have simpliﬁed Hayashi’s bound, at the cost of a slight increase to it.) Suppose that this disconnected diagram separates the link into two subsets K1 and K2 . Then, the restriction of D to each Ki is a diagram Ei for Ki . We may now reverse some of these Reidemeister moves to take D ′ to the distant union of E1 and E2 . We apply this argument r − 1 times, and we end with the distant union of D1 , . . . , Dr . Write L = L1 ∪ · · · ∪ Lr , where each Li is non-split, and when i = j, Li and Lj are separated by a sphere that lies in the complement of L. From D1 , we obtain diagrams D1,1 , . . . , D1,r for L1 , . . . , Lr , by disregarding components not belonging to the relevant ′ Li . Let D1 be the distant union of the diagrams D1,1 , . . . , D1,r . Similarly, we obtain ′ diagrams D2,1 , . . . , D2,r from D2 , and let D2 be their distant union. Let ni,j be the number of crossings in Di,j . Thus, n1,1 + · · · + n1,r ≤ n1 , and n2,1 + · · · + n2,r ≤ n2 . Let mj = n1,j + n2,j . By Theorem 1.1 in the non-split case, there is a sequence of at most m1 mr exp(c ) (m1 ) + · · · + exp(c ) (mr ) ′ ′ Reidemeister moves taking D1 to D2 . By Proposition 7.3, there is a sequence of at (k n1 ) n1 ′ most 2(r − 1) exp (k ) Reidemeister moves taking D1 to D1 , and similar number n ′ taking D2 to D2 . The total number of moves is less than exp(c ) (n). This concludes the proof of Theorem 1.1, assuming Theorem 1.3. ´ 8. Overview of the proof of Mijatovic’s theorem (Simple case) The driving force behind this paper is Theorem 1.3, which is based on work of Mijatovi´. As mentioned in the Introduction, Mijatovi´ in fact proved the following c c related result [17]. Theorem 8.1. Let M be a compact orientable irreducible 3-manifold, with boundary a non-empty collection of tori. Suppose that the closure of each component of the complement of the characteristic submanifold of M satisﬁes at least one of the following conditions: 14 AN UPPER BOUND ON REIDEMEISTER MOVES 25 • it does not ﬁbre over the circle; or • it is not a surface semi-bundle; or • it has at least two boundary components. Let T1 and T2 be two triangulations of M with t1 and t2 3-simplices respectively. Then t t there is a sequence of at most exp(b 1 ) (t1 ) + exp(b 2 ) (t2 ) interior Pachner moves and boundary Pachner moves, followed by an ambient isotopy, followed by a homeomorphism of M to itself that is supported in the characteristic submanifold, that leaves each component of ∂M invariant, and that takes T1 to T2 . Here, b ≤ 2200 . There are two main diﬀerences between the statements of Theorems 8.1 and 1.3. Firstly, boundary Pachner moves are permitted in Theorem 8.1 but not in Theorem 1.3. Secondly, the homeomorphism of M is not required to be the identity on the boundary of M in Theorem 8.1. We will explain how to guarantee these extra requirements in Sections 10 - 13, following a summary of Mijatovi´’s techniques in this and the next c section. We brieﬂy recall what is meant by the characteristic submanifold of M. We ﬁrst consider the case where M is compact, orientable and irreducible, and has boundary a (possibly empty) collection of incompressible tori. A torus properly embedded in M is canonical if it is essential and, moreover, any other essential torus can be isotoped oﬀ it. If one takes one representative for each ambient isotopy class of canonical torus, then these can be chosen to be disjoint. The resulting collection of tori is the JSJ tori, and their union is well-deﬁned up to ambient isotopy. A key result in the theory is that if one cuts M along an open regular neighbourhood of the JSJ tori, the resulting pieces are either Seifert ﬁbred or simple. The union of the Seifert ﬁbred pieces is the characteristic submanifold of M. When M has boundary that is incompressible, but not a union of tori, then one must vary the above deﬁnition. Here, M is still assumed to be compact, orientable and irreducible. One considers essential annuli and tori properly embedded in M, and again such a surface is canonical if any other essential annulus or torus can be isotoped oﬀ it. To form the JSJ annuli and tori, one takes one representative of each isotopy class of canonical annulus and torus, but one then discards certain annuli, called matching annuli, which have Seifert ﬁbred spaces on both sides with matching Seifert ﬁbrations. In this case, the JSJ annuli and tori divide M into pieces that are simple, Seifert ﬁbred or an I-bundle over a surface, and the characteristic submanifold is the union of the Seifert ﬁbred and I-bundle pieces. For further details on JSJ decompositions of 3-manifolds, see [10] and [11]. See also [19]. The idea behind Theorem 8.1 is as follows. As in Theorem 1.3, a ‘canonical triangulation’ Tcan for M is constructed. However, this is slightly diﬀerent from the ′ ′ triangulation Tcan in Theorem 1.3. Like Tcan , Tcan depends on a given triangulation of ∂M. However, its restriction to ∂M does not equal this triangulation. Mijatovi´’s triangulation Tcan is constructed from a collection of surfaces in M. c These include the JSJ tori of M. In each component of the complement of the JSJ tori, the choice of surfaces depends on whether that piece is simple or Seifert ﬁbred. Therefore, in this section, we will focus on the case where M is simple. In the next 26 ALEXANDER COWARD AND MARC LACKENBY section, we will examine the case where M is Seifert ﬁbred. In Section 10, we will consider the general case. Suppose therefore that M is a compact orientable simple Haken 3-manifold satisfying the conditions of Theorem 8.1. A partial hierarchy for M is a sequence M = M0 S1 M1 S2 ··· Sn Mn , where each Mi is a 3-dimensional submanifold of M, each Si is a properly embedded incompressible surface in Mi−1 , and Mi is obtained from Mi−1 by cutting along Si . A hierarchy is a partial hierarchy where the ﬁnal manifold is a collection of 3-balls. At each stage, one keeps track of a boundary pattern Pi for Mi . Here, we are using this term to mean a collection of disjoint simple closed curves and graphs embedded in ∂Mi . This boundary pattern is deﬁned as follows. One considers the union of the surfaces S1 , . . . , Si as a 2-complex in M, so that each Sj has boundary that runs over S1 ∪ · · · ∪ Sj−1 . Each surface Sj is required to be in general position with respect to the previous surfaces, in the sense that its boundary is required to be transverse to the union of the boundaries of the earlier surfaces. In general, one also wants to ensure that when two parts of ∂Sj intersect because they lie on diﬀerent sides of some previous surface, then they also are required to be in general position. However, this situation will never in fact arise in this paper, because we will always cut along separating surfaces. The boundary pattern Pi for Mi is deﬁned to be the image of ∂S1 ∪ · · · ∪ ∂Si in ∂Mi . This boundary pattern is essential which means that ∂Mi − Pi is incompressible in Mi . Mijatovi´’s canonical triangulation Tcan is constructed as follows. The union of ∂M c and the surfaces S1 , . . . , Sn is a 2-complex. Triangulate each face of this complex, by inserting a vertex in its interior, and then coning oﬀ. The boundary of each 3-ball in Mn then inherits a triangulation. Triangulate this 3-ball by coning oﬀ this boundary triangulation. Provided one uses the right hierarchy, the result is Tcan . Not any choice of hierarchy will work here. It is important that whenever Mi−1 is given some triangulation, then Si can be realised as a normal surface in that triangulation with an estimable number of triangles and squares. It is also important that there is a bound on the length n of the hierarchy, in terms of the number of tetrahedra in any given triangulation of M. The reason for this will shortly become apparent. To prove Theorem 8.1 or 1.3, one starts with a triangulation T for M. Let’s also call this triangulation T0 (M) and T0 (M0 ). One realises S1 as a normal surface in this triangulation, with control over the number of triangles and squares. Then one applies Pachner moves to T0 (M), creating a triangulation T1 (M) in which S1 is a subcomplex. This restricts to a triangulation T1 (M1 ) of M1 . Then one repeats, by realising S2 as a normal surface in T1 (M1 ), and so on. At each stage, the boundary pattern Pi is required to be a subcomplex of the triangulation Ti (Mi ). The ﬁnal result is a triangulation Tn (Mn ) of Mn , the components of which patch together to form a triangulation of Tn (M) of M in which S1 ∪ · · · ∪ Sn is a subcomplex. Now one applies Pachner moves to Tn (M) so that the triangulation on the 2-complex S1 ∪ · · · ∪ Sn agrees with that of Tcan . Thus, on each component of Mn , we have two triangulations of the 3-ball which agree on their boundaries. Then Mijatovi´ applies an earlier result c AN UPPER BOUND ON REIDEMEISTER MOVES 27 (Theorem 5.2 of [16]), which allows one to bound the number of interior Pachner moves required to pass between two such triangulations of a 3-ball. The precise choice of hierarchy made by Mijatovi´ is rather delicate. The surfaces c that he uses are designed to ensure that one can give an upper bound on the number of triangles and squares of the normal surface Si in any given triangulation of Mi−1 , and also to provide a bound on the length of the hierarchy. The ﬁrst surface S1 must be chosen with particular care. In order that later stages of the hierarchy can be chosen canonically, it is important that the exterior of S1 is not a union of I-bundles. In other words, S1 must be neither a ﬁbre in a ﬁbration of M over the circle nor a ﬁbre in a surface semi-bundle. The hypotheses on M in Theorem 8.1 are there to ensure that it is possible ﬁnd a properly embedded essential surface satisfying this condition. For when M is a compact orientable irreducible atoroidal 3-manifold with boundary a single incompressible torus, then a theorem of Culler and Shalen [2] gives that M contains a properly embedded connected essential surface that is either separating or closed. Thus, it is not a ﬁbre. It may be a semi-ﬁbre, but our hypotheses on M then ensure that M does not ﬁbre over the circle and so one may instead use a non-separating surface. When M has more than one boundary component and is not homeomorphic to S 1 × S 1 × I, one may use Culler and Shalen’s theorem to ﬁnd an essential properly embedded surface that intersects at most one component of ∂M. Hence, in this case, it is neither a ﬁbre nor a semi-ﬁbre. See Corollary 3.2 in [17] for more details. It is possible to ﬁnd such a surface in normal form, with a bound on its number of triangles and squares using Propositions 4.1 and 4.2 of [17]. At each of the later stages of the hierarchy, one of the following types of surface is used: (1) the boundary of a regular neighbourhood of a closed, connected, properly embedded, π1 -injective surface of maximal Euler characteristic in some component of Mi−1 minus its characteristic submanifold (such a surface is only used in the initial stages when the boundary pattern is empty); (2) a canonical annulus in some component of Mi−1 , and which is also disjoint from the boundary pattern Pi−1 (here, canonical is deﬁned in terms of the boundary pattern Pi−1 ); (3) an incompressible annulus (or two parallel copies of such an annulus) in a component of Mi−1 that is either an I-bundle over a surface or a compression body, which joins diﬀerent boundary components, and which has minimal intersection number with Pi−1 (such a surface is only used when the component of Mi−1 , with its boundary pattern, has no canonical annuli and empty characteristic submanifold); (4) a meridian disc (or perhaps two parallel copies of such a disc) for a handlebody component of Mi−1 , and which has minimal intersection number with Pi−1 (again such a surface is only used when the component of Mi−1 , with its boundary pattern, has no canonical annuli and empty characteristic submanifold). 28 ALEXANDER COWARD AND MARC LACKENBY We refer the reader to [15] for the precise order in which these surfaces are used. There, it is also shown that Si may be realised as a normal surface in Ti−1 (Mi−1 ) with bounded complexity, and a bound on the length of the hierarchy is also given. We now explain how one passes from the triangulation Ti−1 (M) to the triangulation Ti (M). The surface Si is a normal surface in Ti−1 (Mi−1 ). The triangulation Ti (M) is chosen so that Si is simplicial in it. More precisely, Si intersects each tetrahedron of Ti−1 (M) in a collection of triangles and squares. A vertex is inserted into each of these triangles and squares, and then the triangle or square is coned oﬀ. The surface Si divides each face of Ti−1 (M) into discs. A vertex is inserted into each of these discs, and this too is coned oﬀ. Now, each tetrahedron of Ti−1 (M) is divided into balls by Si . A vertex is placed in the interior of each of these, and then the ball is coned oﬀ. The resulting triangulation of M is Ti (M). Its restriction to Mi is Ti (Mi ). We now give Mijatovi´’s sequence of Pachner moves which takes Ti−1 (M) to Ti (M). c This takes place in 6 steps: (1) Perform a (1, 3) boundary Pachner move on each triangle in ∂M, by attaching a tetrahedron to it. Then, in a similar fashion, make a (2, 2) boundary Pachner move for each 1-simplex of Ti−1 (M) in ∂M. (2) Add a vertex into each tetrahedron of Ti−1 (M) (but not the newly attached tetrahedra from Step 1) by performing a (1, 4) Pachner move. Then add a vertex to each triangle of Ti−1 (M) by performing a (1, 4) move on an adjacent tetrahedron and then a (2, 3) move. (3) Subdivide the 1-skeleton of Ti−1 (M) so that it becomes a subcomplex of Ti (M), and keep the triangulation of the 3-simplices of Ti−1 (M) coned. The precise details of how to do this are in Step 2 in Section 5 of [14]. (4) Subdivide the 2-skeleton of Ti−1 (M) to get a subcomplex of Ti (M), and keep the triangulation of the 3-simplices of Ti−1 (M) coned. The process here is described in Lemma 4.2 of [14]. (5) Chop up the tetrahedra of Ti−1 (M) along the normal triangles and squares of Si and triangulate the complementary regions by coning them from points in their interiors. One uses Lemma 5.1 of [14] to do this. (6) Finally, remove the tetrahedra that are not contained in any of the 3-simplices of Ti−1 (M) by performing boundary Pachner moves. We will not need all the details of this process. But the following observation will be important for us. The boundary Pachner moves that are used depend only on the intersection between ∂Si and ∂M. In fact, the way they arise is precisely as follows: (1) A (1, 3) move is performed on each triangle of Ti−1 (M) ∩ ∂M, which attaches a tetrahedron. (2) A (2, 2) move is performed on each edge of Ti−1 (M)∩∂M, which again attaches a tetrahedron. (3) The newly introduced tetrahedra are then modiﬁed using interior Pachner moves. However, these moves are determined entirely by ∂Si ∩ ∂M. (4) Boundary Pachner moves are then performed which remove the tetrahedra not included in Ti (M). Once again, these moves are determined entirely by ∂Si ∩ ∂M. AN UPPER BOUND ON REIDEMEISTER MOVES 29 We term the boundary Pachner moves arising from the above procedure the speciﬁed sequence of boundary Pachner moves. Note that no (3, 1) boundary Pachner moves that attach a tetrahedron were performed. ´ 9. Overview of the proof of Mijatovic’s theorem (Seifert fibred case) In this section, we give an outline of the proof of Theorem 8.1 in the case where M is Seifert ﬁbred, following [16]. In [16], Mijatovi´ also dealt with many closed Seifert ﬁbre spaces, but we will not do c so here. We will assume (as stated in Theorem 8.1) that M has non-empty boundary. We will not consider here the case where the base orbifold of the Seifert ﬁbration has zero Euler characteristic. In other words, we will exclude the case where M is homeomorphic to an I-bundle over a torus or Klein bottle. These spaces required a separate argument in [16] because they admit properly embedded essential annuli which cannot be made vertical in the Seifert ﬁbration after an ambient isotopy. As in the simple case, the goal is to build the triangulation Tcan of M from a collection of surfaces. However, these surfaces do not exactly form a hierarchy. They are as follows: (1) The ﬁrst surface S1 is a union of properly embedded disjoint tori, which bound a collection of solid tori that together form a regular neighbourhood of the singular ﬁbres. The exterior of these solid tori, which we denote by M− , is a union of regular ﬁbres and hence a circle bundle over a surface. (2) The second surface S2 is a union of meridian discs, one for each of the solid tori from (1). At this stage, Mijatovi´ gives the tori S1 a certain triangulation, c arising from these meridian discs and from the regular ﬁbres in the Seifert ﬁbration. We will not dwell on the details of this triangulation, because we will follow a slightly diﬀerent approach in our proof of Theorem 1.3. (3) The third surface is a horizontal section S3 of the circle bundle M− . Now, M− may contain many horizontal sections, even up to ambient isotopy. The section S3 is chosen so that its boundary is normal in the given triangulation of ∂M− and has least weight among all such normal simple closed curves. (4) The fourth surface S4 is a maximal collection of disjoint non-parallel vertical annuli properly embedded in M− , each of which intersects S3 in a single arc. Thus, this fails to be a hierarchy for two reasons. Firstly, S1 is compressible in M. Secondly, S4 is not properly embedded in the exterior of S3 . Nevertheless, S1 ∪ S2 ∪ S3 ∪ S4 is a 2-complex, and the complementary regions of this complex are balls. To construct the triangulation Tcan , ﬁrst a vertex is introduced into the interior of each face of the 2-complex, and this face is coned oﬀ. Then a vertex is placed in the interior of each complementary ball, and this too is coned oﬀ. Just as in this previous section, it is important that this 2-complex is constructible, given an arbitrary triangulation T of M that restricts to the given triangulation T∂M on ∂M. In the previous section, we realised the ﬁrst surface as a normal surface in T with bounded weight (as a function of the number of tetrahedra in T ), and then 30 ALEXANDER COWARD AND MARC LACKENBY we performed a sequence of Pachner moves to T , taking it to a triangulation in which the surface is simplicial. In the case here, it is not immediately clear that the tori S1 can be placed into normal form in T , because they are compressible. Instead, one must construct them in stages. The ﬁrst step is to ﬁnd a maximal collection of disjoint non-parallel essential vertical tori in the Seifert ﬁbration that are in normal form with respect to T . Then Pachner moves are performed on T , after which these tori are simplicial. These tori decompose M into pieces, each of which is Seifert ﬁbred. There are a limited range of possibilities for these pieces. The base orbifold may be a pair of pants containing no singularities; an annulus containing one singular point; a Mobius band with no singularities; or a disc with two singularities. In each case, the piece contains one or two properly embedded essential vertical annuli, which decompose the piece into one or two solid tori. A subset of these solid tori are regular neighbourhoods of the singular ﬁbres, and the boundaries of these solid tori are the required surface S1 . The rest of the argument follows the lines of the simple case fairly closely. One new feature that arises in the Seifert ﬁbred case is that it is not possible to isotope the section S3 to a normal surface with bounded weight. Instead, one must also use homeomorphisms of M that are supported in the interior of M. This is because of the presence of incompressible normal tori. It is for this reason that the conclusions of Theorems 8.1 and 1.3 make reference to a homeomorphism supported in the characteristic submanifold of M. Another new feature in the Seifert ﬁbred case is that S4 is not properly embedded in the exterior of S1 ∪ S2 ∪ S3 . Instead, it is properly embedded in the exterior of S1 ∪ S2 . Thus, one must ensure that the horizontal section S3 intersects each vertical annulus of S4 in a single arc. This is clearly possible topologically, but one must be careful to ensure that it holds, while at the same time making S3 and S4 normal surfaces with bounded complexity. The procedure in the Seifert ﬁbred case for taking the given triangulation T to the canonical triangulation Tcan is very similar to that used in Section 8. In particular, the sequence of boundary Pachner moves that is used depends only on the intersection between the curves ∂S1 , ∂S2 , ∂S3 , ∂S4 and ∂M. 10. The construction of the canonical triangulation In this section, we consider a general compact orientable Haken 3-manifold M, satisfying the hypotheses of Theorem 1.3. In particular, it may have JSJ tori. We start with a given triangulation T∂M for ∂M. The goal is to deﬁne the canonical ′ triangulation Tcan for M which agrees with T∂M on ∂M. However, our ﬁrst step is to build a slightly simpler triangulation Tcan , which does not agree with T∂M on ∂M. We use the terminology Tcan because in the case where M is simple or Seifert ﬁbred, it agrees with Mijatovi´’s triangulations described in the previous two sections. c Again, the aim is to build a 2-complex in M, for which each complementary region is a 3-ball. From this, Tcan is built, by coning oﬀ each face of the complex, and then coning oﬀ each complementary ball. The 2-complex is, as in the previous sections, constructed from a collection of surfaces, in a number of steps. AN UPPER BOUND ON REIDEMEISTER MOVES 31 Step 1. The ﬁrst surface S1 is built from the JSJ tori, which divide the manifold into simple and Seifert ﬁbred pieces. When a JSJ torus has simple pieces on both sides (possibly the same simple piece), two parallel copies of the torus are used in S1 . Similarly, when a JSJ torus has Seifert ﬁbred pieces on both sides, two parallel copies of it are used. However, when a torus has a simple piece on one side and a Seifert ﬁbred piece on the other, only one copy of the torus is used. Step 2. The next surfaces are hierarchies for the simple pieces, as described in Section 8. Step 3. When a JSJ torus has simple pieces on both sides, we have taken two copies of the torus. Between these lies a region homeomorphic to T 2 × I. The boundary of this region has inherited a boundary pattern from the hierarchies in the adjacent pieces. The next surface is a vertical annulus in each T 2 × I region, which intersects the boundary pattern transversely and in as few points as possible. This cuts T 2 × I into a solid torus. The next surface is a meridian disc for each such solid torus, which again intersects the boundary pattern transversely and minimally. Step 4. We now turn to the JSJ tori that have Seifert ﬁbred pieces on both sides. We have taken two copies of each such torus, which bound a region homeomorphic to T 2 × I. We insert two vertical annuli into T 2 × I, with respective slopes those of the regular ﬁbres in the adjacent Seifert ﬁbred pieces. We arrange for these annuli to intersect transversely and minimally. Step 5. We now tackle the Seifert ﬁbred pieces. For each such piece, its boundary components which do not lie in the boundary of M have inherited some boundary pattern, which decompose the tori into discs. We now use a similar sequence of surfaces to that described in Section 9. At one stage, a slight variant of the procedure is required. Recall that in Step 3 in Section 9, a section S3 for the circle bundle M− is chosen, and this is required to have minimal intersection number with the edges of the given triangulation of ∂M− . This triangulation of ∂M− was constructed in Step 2 of Section 9. For those components of ∂M− that are actually components of ∂M, they are just assigned the given triangulation, which is the restriction of T∂M . For those components of ∂M− which bound a solid torus neighbourhood of a singular ﬁbre, a certain triangulation was constructed using the regular ﬁbres and a meridian disc of the solid torus. However, in our case, we pursue a slightly diﬀerent approach. We have already assigned a boundary pattern to each component of ∂M− − ∂M, which ﬁlls the surface. We pick the section S3 to have minimal intersection number with the union of this boundary pattern and the 1-skeleton of ∂M. In fact, the whole purpose of Step 4 above was to ensure that every component of ∂M− − ∂M picks up a boundary pattern that chops up the torus into discs. The union of the above surfaces with ∂M is a 2-complex. Once again, we form Tcan from this 2-complex, by coning its faces and then the complementary 3-balls. Now, the restriction of Tcan to ∂M clearly does not equal T∂M . We now ﬁx this, by ′ introducing Tcan , which is the canonical triangulation required by Theorem 1.3. 32 ALEXANDER COWARD AND MARC LACKENBY Let S1 , . . . , Sn be the above sequence of surfaces, described in Steps 1 - 5. For 0 ≤ i ≤ n, we will keep track of a triangulation Ti (∂M) for ∂M, in which ∂M ∩ (S1 ∪ · · · ∪ Si ) is simplicial. The initial triangulation T0 (∂M) will be the given triangulation T∂M . Each surface Si intersects ∂M in a collection of disjoint simple closed curves and arcs. We realise these as normal curves and arcs in the triangulation Ti−1 (∂M), with the property that they intersect the 1-simplices in as few points as possible. The new triangulation Ti (∂M) is chosen as follows. Each triangular face of Ti−1 (∂M) is divided up by arcs which are subsets of ∂Si . We declare that these arcs are 1-simplices in Ti (∂M). The regions in the complement of these arcs are discs. A vertex is inserted into each such disc, and then the disc is coned oﬀ. The result is Ti (∂M). At this stage we make a speciﬁc choice of 2-dimensional Pachner moves taking Ti−1 (∂M) to Ti (∂M), as follows. A (1, 3) move is performed on each triangle of Ti−1 (∂M). Then, at each of the original edges e of Ti−1 (∂M), a (2,2) move is performed. Then, at each such edge e, an alternating sequence of (1,3) and (2,2) moves is performed. The number of (1,3) moves is equal to the number of points of intersection between ∂Si and e. The result of this is that each of the original triangles of Ti−1 (∂M) has been transformed into a cone. (See Figure 12.) The triangulation of this cone is now modiﬁed without changing its boundary any further, using 2-dimensional Pachner moves, so that it becomes Ti (∂M). We term this the speciﬁed sequence of Pachner moves taking Ti−1 (∂M) to Ti (∂M). In Section 8, a sequence of boundary Pachner moves on M, also called the speciﬁed sequence, was deﬁned. Note that the speciﬁed sequence of boundary Pachner moves given in Section 8 induces the speciﬁed sequence of 2-dimensional Pachner moves deﬁned here. Associated to any sequence of 2-dimensional Pachner moves on a triangulated surface F , there is a 3-dimensional space constructed as follows. One starts with F . Each time a 2-dimensional move is performed, a 3-simplex is attached onto one side of F . Call this the ‘top’ side. This top side is a copy of F with its new triangulation. After a sequence of these moves, the resulting space is approximately a copy of F × I. The bottom has the original triangulation of F , and the top has the new triangulation. We say ‘approximately’ F × I, because this space need not in fact be a 3-manifold. This is because it is possible that some simplices of F are left untouched by the sequence of 2-dimensional Pachner moves, in which case these simplices lie in both the top and bottom copy of F . We call this space a generalised product. When F is the boundary of a 3-manifold, we also call this space a generalised collar. We have deﬁned above a speciﬁed sequence of 2-dimensional Pachner moves which takes T∂M to Tn (∂M). Associated with this sequence, there is a generalised product. Now Tn (∂M) is the boundary triangulation of Tcan . Thus, we can attach the generalised product to Tcan , to create a triangulation of M with boundary triangulation T∂M . This ′ is Tcan , the canonical triangulation for M. 11. From boundary Pachner moves to interior ones Let M be a compact orientable 3-manifold with a triangulation T . Suppose that we are given a sequence of Pachner moves, starting with T = T0 and ending with a triangulation Tn . We want to use this sequence to specify a sequence of interior AN UPPER BOUND ON REIDEMEISTER MOVES 33 arcs of ∂S i triangle of Ti-1(M) (1-3) moves (1-3) and (2-2) moves retriangulate triangle Figure 12 ′ ′ Pachner moves to M, giving triangulations T = T0 , . . . , Tn of M. At each stage, the ′ triangulation Ti of M will contain a triangulated generalised collar on ∂M. If we form the closure of the complement of this generalised collar, the result will be the triangulation Ti . We say that such a sequence of interior Pachner moves and generalised collars are associated with the given sequence of Pachner moves. The case we have in mind is where Tn = Tcan . A sequence of Pachner moves taking T to Tn = Tcan arises from the proof of Theorem 8.1. The associated sequence of interior ′ ′ Pachner moves will take T to Tn = Tcan , and will be the moves required by Theorem 1.3. We start with the motivating case, where Ti is obtained from Ti−1 by a boundary Pachner move which attaches a simplex. Suppose, for simplicity, that this has the eﬀect ′ of performing a (1, 3) move on the boundary. Now, embedded within Ti−1 is a copy of Ti−1 . The boundary Pachner move attaches on a single tetrahedron to a triangle ′ in Ti−1 . A copy of this triangle is in Ti−1 , lying between Ti−1 and the generalised ′ collar. What we do is insert two tetrahedra into Ti−1 at the location of this triangle. These two tetrahedra are glued to each other along three faces. The tetrahedron that is adjacent to the generalised collar is included in the generalised collar of Ti′ . This procedure which inserts the two tetrahedra can clearly be achieved by interior Pachner ′ moves. One simply performs a (1, 4) move on the tetrahedron of Ti−1 which is adjacent replacemen 34 M with triangulation T i-1 ∂M (1-3) boundary Pachner move ALEXANDER COWARD AND MARC LACKENBY Ti-1 ’ Ti-1 generalised collar (1-4) move (3-2) move include these tetrahedra in Ti include this tetrahedron in the generalised collar Figure 13 to the triangle and which is not part of the generalised collar. Then one performs a (3, 2) move. This is illustrated in Figure 13. We now deal with some easy cases. ′ When an interior Pachner move is performed on Ti−1 , we do the same move to Ti−1 and do not change the generalised collar. When a boundary Pachner move is performed on Ti−1 , and this removes a tetrahe′ dron, then we do not perform a Pachner move to Ti−1 . Instead, we simply enlarge its generalised collar to include the relevant tetrahedron. There are still two cases to consider. The triangulation Ti may be obtained from Ti−1 by a boundary Pachner move that adds a tetrahedron, and which performs a (2, 2) move or a (3, 1) move on the boundary. We will not in fact consider the (3, 1) case, because it is more complicated, and we will not need it in this paper. So, suppose that a (2, 2) move is performed on the boundary of Ti−1 . Two triangles are involved ′ in this move, and there are two corresponding triangles in Ti−1 . These triangles are glued along an edge, and their union is a disc (possibly with identiﬁcations along its boundary). What we do is ‘blow air’ into this disc, creating two copies of the disc. AN UPPER BOUND ON REIDEMEISTER MOVES 35 Figure 14. A sequence of (2, 3) moves followed by a (3, 2) move In the space between these two discs, we insert two tetrahedra. These two tetrahedra are glued to each other along two triangles. The tetrahedron that is adjacent to the ′ generalised collar of Ti−1 is included in the generalised collar of Ti′ . The other new tetrahedron is included in the copy of Ti in Ti′ . ′ We need to explain how the insertion of these two tetrahedra, which takes Ti−1 to Ti′ , can be achieved using interior Pachner moves. First apply a (1, 4) to every ′ tetrahedron of Ti−1 . After this, the tetrahedra incident to the interior of any face are distinct. Consider the two triangles involved in the boundary Pachner move. These two triangles are adjacent to distinct tetrahedra because of the (1, 4) moves we have just applied. Consider these two tetrahedra. Apply some (2, 3) moves until they are adjacent. Now apply two (2, 3) moves in the resulting adjacent tetrahedra to insert the two tetrahedra in the place of the two triangles that get split open. This process is illustrated in Figure 14. Now apply (3, 2) moves and (4, 1) moves to undo the initial moves. Note that if Ti contains t tetrahedra, then the number of interior Pachner moves in this process is at most 4t. We close with an important observation. The generalised collar constructed above, ′ which is a subset of Tn , only depends on the boundary Pachner moves in the original sequence. Suppose that T is a triangulation of M, and that its restriction to ∂M is the given triangulation T∂M . Suppose that we have found a sequence of Pachner moves which 36 ALEXANDER COWARD AND MARC LACKENBY takes T to Tcan . Suppose also that the boundary Pachner moves in this sequence precisely induce the speciﬁed sequence of 2-dimensional Pachner moves on ∂M, deﬁned in Section 10. Then we deduce that the associated sequence of interior moves takes T ′ to Tcan . This important observation will be crucial in our proof of Theorem 1.3, which we now come to. 12. Proof of Theorem 1.3 We are given a triangulation T for ∂M, and we need to use interior Pachner moves, ′ plus possibly a homeomorphism supported in the interior of M, to take T to Tcan . This will be achieved by performing a sequence of Pachner moves that takes T to Tcan , and so that the boundary Pachner moves in this sequence precisely induce the speciﬁed sequence in Section 10. Then, as we observed at the end of the previous section, the ′ associated sequence of interior Pachner moves will take T to Tcan , as required. In Mijatovi´’s proof and Section 8, a sequence of Pachner moves was given, which c takes T to Tcan . But we need to ensure that the boundary Pachner moves in this section are the speciﬁed sequence from Section 10. Let S1 , . . . , Sn be the sequence of surfaces used in the deﬁnition of Tcan . Suppose that we have subdivided the triangulation T to Ti−1 (M) using Pachner moves. Then S1 , . . . , Si−1 are simplicial in Ti−1 (M), and so Ti−1 (M) restricts to a triangulation Ti−1 (Mi−1 ) of Mi−1 . We want to realise Si as a normal surface in Ti−1 (Mi−1 ) with bounded complexity. The precise result that we use depends on what type of surface Si is. For example, it may be the JSJ tori, or it may be a surface where −χ(Si ) is minimised, or it may be one of several other possibilities. However, in each case, there 2 is an ambient isotopy taking Si to a normal surface with at most 2350t triangles and squares, where t is the number of tetrahedra of Ti−1 (Mi−1 ). (See Lemma 4.5 in [15].) However, there is a complication. In Section 10, we have already speciﬁed the intersection between Si and ∂M. Recall that we have already chosen the sequence of triangulations T∂M = T0 (∂M), T1 (∂M), . . . , Tn (∂M), where the restriction of Ti−1 (M) to ∂M is Ti−1 (∂M). The simple closed curves and arcs Si ∩∂M need to be the speciﬁed normal arcs in Ti−1 (∂M). However, when the usual normalisation procedure is applied to Si in Ti−1 (Mi−1 ), ∂Si may need to be moved. We therefore require the following result, which we will prove in Section 13. Theorem 12.1. Let M be a compact orientable irreducible 3-manifold with an essential boundary pattern P . Let T be a triangulation of M, in which P is simplicial, and which consists of t tetrahedra. Let F be a normal surface properly embedded in M, essential with respect to P , consisting of n normal discs. Let X be the closure of the union of some non-adjacent components of ∂M − P . Suppose that ∂F ∩ X is ambient isotopic to a collection of normal curves and arcs c in X, and that c has minimal weight in its ambient isotopy class in X. Then F is ambient isotopic in M to a normal surface F ′ whose boundary agrees with c in X and which consists of at most 8000n4 t2 normal discs. Recall that ∂Si has been chosen so that its intersection with ∂M minimizes weight in its ambient isotopy class. Let c be ∂Si ∩∂M. Now, according to Mijatovi´’s arguments, c AN UPPER BOUND ON REIDEMEISTER MOVES 37 Si can be isotoped in Mi to a normal surface with a bounded number of normal discs, n say. This isotopy may move ∂Si . By the above result, we may ﬁnd a normal surface isotopic to Si , with boundary that agrees with Si on ∂M, and with at most 8000n4t2 normal discs, where t is the number of tetrahedra in Ti−1 (Mi−1 ). We now apply Pachner moves, as described in Section 8, taking Ti−1 (M) to Ti (M). As explained in Section 8, the boundary Pachner moves that are used only depend on the intersection between ∂M and Si . And we have ensured that these are ﬁxed as in Section 10. Thus, the boundary Pachner moves precisely induce the speciﬁed sequence of 2-dimensional Pachner moves on ∂M deﬁned in Section 10. Hence, the associated ′ sequence of interior moves creates Tcan , as required. We close this section by bounding the number of Pachner moves used. To provide ′ such a bound, it suﬃces to bound the number of moves required to pass from Ti−1 (M) to Ti′ (M), and also to bound the number n, where S1 , . . . , Sn is the sequence of surfaces used to deﬁne Tcan . We start with the former estimate. Suppose that Ti−1 (M) has t tetrahedra. In [15], Mijatovi´ argues that at most c exp(2) (t) Pachner moves are required to pass to the next triangulation. However, as we have seen, his triangulations and ours are a little diﬀerent. Nevertheless, we will ′ now show that we also can pass from Ti−1 (M) to Ti′ (M) using at most exp(2) (t) interior Pachner moves. Let us focus on the case where Si is a surface from Step 2. In other words, suppose that it is part of a hierarchy in one of the simple pieces. Proposition 4.2 of [17] or Lemma 4.5 of [15] gives that Si can be realised as a normal surface in Ti−1 (Mi−1 ) 2 with at most 2350t triangles and squares. However, because the intersection Si ∩ ∂M 2 is speciﬁed in advance, then in fact we only get a bound of at most 8000 21400t t2 triangles and squares, using Theorem 12.1. Then using Lemma 4.1 in [16], there is a 2 sequence of at most 160000 21400t t3 Pachner moves taking Ti−1 (M) to Ti (M), in which Si is simplicial. However, boundary Pachner moves may be used in this process, and so we need to bound the number of interior Pachner moves required in the associated ′ sequence that takes Ti−1 (M) to Ti′ (M). The boundary Pachner moves in this sequence are precisely those from the speciﬁed sequence described in Section 8. Recall that ﬁrst a boundary (1, 3) move is performed on each triangle of Ti−1 (M) ∩ ∂M. The number of such moves is at most 4t, and hence the number of interior moves in the associated sequence is at most 8t. Then a (2, 2) move is performed along each edge of Ti−1 (M) ∩ ∂M. There are at most 6t such edges. Using the bound at the end of Section 11, the number of interior Pachner moves in the associated sequence is at most 6t(4t). After this, the only boundary Pachner moves that are performed are those that remove tetrahedra, and these do not create any interior moves in the associated ′ sequence. So, the number of interior Pachner moves taking Ti−1 (M) to Ti′ (M) is at most 8t + 6t(4t) + 160000 21400t t3 ≤ exp(2) (t), as claimed. Note that the inequality holds because the values of t for which we are making this estimate are suﬃciently large. The number of tetrahedra in Ti′ (M) is also at most exp(2) (t). 2 38 ALEXANDER COWARD AND MARC LACKENBY An easy induction gives that, for i ≥ 1, the number of tetrahedra in Ti′ (M) is at ′ most exp(2i) (t), and that the number of Pachner moves required to take Ti−1 (M) to Ti′ (M) is also at most exp(2i) (t). Thus, all we need to do now is bound the number n of surfaces in the sequence S1 , . . . , Sn . In Step 1, only one surface (which may be disconnected) is used. The number of surfaces in Step 2 is at most 2160t , by a bound of Mijatovi´ in Section 5 of c [15]. In Step 3, two surfaces are used. In Step 4, two surfaces are also used. The ﬁnal step is Step 5, which deals with the Seifert ﬁbred pieces. Only four surfaces are used, but in fact the ﬁrst surface should be counted as two, because it is built in two stages. So, we can certainly take n ≤ 2161t . t ′ Thus, the number of interior Pachner moves taking T to Tcan is at most exp(a ) (t), where a = 2162 . This proves Theorem 1.3. 13. Adjusting boundaries of normal surfaces In this section we prove Theorem 12.1. Our strategy will be to isotope F so that its boundary is in the correct place and has controlled edge degree. Then we shall apply to the resulting surface a standard normalization procedure that doesn’t increase edge degree and which, by the hypotheses on c, does not aﬀect the intersection with X. The bound on the edge degree of this surface will yield the required bound on the number of normal discs in F ′ . Observe that ∂F ∩ X certainly consists of at most 4n normal arcs. Hence c consists of at most 4n normal arcs. Without loss of generality pick c so that every normal arc of c intersects each normal arc of ∂F in at most one point. Then the number of intersection points between ∂F and c is at most (4n)2 = 16n2 . Label the components of ∂F ∩ X as {α1 , . . . , αm } and the components of c as {β1 , . . . , βm } so that there is an ambient isotopy of X that takes αi to βi for each i. Consider α1 and β1 . These intersect in at most 16n2 points. We will now isotope F about in a small collar neighborhood of X to obtain a new surface where the image of α1 is equal to β1 . This is achieved as follows. Note that for convenience we will refer to the surfaces obtained from F by our isotopies as F , with the same notation for the components of ∂F ∩ X. By Lemma 3.1 of [6] applied to the double of X along its boundary, either α1 and β1 are disjoint, or there is a bigon of α1 ∪ β1 in the interior of X that bounds a disc B whose boundary consists of a subarc of α1 and a subarc of β1 and whose interior is disjoint from α1 ∪ β1 , or there is a triangle, B ′ , in X whose boundary consists of a subarc of ∂X, a subarc of α1 and a subarc of β1 and whose interior is disjoint from α1 ∪ β1 . In the latter two cases, we would like to isotope α1 across B or B ′ , but we are impeded by the fact that there may be arcs of ∂F other than α1 running though B. Pick one of these arcs that is outermost on B or B ′ and label the disc it cuts oﬀ G. This is a bigon or triangle whose interior is disjoint from ∂F ∪ c and whose boundary consists of a subarc, σ, of αi , for some i, a subarc, τ , of β1 and possibly a subarc, γ, of ∂X. Now isotope F by sliding αi across G in a small collar neighborhood of ∂M to form a new surface. More precisely, choose a product structure X × I, where I = [0, 1], on a small regular neighborhood of X in M so that (∂F ∩ X) × {0} = ∂F ∩ X, the components of F ∩ (X × I) are vertical in X × I and the arcs of 1-skeleton emanating AN UPPER BOUND ON REIDEMEISTER MOVES 39 away from X into the interior of M or parts of ∂M outside X are at most a small deviation away from being vertical. Also suppose that (∂X × I) ∩ ∂M = ∂X × I. The new surface obtained by sliding F across G is then obtained by removing σ × I, inserting (G × {1}) ∪ (τ × I) and then pushing the resulting surface a little further to make it disjoint from G. Note that performing this operation increases the edge degree of F by at most 12t + 4n, the ﬁrst term being an upper bound on the number of possible new intersections of F with the arcs of 1-skeleton not entirely in X, and the second term being an upper bound on the number of new intersections of F with arcs entirely in X. Repeat this operation to remove all arcs of intersection of B or B ′ with ∂F and then slide F across B or B ′ itself. Repeat this proceed until α1 and β1 are disjoint. The edge degree has been increased by at most 16n2 (12t + 4n). Now, F has been arranged so that α1 is disjoint from β1 . Hence α1 and β1 cobound an annulus or, together with two arcs of ∂X, a disc. Call this annulus or disc A. We wish to isotope α1 across A to make α1 equal to β1 . This time we may be impeded by arcs of intersection of ∂F with A, which may be removed as before, or by entire curves or arcs of intersection of ∂F with A, which may be removed by sliding across annuli or discs in a fashion similar to sliding across bigons and triangles. The eﬀect of the removal of bigons and triangles on edge degree is already incorporated in our previous estimate. Sliding across an annulus or disc increases edge degree by at most 12t + 4n and we need to do this at most m − 1 ≤ 4n − 1 times before ∂F is disjoint from the interior of A. The ﬁnal isotopy we perform at this stage is across A itself, except we do not make the ﬁnal push away from A. Thus the edge degree since our last estimate has increased by at most 4n(12t + 4n). To recap, we have isotoped F about so that α1 = β1 and so that the edge degree of F has increased by at most (16n2 + 4n)(12t + 4n). We may now repeat the above procedure with the other components of ∂F ∩ X, treating them one at a time and keeping components ﬁxed once they have been isotoped to equal the corresponding component of c. Thus we form a surface whose boundary components intersect X in a way that agrees with c and whose edge degree increases overall by at most m(16n2 + 4n)(12t + 4n) ≤ 4n(16n2 + 4n)(12t + 4n). The initial surface F , before any isotopies were performed, had edge degree at most 4n. 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