aa r X i v : . [ m a t h . G T ] A ug LINKS WITH SPLITTING NUMBER ONE
MARC LACKENBY †
1. Introduction
One of the knot invariants that is least well understood is unknotting number. This is definedto be the minimal number of crossing changes that one can apply to some diagram of the knotin order to unknot it. For any given diagram of a knot K , it is of course easy to determine theminimal number of crossing changes that one can apply to it in order to unknot it, by using oneof the several known algorithms to detect the unknot. However, one has no guarantee in generalthat there is not some more complicated diagram of K that can be unknotted using fewer crossingchanges. Many techniques have been developed to find lower bounds on the unknotting number ofa knot, for example, using the Alexander module [34], the Goeritz form [23, 44], gauge theory [45]and Heegaard Floer homology [29, 35, 36]. However, no known technique is perfect, and in factthere are many explicit knots for which the unknotting number is not known [24]. A satisfactoryresolution will only be found when an algorithm that determines the unknotting number of a knotis discovered. But this appears to be a very long way off. In fact, it is conceivable that no suchalgorithm exists. It is not even known whether one can decide algorithmically whether a knot hasunknotting number one.In this paper, we explore some natural generalisations of unknotting number to links with morethan one component. One might consider the unlinking number u ( L ) of a link L, which is theminimal number of crossing changes required to turn it into the unlink. But it turns out that itis just as natural to consider the splitting number s ( L ), which is the minimal number of crossingchanges required to turn it into a split link. (A link is split if there is an embedded 2-sphere disjointfrom the link with link components on both sides.) Some authors [5] have also analysed a variantof splitting number, where one only considers crossing changes between distinct components of thelink. The minimal number of such crossing changes required to create a split link we denote by s d ( L )(where d stands for ‘distinct’). Other authors [25] have also required the resulting link to be totallysplit , which means that there is a union of disjoint balls containing the link, such that each ballcontains a single component of the link in its interior. We say that the total splitting number ts ( L )is the minimal number of crossing changes required to make the link totally split. Again, one canconsider only crossing changes between distinct link components and we denote the resulting variantof total splitting number by ts d ( L ). Our main result is that, under some fairly mild hypotheses,there are algorithms to determine whether any of these quantities is 1 for a given link. Theorem 1.1.
There is an algorithm to solve the following problem. The input to the algorithm is alink L in S , given either by a diagram or by a triangulation of S with L as a specified subcomplex.The link L must be hyperbolic and 2-string prime. It is required to have at least two componentsand if it has exactly two components, these must have zero linking number. The output is an answer † Partially supported by EPSRC grant EP/R005125/11 o each of the following questions:(i) Is u ( L ) = 1 ?(ii) Is s ( L ) = 1 ?(iii) Is s d ( L ) = 1 ?(iv) Is ts ( L ) = 1 ?(v) Is ts d ( L ) = 1 ? Recall that a link L is if, for each 2-sphere S in S that intersects L transverselyin four points, S − L admits a compression disc in the complement of L . When L is hyperbolic,then it is 2-string prime if and only if its branched double cover is hyperbolic or a small Seifert fibrespace. (This is explained in Section 4.) This condition can be readily verified both in theory [20,43] and in practice [48].The linking number hypothesis when L has two components is a slightly unfortunate one.However, it is not as restrictive as it first may seem. When s ( L ) = 1, then the two components of L must have linking number 0, 1 or −
1, for the following reason. When both components of L areinvolved in the crossing change, then the linking number is ±
1, since the crossing change alters thelinking number by one. On the other hand, when the crossing change moves some component of L through itself, then this does not change the linking number, and so this must be zero.The most notable hypothesis in Theorem 1.1 is that the link L has more than one component.As mentioned above, it remains the case that there is no known algorithm to decide whether a knothas unknotting number one.Whenever one has an algorithmic result such as the one presented above, a finiteness theoremtends to come for free. In this case, the question that we can address is: are there only finitelymany ways to split a link by a crossing change? Of course, one needs a way to compare two crossingchanges, which may occur in different diagrams. There is a natural method of doing this usingsurgery. Given any crossing in some diagram of L , we may encircle the two sub-arcs of L near thecrossing by a simple closed curve C , as shown in Figure 1. This bounds an embedded disc D suchthat D ∩ L is two points in the interior of D . Such a disc D is called a crossing disc . The boundarycurve C of a crossing disc is called a crossing circle . In the interior of the crossing disc D , there is anembedded arc joining the two points of D ∩ L . This is the crossing arc associated with D . Changingthe crossing is achieved by ± C . We say that two crossing changes are equivalent if their associated crossing circles are ambient isotopic in the complement of L and the associatedsurgery coefficients are equal. Theorem 1.2.
Let L be as in Theorem 1.1. If s ( L ) = 1 , then, up to equivalence, there are onlyfinitely many ways to turn L into a split link by performing a crossing change. In fact, if t is thenumber of tetrahedra in a triangulation of S with L as a subcomplex, then the number of distinctways of creating a split link from L by a crossing change is at most k t , for some universal computableconstant k . Hence, the number of ways is at most k c ( L ) where c ( L ) is the crossing number of L .Moreover, there is an algorithm to find all these crossing changes. L crossing -1 surgeryalong C circle C Figure 1An algorithm to compute the constant k is given in Section 11, building on Section 11 of [22],although it would be technically challenging to implement.Although the algorithm given by Theorem 1.1 is not very efficient, one can often apply thetechniques behind it quite practically. For example, we can obtain the following result. Theorem 1.3.
Any crossing change that turns the Whitehead link into a split link is equivalent tochanging a crossing in some alternating diagram.
An outline of the paper is as follows. In Section 2, we recall the operation of trivial tanglereplacement. This is a generalisation of a crossing change, and is in fact the central object of studywithin this paper. It is well known that trivial tangle replacement can be studied by analysingthe double cover of the 3-sphere branched over the link, via the Montesinos trick. We recall therelevant theory in Section 2. In Section 3, we compare trivial tangle replacement with crossingchanges, focusing in particular on the notions of equivalence in each case. In Section 4, we give acharacterisation of the hyperbolic links in the 3-sphere that are 2-string prime, in terms of theirbranched double covers. We also consider double covers branched over sublinks of the link, whichare the 3-manifolds that play a central role in the proof of Theorem 1.1. In Section 5, we give anoverview of the general set-up of our algorithm. This involves constructing certain double branchedcovers M and then searching for exceptional surgery curves in M . This second step uses earlier workof the author [22]. The algorithm divides according to whether M is Seifert fibred or hyperbolic.In Section 6, the Seifert fibred case is analysed. In Section 7, the hypotheses of the main theoremin [22] are verified. In Section 8, we analyse the mapping class group of finite-volume hyperbolic3-manifolds, mostly from an algorithmic perspective. In Section 9, we show that all the problemsin Theorem 1.1 are decidable. In Section 10, we give an overview of the work in [22]. This leads tothe finiteness result, Theorem 1.2, in Section 11. In Section 12, we analyse the Whitehead link andwe classify the crossing changes that can be applied to the link to turn it into a split link.
2. Tangle replacement
In this section, we recall the operation of tangle replacement, and the well-known Montesinostrick [32].A tangle is a 1-manifold A properly embedded within a 3-ball B . When A has no closedcomponents, and so is a collection of k arcs for some positive integer k , it is termed a k -string trivial if there is homeomorphism between B and D × I taking A to P × I ,for some finite collection of points P in the interior of D . In the case of a trivial 2-string tangle,its core is an arc α × {∗} , where α is an embedded arc in the interior of D joining the two pointsof P , and ∗ is a point in the interior of I . It is in fact the case that a trivial 2-string tangle has aunique core up to isotopy of B that leaves A invariant.Let M be a compact orientable 3-manifold with (possibly empty) boundary. Let L be a compact1-manifold properly embedded in M . Let α be an arc embedded in the interior of M such that L ∩ α = ∂α . Let B be a regular neighbourhood of α in M , which intersects L in a trivial 2-stringtangle in which α is a core arc. Suppose that we remove this tangle and insert into M another trivialtangle with the same endpoints. The result is a new 1-manifold in M , which we say is obtained from L by tangle replacement along α . The possible trivial tangles that we may insert are parametrisedas follows. On ∂B − L , there is a unique isotopy class σ of essential simple closed curves that bounda disc in the complement of the new tangle. We term this the tangle slope . The link that resultsfrom L by this tangle replacement is denoted L σ .We say that the distance ∆( σ, σ ′ ) between two tangle slopes σ and σ ′ on ∂B − L is equal tohalf the minimal intersection number between two representative simple closed curves. Any simpleclosed curve on ∂B − L is separating, and hence any two curves have even intersection number.Therefore, the distance between slopes is always an integer.Let A be a trivial 2-string tangle in the 3-ball B . Then there is a unique double cover V of B branched over A . It is well known that V is a solid torus. This is because V is of the form A × I ,where A is the annulus that is the double cover of the disc branched over two points.Consider the link L σ obtained from L by tangle replacement. Suppose that M admits a doublecover ˜ M branched over L . Then there is a corresponding double cover ˜ M σ branched over L σ whichis defined as follows. The inverse images of B and M − int( B ) in ˜ M give double covers branchedover L ∩ B and L − int( B ) respectively. Similarly, the inverse image of ∂B is a double cover of the2-sphere branched over four points. Now there is a unique double cover of S branched over fourpoints. This is a torus T . Hence, any homeomorphism ∂B → ∂B that sends ∂B ∩ L to ∂B ∩ L liftsuniquely to a homeomorphism T → T . One may view the tangle replacement as simply attaching B to M − int( B ) via some homeomorphism that leaves ∂B ∩ L invariant. Lifting this homeomorphismto the branched double covers gives a gluing map, via which we may construct the double cover ˜ M σ branched over L σ . Since the double cover of B branched over L ∩ B is a solid torus, ˜ M σ and ˜ M are related by Dehn surgery. The surgery curve in ˜ M is the inverse image of the arc α . It is easyto check that the distance between between the two surgery slopes, one giving ˜ M σ and the othergiving ˜ M , is equal to the distance ∆( σ, µ ) between σ and the meridian slope µ of α .The above use of branched double covers leads to a very useful method of parametrising slopesof trivial tangles. We will consider trivial tangles A within the 3-ball B , where ∂B ∩ A is a given setof four points. The tangle is determined by the unique isotopy class of essential curves in ∂B − A that bound a disc in the complement of A . The 2-sphere ∂B admits a unique double cover branchedover ∂B ∩ A , which is a torus T . The elevation of the simple closed curve in ∂B − A that bounds4 disc in B − A is an essential simple closed curve in T . One may parametrise the tangle B ∩ A bymeans of this slope. It is possible to show (for example [4]) that this slope determines the trivialtangle up to an isotopy of B fixed on ∂B . Moreover, each slope is realised by some tangle. Thus,trivial tangles are in one-one correspondence with slopes on T . One can pick a basis { λ, µ } for thehomology of H ( T ), and in the usual way, the slope with class ± ( pλ + qµ ) in H ( T ) is representedby p/q ∈ Q ∪ {∞} . Given p/q ∈ Q ∪ {∞} , one may explicitly construct the associated tangle, asfollows. Let [ c , . . . , c n ] denote a continued fraction expansion for p/q , where each c i ∈ Z . Then theassociated tangle is shown in Figure 2, with the two possibilities shown depending on whether n iseven or odd. c crossings -c crossings c crossings -c crossings B c crossings -c crossings c crossings B Figure 2In the above figure, each box contains a line of crossings in a row, called a twist region .Conventionally, in a twist region with a ‘positive’ number of crossings, they are twisted in a clockwise-fashion. However a box with a ‘negative’ number c of crossings is in fact a string of | c | crossingstwisted in an anti-clockwise fashion.We will be considering the operation of tangle replacement throughout this paper. At variouspoints, it will be important for us to consider when tangle replacement can change a trivial tangleto another trivial tangle. More specifically, suppose that A is a trivial 2-string tangle in the 3-ball B , and suppose that α is an embedded arc in the interior of B such that α ∩ A = ∂α . Supposethat tangle replacement along α changes A into another trivial 2-string tangle A ′ . Then what arethe possible locations for α , what are the possible tangle replacements, and what is the relationshipbetween the slopes of A and A ′ ? Fortunately, all of these questions have been given a precise answerby Baker and Buck [1], by use of branched double covers and surgical methods of Gabai [9]. Thesituation is simplest to state when the distance of the tangle replacement is at least two, as follows(see Theorems 1.1 and 3.1 in [1]). Theorem 2.1.
Let A be a trivial 2-string tangle with slope ∞ in the 3-ball B . Suppose that α isan embedded arc in the interior of B such that α ∩ A = ∂α . Suppose that distance d ≥ tanglereplacement along α changes A into another trivial 2-string tangle A ′ with slope p/q . Then one ofthe following holds:(i) α is the core arc of the tangle A ;(ii) p/q = (1 ± dab ) / ± da , for coprime integers a and b . Moreover, if a/b has continued fractionexpansion [ c , . . . , c n ] , then there is an isotopy of B , fixed on ∂B , taking B ∩ A to the trivialtangle with continued fraction expansion [0 , c , . . . , c n , , − c n , . . . , − c ] , and taking α to the rossing arc of the central twist region labelled . The tangle replacement simply replaces the crossings with ± d . (See Figure 3.) -c crossings c crossings B α c crossings -c crossings Figure 3Using methods similar to those of Baker and Buck, we can obtain the following result.
Theorem 2.2.
Let A be a 2-string tangle in the 3-ball B such that ∂B − ∂A is compressible in thecomplement of A . Let α be an embedded arc in the interior of B such that α ∩ A = ∂α . Supposethat ∂B − ∂A is incompressible in the complement of A ∪ α . Let A ′ be obtained from A by trivialtangle replacement along α with distance at least two. Then the following hold.(i) There is no 3-ball in B with boundary disjoint from A ′ and that encloses a closed componentof A ′ .(ii) If ∂B − ∂A ′ is compressible in the complement of A ′ , then A and A ′ are trivial tangles, andhence (i) or (ii) of Theorem 2.1 holds.Proof. Let M and M ′ be the double covers of B , branched over A and A ′ respectively. These differby surgery along a curve K in M that is the inverse image of α . The distance between the surgeryslope and the meridian slope is equal to the distance of the tangle replacement, which is at least twoby assumption. Note that M has compressible boundary, since the inverse image of a compressiondisc for ∂B − ∂A contains a compression disc for ∂M . Hence, M is either a solid torus or reducible.In fact, A is obtained from a trivial tangle by possibly tying a little knot in one or both of its strings.Hence, M is the connected sum of a solid torus with two rational homology 3-spheres, one or bothof which may be 3-spheres. In particular, M contains no non-separating 2-sphere, and so the sameis true of M − int( N ( K )).On the other hand, ∂M is incompressible in the complement of K , for the following reason.If there were a compression disc for ∂M in the complement of K , the equivariant disc theorem[31] would provide one or two disjoint compression discs for ∂M in the complement of K that areinvariant under the involution of M . These descend to a compression disc D for ∂B − ∂A in thecomplement of α and that intersects A in at most one point. This disc D cannot be disjoint from A by hypothesis. Hence, it intersects A in a single point. Its boundary lies in the 2-sphere ∂B and so bounds discs in ∂B . The union of either of these discs with D forms a 2-sphere which, forparity reasons, must intersect A an even number of times. Therefore, ∂D bounds a disc in ∂B ∂A once. The inverse image of ∂D in ∂M therefore bounds a disc in ∂M , whichcontradicts the fact that it is the boundary of a compression disc.To prove (i), suppose that B contains a ball with boundary disjoint from A ′ that encloses aclosed component of A ′ . The inverse image of this ball in M ′ is a connected 3-manifold with twospherical boundary components. The complement of this manifold in M ′ is connected, and so wededuce that M ′ contains a non-separating 2-sphere. We now apply Scharlemann’s theorem [42]. Thisimplies that a compact orientable irreducible 3-manifold with toroidal boundary cannot be Dehnfilled along slopes with distance at least two, and where one filling gives a manifold with compressibleboundary and the other filling gives a reducible 3-manifold. In our situation, M − int( N ( K )) neednot be irreducible, but it is a connected sum of irreducible 3-manifolds, one of which contains ∂N ( K ).Let X be this summand. This summand must contain ∂M , as otherwise the meridional Dehn fillingof M − int( N ( K )) could not produce a 3-manifold with compressible boundary. So, M − int( N ( K ))is the connected sum of X with a rational homology 3-sphere (which may be a 3-sphere). When X isfilled to give a summand of M ′ , this summand must contain a non-separating sphere in M ′ . Hence,when X is Dehn filled in two different ways with distance at least two, one filling gives a 3-manifoldwith compressible boundary, and the other filling gives a reducible 3-manifold. This contradictsScharlemann’s theorem.We now prove (ii). We observed above that M has compressible boundary. Suppose that ∂B − ∂A ′ is compressible in the complement of A ′ . Then M ′ also has compressible boundary.Theorem 2.4.4 of [7] therefore applies. With the assumption that the distance of the surgery isat least two, it implies that M − int( N ( K )) is either a copy of T × I or a ‘cable space’. In theformer situation, every Dehn filling of K gives a solid torus. So, suppose that M − int( N ( K )) is acable space. This is a Seifert fibred space with annular base space and with one singular fibre. Ifone were to fill ∂N ( K ) along a slope that has distance one from the regular fibre, the result is asolid torus. In all fillings with distance at least two from the regular fibre, the resulting manifoldhas incompressible boundary. Since we are filling ∂N ( K ) along slopes with distance at least twoand we obtain manifolds M and M ′ with compressible boundary, we deduce that the slopes giving M and M ′ have distance 1 from the regular fibre. Thus, M and M ′ are both solid tori. Since M is the branched double cover over A , it admits a (piecewise-linear) involution. Piecewise-linearinvolutions of the solid torus have been classified (see Theorem 4.3 in [13]). Up to conjugacy by apiecewise-linear homeomorphism, there is just one orientation-preserving piecewise-linear involutionwith fixed-point set homeomorphic to two intervals. Therefore, A is a trivial tangle. Similarly, A ′ is a trivial tangle. We are therefore in the setting of Theorem 2.1, and hence (i) or (ii) of Theorem2.1 holds.
3. Crossing arcs versus crossing circles
A crossing change to a link L can be viewed in two ways: as a special type of tangle replacementand as a special type of Dehn surgery. In this section, we will explore how these two alternativeviewpoints are related. 7angle replacement was discussed in Section 2. One starts with an embedded arc α such that α ∩ L = ∂α . A regular neighbourhood of α in S intersects L in a trivial tangle. The crossing changeis implemented by removing this tangle and inserting another trivial tangle, with the property thatthe new and the old tangle slopes have distance exactly 2. There are infinitely many possible tanglereplacements of this form, as shown in Figure 4. Each corresponds to changing a crossing in somediagram of L . n crossings α Figure 4This ambiguity is a slightly unfortunate one. It could, of course, be rectified by requiring α tobe framed in some way. More precisely, one could specify not just the arc α but also an explicitidentification between its regular neighbourhood B and D × I , so that B ∩ L is sent to verticalarcs in the product structure. One would then be able to specify the precise tangle replacement bygiving the slope of the new tangle as an explicit fraction.This is somewhat cumbersome and so it is more usual to specify crossing changes via surgeryalong crossing circles, as described in Section 1. In this section, we investigate the following questions.If a crossing change is specified by tangle replacement along an arc, then how many crossing circlesdoes this give rise to? If a crossing change is specified by surgery along a crossing circle, how manyassociated crossing arcs are there?The second of the above questions has a possibly surprising answer. A crossing circle can giverise to an arbitrarily large number of distinct crossing arcs. The point is that, to obtain a crossingarc from a crossing circle, one must choose a crossing disc D . The associated crossing arc is then theembedded arc in D joining the two points of D ∩ L . But a crossing circle may bound many, quitedifferent crossing discs. An example is given in Figure 5, where a single crossing circle gives rise to4 different crossing arcs. This example generalises in an obvious way to arbitrarily many differentcrossing arcs.Let us now pass to the first of the above questions. When a crossing change is specified bya tangle replacement along α , then there are actually two associated crossing circles, as shown inFigure 6. 8 ssociatedcrossing arcscrossingcircle Figure 5 α Figure 6Implicit in the above statement is that no other crossing circles arise from this tangle replace-ment. We now make this more precise.
Lemma 3.1.
When a crossing change to a link L is achieved by tangle replacement along an arc α ,this gives rise to precisely two crossing circles, so that surgery along either of these crossing circlesimplements this crossing change.Proof. Note first that the two crossing circles shown in Figure 6 are not equivalent. In other words,they are not related by an ambient isotopy that preserves the link L . This is because one can pickany orientation on L and then one of these crossing circles has zero linking number with L and otherhas linking number ± α , we must thicken α to a disc. There are infinitely many ways of doingthis, that are parametrised by an integer n ∈ Z . We denote the boundary of this disc by C n , asshown in Figure 7. = α C nn crossings Figure 7When ± C n , we obtain the tangle with slope n ± , according toFigure 2. Thus, we see that, to obtain a specific slope, in other words to obtain a specific tanglereplacement, there are exactly two choices of n . Specifically, we could perform − C n +1 or +1 surgery along C n . 9 . Simplicity of the branched double covers Theorem 1.1 only applies to links that are hyperbolic and 2-string prime. It is reasonable toask whether it is possible to easily determine whether this condition holds. In the following result,we give an alternative characterisation in terms of the geometry of the branched double cover. Thisis easily checked in practice (using Snappea for example [48]) and can be determined algorithmically[17, 20, 26, 43]. We also examine branched double covers over sublinks of the link L , and derive aresult that will be useful in the proof of Theorem 1.1. Proposition 4.1.
Let L be a hyperbolic link in the 3-sphere.(i) Then L is 2-string prime if and only if its branched double cover is hyperbolic or a small Seifertfibre space.(ii) Suppose that L is 2-string prime, and let L ′ be a sublink of L . Then the double cover of S − int( N ( L − L ′ )) branched over L ′ is hyperbolic or a small Seifert fibre space. Recall that a Seifert fibre space is small if it contains no essential embedded torus. In particular,the 3-sphere is a small Seifert fibre space, as is any lens space.
Proof.
Note that the forwards implication in (i) is a special case of (ii) with L ′ = L . So, we initiallyfocus on (ii). Let M be the double cover of S − int( N ( L − L ′ )) branched over L ′ . To verify that M is hyperbolic or a small Seifert fibre space, we appeal to the solution to the GeometrisationConjecture [37, 38, 39]. So, if M is not hyperbolic or a small Seifert fibre space, then it is toroidalor reducible. Suppose first that M is reducible. Then the equivariant sphere theorem (see Theorem3 in [30] and its proof) implies that there are one or two embedded disjoint essential spheres thatare invariant under the covering involution. Their union descends to a 2-sphere in S either thatis disjoint from L or that intersects L in two points. If the sphere is disjoint from L , then it hascomponents of L on both sides, since its inverse image in M is essential. If the sphere intersects L in two points, then this forms an essential annulus properly embedded in the exterior of L . In bothcases, L fails to be hyperbolic. Suppose now that M is toroidal. Then the equivariant torus theorem(Corollary 4.6 in [13]) gives one or two embedded disjoint essential tori that are invariant underthe covering involution. These descend to an essential torus in the exterior of L or to an essential4-times punctured sphere with meridional boundary. In the former case, this implies that L is nothyperbolic. In the latter case, L is not 2-string prime.We now prove the backwards implication in (i). Let M be the double cover of S branchedover L . Suppose that M is hyperbolic or a small Seifert fibre space. Let S be a 2-sphere in S thatintersects L in four points, such that S − L has no compression disc in the complement of L . Theinverse image of this 2-sphere in M is a torus T , which must be compressible by our hypothesis about M . By the equivariant disc theorem (Theorem 7 in [31]), there are one or two disjoint compressiondiscs for T that are invariant under the involution of M . These descend to a compression disc D for S that intersects L in at most one point. This cannot be disjoint from L by our assumption about S . On the other hand, if D intersects L in a single point, then ∂D separates S into two discs, oneof which contains a single point of L ∩ S . We deduce in that case that the inverse image of D was10ot a compression disc for T , which is a contradiction.
5. The general set-up
Let L be our given link in S . Suppose that a crossing change to L transforms it into a splitlink L ◦ . Associated with this crossing change is a crossing circle C in the complement of L thatbounds a crossing disc D . Running between the two points of L ∩ D is the crossing arc α .Let L ′ be the union of the components of L containing L ∩ D . Thus, L ′ has one or twocomponents. We let M be the manifold obtained from S − int( N ( L − L ′ )) by taking the doublecover branched over L ′ . More precisely, we consider the double cover of S − int( N ( L )) determinedby the homomorphism π ( S − int( N ( L )) → Z / L ′ . Then M is obtained from this cover by Dehn filling each component of the inverse imageof ∂N ( L ′ ) using slopes that are elevations of meridians.Suppose that L ◦ is obtained from L by performing surgery along C via the slope ±
1. Let L ′◦ ⊂ L ◦ be the image of L ′ after this surgery. Let M ◦ be the double cover of S − int( N ( L ◦ − L ′◦ ))branched over L ′◦ . Then M ◦ is obtained from M by Dehn surgery along a curve K . Moreover, if µ is the meridional slope on ∂N ( K ) and σ is the surgery slope, then ∆( σ, µ ) = 2.There are two main reasons why we use this set-up. Firstly, the distance between the surgeryslope σ and the meridian slope µ is more than 1. Secondly, we will see that H ( M − int( N ( K )) , ∂M ) =0. These two seemingly technical points are important because Dehn surgery theory works mostsmoothly when they hold. In particular, they are hypotheses in the following theorem of the author[22]. Let M be a compact orientable 3-manifold with ∂M a (possibly empty) union of tori. Let K be a knot in M , and let σ be a slope on ∂N ( K ) other than the meridional slope µ . Let M K ( σ ) bethe manifold that is obtained by Dehn surgery along K via the slope σ . Then σ is an exceptionalslope and K is an exceptional surgery curve if any of the following holds:(i) M K ( σ ) is reducible,(ii) M K ( σ ) is a solid torus, or(iii) the core of the surgery solid torus has finite order in π ( M K ( σ )).Also, σ and K are norm-exceptional if there is some z ∈ H ( M − int( N ( K )) , ∂M ) that maps to anelement z σ ∈ H ( M K ( σ ) , ∂M K ( σ )), such that the Thurston norm of z σ is less than the Thurstonnorm of z . Theorem 5.1.
There is an algorithm that takes, as its input, a triangulation of a compact connectedorientable 3-manifold M , with ∂M a (possibly empty) union of tori. The output to the algorithmis a list of all knots K within M and all slopes σ on ∂N ( K ) with all the following properties:(i) M − int( N ( K )) is irreducible and atoroidal, and H ( M − int( N ( K )) , ∂M ) = 0 ; ii) σ is an exceptional or norm-exceptional slope on ∂N ( K ) , such that ∆( σ, µ ) > , where µ is themeridian slope on ∂N ( K ) .In particular, there are only finitely many such knots K and slopes σ . The way that the algorithm lists the possibilities for K is described in Section 10. It is straight-forward to then realise each possibility for K as a subcomplex of a suitable iterated barycentricsubdivision of the given triangulation of M . (See Theorem 10.2 and the discussion after it.)Note that in our setting, σ is an exceptional surgery slope on ∂N ( K ). This is because Dehnfilling M − int( N ( K )) along σ gives the manifold M ◦ . This is the branched double cover of S − int( N ( L ◦ − L ′◦ )) branched over L ′◦ . The splitting sphere in the complement of L ◦ lifts to reducingspheres in M ◦ .Thus, roughly speaking, the algorithm required by Theorem 1.1 proceeds by constructing thefinitely many possibilities for M , then using Theorem 5.1 to find all the exceptional surgery curves K in M satisfying the hypotheses of Theorem 5.1, and then determining whether any of thesedescend to a crossing arc α for L . It thereby builds a finite list of possibilities for α and for eachsuch possibility, it provides the associated tangle replacement slope. In each case, we perform thistangle replacement and determine whether the resulting link is split, totally split or the unlink, asappropriate.
6. The Seifert fibred case
As in the previous section, M is the double cover of S − int( N ( L − L ′ )) branched over L ′ .We saw in Proposition 4.1 that the manifold M is hyperbolic or a small Seifert fibre space. Inthis section, we deal with the case where M is Seifert fibred. In this setting, the list of potentialexceptional surgery curves is very simple, as given by the following result. Theorem 6.1.
Let M be a Seifert fibred 3-manifold with non-empty boundary and let K be aknot in M such that M − int( N ( K )) is irreducible and atoroidal and H ( M − int( N ( K )) , ∂M ) = 0 .Suppose that σ is an exceptional slope on ∂N ( K ) such that ∆( σ, µ ) > , where µ is the meridionalslope. Then K is isotopic to a singular fibre of M and σ is the slope of the regular fibres, when N ( K ) is a fibred regular neighbourhood of K . In particular, M − int( N ( K )) is Seifert fibred. We will defer the proof of this until Section 10.The above result says that the exceptional surgery curve K is isotopic to a singular fibre of M .However, the manifold M that we are considering comes with an involution that preserves K , andthere is no a priori reason why this isotopy should be equivariant with respect to the involution. Wedeal with this as follows. Addendum 6.2.
Let M , K , σ and µ be as in Theorem 6.1. Let τ be a piecewise-linear involutionof M that leaves K invariant. Then there is a Seifert fibration of M that is invariant under τ andthat has K as a singular fibre. Again, σ is the slope of the regular fibres when N ( K ) is fibred regular eighbourhood of K . Before we prove this, we quote the following theorem.
Theorem 6.3.
Let M be a Seifert fibre space that admits a piecewise-linear involution τ . Then itadmits a Seifert fibration that is invariant under τ . This was proved by Tollefson [47], but under the assumption that if the base space of M is a 2-sphere, then it has at least four singular fibres. The case excluded by Tollefson requires the OrbifoldTheorem [2, 6]. The quotient M/τ is an orbifold. Since M is Seifert fibred, and its base space isa 2-sphere with at most 3 singular points, then M/τ is orbifold-irreducible and orbifold-atoroidal.Hence by the Orbifold Theorem, it is either hyperbolic or Seifert fibred. But if
M/τ were hyperbolic,then so would M be, which is impossible. Thus, M/τ is Seifert fibred. Its Seifert fibration lifts to aSeifert fibration of M that is invariant under τ . Proof of Addendum 6.2.
We assume Theorem 6.1, which will be proved in Section 10. By Theorem6.1, M − int( N ( K )) is Seifert fibred. It admits an involution, which is the restriction of τ . Hence,by Theorem 6.3, it admits a Seifert fibration that is invariant under the involution. The slope ofthe regular fibres must be the exceptional slope σ , because filling along any other slope gives aSeifert fibre space. Moreover, this Seifert fibre space cannot be a solid torus, by our hypothesis that H ( M − int( N ( K )) , ∂M ) = 0. Thus, the Seifert fibration on M − int( N ( K )) extends to a Seifertfibration on M that is invariant under the involution τ . Since the meridional slope µ is assumed tohave distance at least two from the slope of the regular fibre σ , we deduce that the surgery curve K is a singular fibre. Remark 6.4.
Note that Addendum 6.2 and Theorem 6.3 assert the existence of some
Seifertfibration that is invariant under τ . However, most Seifert fibre spaces have a unique Seifert fibrationup to isotopy. Indeed, this is true of any Seifert fibre space with non-empty boundary other than T × I , the twisted I -bundle over the Klein bottle and the solid torus (see Theorem VI.18 of [14]for example). This is important, because once one has identified a Seifert fibration on the manifold M in Theorem 6.1, then we know that K is isotopic to a singular fibre in this Seifert fibration. Inthe setting of Addendum 6.2, K is also assumed to invariant under the piecewise-linear involution τ , and hence it descends to a 1-manifold in the orbifold M/τ . We would like to know that K issimilarly well-defined up to isotopy in M/τ . Fortunately, this is provided to us by the followingresult of Bonahon and Siebenmann (see Theorem 2 in [3]) which establishes the uniqueness up toisotopy of Seifert fibrations on many orbifolds.
Theorem 6.5.
Let
M/τ be a Seifert fibred orbifold with non-empty boundary. Suppose that thebase orbifold of the Seifert fibration is not finitely covered by a disc or an annulus. Then the Seifertfibration on
M/τ is unique up to isotopy that preserves the singular locus of the orbifold throughout.
We will need the following constructive version of Theorem 6.3.
Theorem 6.6.
There is an algorithm that takes, as its input, a triangulation T of a Seifert fibrespace M with non-empty boundary and an involution τ of M that preserves T . The output of the lgorithm is a union of disjoint simple closed curves that are the singular fibres in a Seifert fibrationof M that is invariant under τ . The algorithm also produces a regular neighbourhood of thesesingular fibres, together with a slope on each of these solid tori that represents a regular fibre.Proof. In Algorithm 8.1 of [17], Jaco and Tollefson provided an algorithm to determine whether aHaken manifold with incompressible boundary is Seifert fibred. It also produced the informationrequired by the theorem: the singular fibres, together with the slopes of regular fibres on the bound-ary of their solid toral neighbourhoods. However, we need to perform a version of this procedureequivariantly.If necessary, we first subdivide the triangulation T to a triangulation T ′ so that the fixed-pointset of τ is simplicial in T ′ .If M is a solid torus, then this may be algorithmically determined, for example using Theorem6.2 in [17]. In this case, M admits a Seifert fibration that is invariant under τ . It has at most onesingular fibre, that is a core of M . Hence, our algorithm must simply find the slope of the regularfibres on ∂M . But this slope may be taken to be any non-meridional slope that is invariant underthe involution, and this may easily be determined algorithmically.If M is homeomorphic to T × I , this may also be algorithmically determined, using Algorithm8.1 in [17]. In this case, our algorithm ends by declaring that M has no singular fibres.So, we may assume that M is neither a solid torus nor T × I . It therefore contains an essentialproperly embedded annulus. According a theorem of Kobayashi (Theorem 1 in [19]), it containssuch an annulus that is either invariant under τ or disjoint from its image under τ . We need to showthat in addition, this annulus A can be realised as a normal surface, with control over its number ofintersections with the 1-skeleton of T ′ .By the PL-minimal surface theory of Jaco and Rubinstein, there is a PL-least area surface inthe isotopy class of A , which we will also call A . By definition, this is normal in T ′ . By Theorem 7of [16], A is either disjoint from its image under τ or it equals its image. Let ˜ A be A ∪ τ A , which isequal either to A or to the disjoint union of A and τ A .Normal surface theory [12, 15, 28] gives that there is a finite constructible collection of normalsurfaces F , . . . , F n in T ′ that are fundamental. The normal surface ˜ A is a normal sum k F + . . . + k n F n . We will show that ˜ A can be chosen so that each k i is at most 8. By Theorem 4.1.36 of [28](see also Theorem 2.2 of [15]), any normal summand of ˜ A must be incompressible and boundary-incompressible, and no summand can be a sphere or disc. Since Euler characteristic is additiveunder normal summation, any fundamental surface F i that is a summand of ˜ A must be an annulusor M¨obius band. If F i is a M¨obius band, then 2 F i is an annulus. Hence if some k i >
1, then˜ A has an annulus A ′ as a summand which is either fundamental or twice a fundamental surface.The surface τ A ′ is also a normal surface. It is also a summand for ˜ A . Hence, A ′ and τ A ′ havecompatible normal co-ordinates in the sense that no tetrahedron of T ′ contains a quadrilateral of A ′ and a quadrilateral of τ A ′ that are not normally isotopic. One may therefore form the normal sum A ′ + τ A ′ and obtain an embedded normal surface. Now A ′ + τ A ′ is a summand of ˜ A + τ ˜ A = 2 ˜ A .14herefore by Theorem 4.1.36 in [28], no component of the surface A ′ + τ A ′ is a sphere or disc. Hence,it is a union of annuli and M¨obius bands. By Theorem 4.1.36 in [28], these are incompressible andboundary-incompressible. Note that A ′ + τ A ′ is invariant under τ up to normal isotopy. In fact, A ′ + τ A ′ has a normal representative that is actually invariant under τ by Theorem 2 in [16]. Picka component of A ′ + τ A ′ . It is either invariant under τ or disjoint from its image. If this is anannulus, then it is the required surface, because it is a sum of at most 4 fundamental surfaces. Onthe other hand, if this component of A ′ + τ A ′ is a M¨obius band, then its normal sum with itself isthe required annulus.Thus, by searching through normal surfaces of the form k F + . . . + k n F n , where each k i ≤ A that is either invariant under τ or disjoint from itsimage under τ . We now subdivide the triangulation T ′ equivariantly so that A ∪ τ A is a subcomplexof it. We can cut along A ∪ τ A , to form a triangulation of a new Seifert fibred manifold with toralboundary components. Repeating in this way, we eventually decompose our Seifert space into aunion of fibred solid tori. In their boundary are a collection of annuli, which are copies of the lastannulus or annuli that we decomposed along to form the relevant solid torus. The slopes of theseannuli are the slopes of the regular fibres. We can thereby determine, for each of these solid tori,whether they have a singular fibre as a core curve. Our algorithm ends by outputting these solidtori, their core curves and the slopes of the regular fibres on their boundary.
7. Verifying the hypotheses of Theorem 5.1.
We now spend some time verifying that the hypotheses of Theorem 5.1 do hold in our setting.We need to verify that H ( M − int( N ( K )) , ∂M ) = 0. Now it is not hard to check that H ( M − int( N ( K )) , ∂M ) is a subgroup of H ( M, ∂M ), and its rank is either equal to that of H ( M, ∂M ) or one less. By Poincar´e duality, the rank of H ( M, ∂M ) is equal to the first Bettinumber of M , and this is at least the number of toral boundary components. Each component of L − L ′ gives rise to one or two components of ∂M . So, when | L | ≥
4, then | ∂M | ≥ H ( M − int( N ( K )) , ∂M ) = 0.When | L | = 2, we are assuming that the two components of L have zero linking number. So,the crossing disc D that C bounds must intersect a single component of L . For otherwise, thecrossing change would modify the linking number by ±
1, and the result could not be a split link.So, | L ′ | = 1. Moreover, the linking number between L ′ and L − L ′ is zero, and so M has twoboundary components. Once again we deduce that H ( M − int( N ( K )) , ∂M ) = 0.The final case is where | L | = 3. This is a little more delicate. The disc D can intersect atmost two components of L , and so there is a component L that is disjoint from D . Let α be thecrossing arc in D running between the two points of D ∩ L . Let S be a Seifert surface for L . Since α is an arc, we may slide any points of α ∩ S along α and off it. This may introduce new points ofintersection between S and L − L but S remains a Seifert surface for L , and so we may assumethat S is disjoint from α . The inverse image of α in M is the surgery curve K . The inverse image15f S in M is a properly embedded orientable non-separating surface. This represents a non-trivialelement of H ( M − int( N ( K )) , ∂M ), which verifies that this group is non-zero.There are two remaining hypotheses in Theorem 5.1: that M − int( N ( K )) is irreducible andatoroidal. If M − int( N ( K )) is reducible, then the equivariant sphere theorem states that there areone or two disjoint reducing 2-spheres in M − int( N ( K )) that are invariant under the involution.They descend to a 2-sphere in S − int( N ( L − L ′ )) that intersects L ′ in either two or zero points.Hence, by the hypothesis that L is hyperbolic, this sphere S bounds a ball B so that B ∩ L is eitherempty or a trivial 1-string tangle. The sphere or spheres lie in M − int( N ( K )), and so their image S is disjoint from N ( α ). Therefore, α must lie in B , because otherwise each component of the inverseimage of B is a ball in M − int( N ( K )). The tangle replacement occurs within B , and so B ∩ L is replaced by a possibly non-trivial 1-string tangle. However, this cannot make the link L ◦ split,which is a contradiction.Note that here, we used the fact that we are performing a crossing change to L . Thus, thisargument does not immediately extend to other tangle replacements. However, there is anotherargument that works in this more general setting. Suppose that tangle replacement is performedalong α and that this turns the trivial 1-string tangle B ∩ L into a tangle that is split. Then on passingto the branched double cover, we deduce that surgery along on a knot in the 3-ball creates a manifoldcontaining a non-separating sphere. Hence, the distance between the surgery slope and the meridianslope is 1. So, if we assume that the distance is more than 1, then we reach a contradiction. In fact,by using Gabai’s proof of the Property R conjecture [8] and the solution to the Smith conjecture[33], we would be able to classify the possible tangle replacements even in the distance 1 case.Suppose now that M − int( N ( K )) is toroidal. The equivariant torus theorem gives that thereare one or two disjoint essential embedded tori in M − int( N ( K )) that are invariant under theinvolution. They descend either to a sphere in S − int( N ( L − L ′ )) that intersects L ′ in four points,or to a torus disjoint from L . We consider these two cases separately.Suppose that the torus or tori in M − int( N ( K )) project to an embedded torus T in S − int( N ( L )). We are assuming that L is a hyperbolic link, and so T must bound a solid torus inthe complement of L , or must lie within a 3-ball in the complement of L , or must be parallel to acomponent of ∂N ( L ).Suppose that T bounds a solid torus in the complement of L . Since its inverse image in M isdisjoint from K , T is disjoint from α . As α starts and ends on L , it is therefore disjoint from thesolid torus. Hence, the torus was not essential in M − int( N ( K )), which is a contradiction.Suppose that T lies within a 3-ball in the complement of L but does not bound a solid torusin the complement of L . Then T separates S into two components, one of which is disjoint from L and is homeomorphic to the exterior of a non-trivial knot. The other component must be a solidtorus containing L . A meridian disc for this solid torus is disjoint from L , because the torus lies ina 3-ball disjoint from L . Hence, every curve on T has zero linking number with every component of L . Therefore, the inverse image of T in the branched double cover M is two tori. Together these16ound the inverse image V of the solid torus. The surgery curve K lies in V , as otherwise ∂V iscompressible. Note that V − int( N ( K )) is irreducible as otherwise the equivariant sphere theoremimplies that there is an essential sphere in S − int( N ( L ∪ α )), which would imply that L is split,contrary to assumption. Since T is compressible in S − int( N ( L )), V has compressible boundary.Since V has more than one toral boundary component, it is therefore reducible. When surgery on M is performed along K , a reducible manifold M ◦ is created. Let V ◦ be the submanifold of M ◦ thatcomes from V ; so V ◦ is obtained from V by surgery along K . Since M ◦ is reducible, we deduce that ∂V ◦ is reducible or has compressible boundary. Again because V ◦ has more than one toral boundarycomponent, it must be reducible. A theorem of Gordon and Luecke [11] then states that whenan irreducible 3-manifold (in this case, V − int( N ( K ))) with a toral boundary component is Dehnfilled in two different ways to obtain reducible 3-manifolds (in this case, V and V ◦ ), the distancebetween the surgery slopes is one. This contradicts our assumption that the distance of the tanglereplacement is at least two.Finally consider the case where T is parallel to a component of ∂N ( L ). Then T bounds a solidtorus W in S , that intersects L in a single core curve. Since T is the image of a torus disjoint from K , T is disjoint from α . If α is disjoint from the solid torus bounded by T , then the inverse imageof T in M − int( N ( K )) is boundary parallel, which is contrary to hypothesis. So, α lies in the solidtorus bounded by T . Now, T does not admit a compression disc in S − int( W ) that is disjoint from L . This is because T would then be compressible in the complement of L ∪ α , and hence the inverseimage of T in M − int( N ( K )) would be compressible. After the tangle replacement, T continues tobound a solid torus in S , but its intersection with the new link need not be a core curve. However,it still has winding number one, and so T remains incompressible in the complement of the new link L ◦ . Now this link complement contains an essential sphere, because L ◦ is split. After modifying this2-sphere appropriately, we can make it disjoint from the incompressible torus. It then is disjoint fromthe solid torus, and hence is disjoint from the inserted tangle. It therefore corresponds to a 2-spherein the complement of L that separates components of L . Hence, L is split, which contradicts thehypothesis that it is hyperbolic.Thus, we have shown that if M − int( N ( K )) contains an essential torus, then there is one that isinvariant under the involution and this descends to a 2-sphere that intersects L in four points. Sincethis 2-sphere is separating in S , we deduce that the essential invariant torus in M − int( N ( K ))is necessarily separating. We pick an essential invariant torus T that is furthest from K , in thefollowing sense. If T ′ is another essential invariant torus in M − int( N ( K )) that is disjoint from T but not parallel to T , then T ′ lies in the component of M − int( N ( K ∪ T )) containing ∂N ( K ).Let S be the image of T in S . Since L is 2-string prime, this bounds a 3-ball B that contains acompression disc for S − L disjoint from L . Since T was disjoint from K , its image S is disjointfrom α . Therefore, α must lie in B , because otherwise the inverse image T of S is compressible in M − int( N ( K )). Thus, the tangle A = B ∩ L becomes a new 1-manifold A ◦ = B ∩ L ◦ .We claim that ∂B − ∂A ◦ is compressible in the complement of A ◦ . For if it is incompressible,then one may find a splitting sphere for L ◦ that is disjoint from it. This splitting sphere cannot liein B , by Theorem 2.2 (i). Hence, the splitting sphere lies in the complement of B , and therefore17orms a splitting 2-sphere for L , contrary to assumption.Hence, by Theorem 2.2 (ii), A ◦ is a trivial 2-string tangle. So conclusion (i) or (ii) of Theorem2.1 holds. We can view the removal of A and the insertion of A ◦ as tangle replacement along thecore arc β of A . In conclusion (i) of Theorem 2.1, β equals α . But in conclusion (ii), β is differentfrom α . Let K ′ be the inverse image of β in M . Then M ◦ is obtained from M by Dehn surgeryalong K ′ . The distance between the surgery slope and the meridian slope is equal to the distancebetween the slopes of A and A ◦ , and by Theorem 2.1, this is at least d ≥
2. Note that by our choiceof T , M − int( N ( K ′ )) is atoroidal. Hence, we may apply Theorem 5.1 to K ′ instead of K .Thus, we have verified the hypotheses from Theorem 5.1.
8. The mapping class group of a hyperbolic 3-manifold
Any branched double cover comes equipped with an involution. Therefore, in this section, weanalyse the mapping class group of a compact orientable 3-manifold X , by which we mean the groupof homeomorphisms of X , up to isotopy. As we have dealt with the Seifert fibred case in Section 6,the manifolds X that we will consider will be hyperbolic. It is well known that the mapping classgroup of such a manifold is finite and computable. Indeed, we have the following result. Theorem 8.1.
Let X be an orientable finite-volume hyperbolic 3-manifold. Then the mappingclass group of X is finite. Moreover, there is an algorithm that takes, as its input, a triangulation T for X and returns the following:(i) a finite sequence of Pachner moves taking T to a triangulation T ′ ;(ii) a finite group of symmetries of T ′ , which forms a realisation of the mapping class group of X . This result is well known, and this is not the place to explain it in detail. It uses essentially thesame methods as the solution to the homeomorphism problem for compact orientable 3-manifolds.See for example [20] or [43]. A statement of the computability of the mapping class group of X isgiven in Theorem 8.3 of [20] for example, and the proof there gives (i) and (ii) of Theorem 8.1.It is also worth pointing out that, in the situations where we want to apply Theorem 8.1, ∂X is non-empty, and in this case, there is a nice algorithm to solve the problems in Theorem 8.1, asfollows.It was shown by Petronio and Weeks [40] that when ∂X is a non-empty collection of tori, X admits a hyperbolic structure if and only if it has an ideal triangulation that admits a ‘partiallyflat’ solution to the hyperbolic gluing equations. Thus, one first transforms the given triangulationinto an ideal one. Then one applies all possible 2-3 and 3-2 Pachner moves to this, to create a listof ideal triangulations for X . Then, for each triangulation in this list, one applies all possible 2-3and 3-2 Pachner moves, and so on. In this way, an ever-increasing list of ideal triangulations for X is created. It is a theorem of Matveev [27] that any ideal triangulation for X will eventuallyappear in this list. As this is being produced, the algorithm checks whether each ideal triangulation18dmits a partially flat solution to the gluing equations. If it does, this is the required hyperbolicstructure. From this, one can compute the Epstein-Penner decomposition, using the algorithmof Weeks [49]. The Epstein-Penner decomposition is a way of building X out of hyperbolic idealpolyhedra via isometries between their faces. It has the key property that the mapping class groupof X is precisely the group of combinatorial automorphisms of these polyhedra that respect the faceidentifications. One can then easily decompose the ideal polyhedra into a triangulation T ′ satisfyingthe requirements of Theorem 8.1.The computation of the mapping class group given in Theorem 8.1 can be made effective inthe following sense. Theorem 8.2.
Let X be an orientable finite-volume hyperbolic 3-manifold. Let h and h be finiteorder piecewise-linear homeomorphisms of X , given as combinatorial automorphisms of triangula-tions T and T of X , together with a finite sequence of Pachner moves relating T and T . Thenthere is an algorithm to determine whether h and h are equal in the mapping class group of X .Proof. By a theorem of Gabai (Theorem 1.2 in [10]), two homeomorphisms of a finite-volumehyperbolic 3-manifold X are homotopic if and only if they are isotopic. Thus, determining whether h and h are equal in the mapping class group is equivalent to determining whether h h − is homotopicto the identity. This is equivalent to the induced map on π ( X ) being an inner automorphism. Thiscan be determined as follows. Pick a generating set γ , . . . , γ n for π ( X ). The hyperbolic structureon X , which has been determined using [20] or [43], realises γ , . . . , γ n as elements A , . . . , A n ofPSL(2 , C ). The homomorphism induced by h h − sends A , . . . , A n to A ′ , . . . , A ′ n in PSL(2 , C ). Ifsome A ′ i is conjugate to A i , the conjugating element would have to send the fixed-point set for A ′ i inthe sphere at infinity to the fixed-point set of A i . Hence, once we consider a couple of loxodromic A i with disjoint fixed-point sets (which can readily be arranged by a minor adjustment to the generatingset), then there are only finitely many possible elements of PSL(2 , C ) that can conjugate the orderedset A ′ , . . . , A ′ n to A , . . . , A n . If none of these lies in π ( X ), then this can be determined and hence,it can be deduced that the homomorphism induced by h h − is not an inner automorphism. Onthe other hand, if it is an inner automorphism, then an exhaustive search through the possibleconjugating elements of π ( X ) will eventually establish that it is indeed an inner automorphism.We also note that equality in the mapping class group for such homeomorphisms is equivalentto something rather stronger. Theorem 8.3.
Let X be an orientable finite-volume hyperbolic 3-manifold. Then two finite orderpiecewise-linear homeomorphisms h and h of X are isotopic if and only if there is a piecewise-linearhomeomorphism φ of X that is isotopic to the identity and that satisfies h = φ − h φ . Moreover,there is an algorithm to find such a homeomorphism φ when one is given h and h as combinatorialautomorphisms of triangulations T and T of X , together with a finite sequence of Pachner movesrelating T and T .Proof. Let O and O be the orbifolds X/ h h i and X/ h h i . These are orbifold-irreducible andorbifold-atoroidal since X is irreducible and atoroidal. Hence, they admit hyperbolic structures by19he Orbifold Theorem [2, 6]. These lift to hyperbolic metrics g and g on X . By Gabai’s theorem(which is the analogue of the Smale Conjecture for hyperbolic 3-manifolds, Theorem 7.3 [10]), thereis an isotopy between g and g . In other words, there is a 1-parameter family of diffeomorphisms φ t ( t ∈ [0 , φ is the identity and φ ∗ g = g . Let φ = φ . Now, h and h are isometrieswith respect to g and g respectively. Hence, φ − h φ is isometry with respect to φ ∗ g = g . It ishomotopic to the isometry h of g . By Mostow rigidity, two homotopic isometries are equal, andhence φ − h φ = h , as required.Suppose now that we are given isotopic, finite order, piecewise-linear homeomorphisms h and h of X , given as combinatorial automorphisms of triangulations T and T of X , together with afinite sequence of Pachner moves relating T and T . We then know that φ , satisfying the aboveconditions, exists. We must give an algorithm to find it. Note that φ descends to a homeomorphism φ : O → O that respects the singular locus. This satisfies ( φ ) ∗ ( p ) ∗ π ( X ) = ( p ) ∗ π ( X ) where p : X → O and p : X → O are the quotient maps and ( p i ) ∗ : π ( X ) → π ( O i ) are the inducedhomomorphisms at the level of orbifold fundamental groups. Conversely, given a homeomorphism φ between O and O respecting the singular locus and satisfying ( φ ) ∗ ( p ) ∗ π ( X ) = ( p ) ∗ π ( X ), wemay lift it to a homeomorphism φ : X → X such that φ − h φ = h k for some k ∈ Z . Moreover, when φ is isotopic to the identity, then we may take k = 1. When h and h are given as combinatorialautomorphisms of triangulations T and T of X , we may subdivide these to triangulations T ′ (respectively, T ′ ) with the property that a simplex is invariant under h (respectively h ) if andonly if it is in the fixed-point set of h (respectively h ). These descend to triangulations of O and O , and we may then start to search for homeomorphisms φ between them, by exhaustivelytrying sequences of Pachner moves. We will eventually find such a homeomorphism that lifts to ahomeomorphism φ : X → X that is isotopic to the identity and that satisfies φ − h φ = h .
9. The algorithm to detect links with splitting number one
In this section, we provide the algorithm required by Theorems 1.1 and 1.2.We have already seen that when L admits a tangle replacement along an arc α that createsa split link, then there is an associated exceptional surgery curve K in the manifold M . Whenthe distance of the tangle replacement is at least 2, this satisfies the hypotheses of Theorem 5.1,and so this theorem provides a list of all possibilities for K up to isotopy of M . But it may notbe clear whether a knot K ′ provided by Theorem 5.1 is isotopic to a curve that is invariant underthe involution on M . Even if it is, it is not clear whether it has several different representativesin its isotopy class in M , each of which is invariant under the involution, but which descend tonon-isotopic arcs in the exterior of L . Thus, it is not immediately clear how to create a finite list ofall possibilities for the arc α . To circumvent this problem, we argue as follows.The construction of M as a branched double cover provides a piecewise-linear involution τ of M that restricts to a piecewise-linear involution of M − int( N ( K )). Hence, if K ′ is isotopic to K ,then M − int( N ( K ′ )) also admits a piecewise-linear involution. So, for each of the knots K ′ providedby Theorem 5.1, we check whether M − int( N ( K ′ )) admits a piecewise-linear involution. The main20ase that we will consider is where M and M − int( N ( K )) are hyperbolic. In this situation, theirmapping class groups are finite and computable using Theorem 8.1. So, we can decide whether M − int( N ( K ′ )) admits a piecewise-linear involution, and we can find an explicit representative foreach such involution. We are only interested in involutions that extend to an involution of M thatis isotopic to τ . The following lemma asserts that, if we know the involution of M − int( N ( K ′ )) andif we know that it extends to the involution τ on M (up to isotopy), then this is enough to be ableto recreate the image of K in the orbifold M/τ . Recall that
M/τ is the orbifold with underlyingmanifold S − int( N ( L − L ′ )) and with singular set equal to L ′ . The image of K in M/τ is therequired arc α . Lemma 9.1.
Let τ be a piecewise-linear involution of a hyperbolic 3-manifold M that leaves aknot K invariant, where M − int( N ( K )) is hyperbolic. Let ρ be a piecewise-linear homeomorphismof M that is isotopic to the identity, taking K to a knot K ′ . Let η be another piecewise-linearinvolution of M that leaves K ′ invariant. Suppose that the restrictions of ρτ ρ − and η are isotopichomeomorphisms of M − int( N ( K ′ )) . Suppose also that there is a piecewise-linear homeomorphism φ of M that is isotopic to the identity and that satisfies η = φ − τ φ . Then there is a piecewise-linear homeomorphism M/τ → M/τ respecting the singular locus of this orbifold and taking
K/τ to φ ( K ′ ) /τ . Moreover, this homeomorphism is isotopic to the identity on the components of ∂M/τ that are disjoint from the singular set.Proof. Since ρτ ρ − and η are isotopic piecewise-linear homeomorphisms of M − int( N ( K ′ )), thenby Theorem 8.3, there is a piecewise-linear homeomorphism ψ of M − int( N ( K ′ )), isotopic to theidentity, such that ρτ ρ − = ψ − ηψ . This extends to a piecewise-linear homeomorphism ψ of M such that ψ ( K ′ ) = K ′ . Thus, we have the following commutative diagram:( M, K ) ρ −→ ( M, K ′ ) ψ −→ ( M, K ′ ) φ −→ ( M, φ ( K ′ )) y τ y ρτρ − y η y τ ( M, K ) ρ −→ ( M, K ′ ) ψ −→ ( M, K ′ ) φ −→ ( M, φ ( K ′ ))Hence, the composition φψρ descends to a homeomorphism M/τ → M/τ taking singular locus tosingular locus and taking
K/τ to φ ( K ′ ) /τ . Since φψρ is isotopic to the identity, its action on eachcomponent of ∂M/τ that is disjoint from the singular set is isotopic to the identity.Thus, the algorithm required by Theorems 1.1 and 1.2 is as follows:1. If L is provided by a diagram, then we use this to build a triangulation of S in which L issimplicial.2. Pick all sublinks L ′ of L consisting of one or two components. If | L | = 2, then we require that | L ′ | = 1. There are only finitely many choices for L ′ and so let us focus on just one such choice.3. Construct a triangulation T of the double cover of S − int( N ( L − L ′ )) branched over L ′ . Denotethis manifold by M . Note that M has non-empty boundary.21. By Proposition 4.1, M is either hyperbolic or a small Seifert fibre space. Using Algorithm 8.1in [17], determine which of these cases holds. The algorithm divides into these two cases. The hyperbolic case
1. Use Theorem 5.1 to produce a finite list of knots K in M , with slopes σ on ∂N ( K ) that satisfythe hypotheses of Theorem 5.1. Each knot K can be given as a subcomplex of some iteratedbarycentric subdivision of T . Let us now focus on just one choice of K and σ .2. From the way that K is given, it is easy to build a triangulation T ′ for M − int( N ( K )). Forexample, one can take two further barycentric subdivisions of the triangulation of M , and thenremove the simplices that are incident to K .3. Use Theorem 4.1.12 in [28] or Theorem 5.2 in [17] to determine whether M − int( N ( K )) isirreducible and use Theorem 6.4.10 in [28] or Algorithms 8.1 and 8.2 in [17] to determinewhether it is atoroidal. If it is reducible or toroidal, then discard it and move on to the nextchoice of K and σ . So let us assume that M − int( N ( K )) is irreducible and atoroidal. It istherefore hyperbolic or Seifert fibred, by the solution to the Geometrisation Conjecture. Infact, it cannot be Seifert fibred, because it can be Dehn filled to form the hyperbolic manifold M .4. Compute the mapping class group of M − int( N ( K )) using Theorem 8.1. This gives a triangu-lation T ′′ of M − int( N ( K )) and a group of symmetries of T ′′ that realises the mapping classgroup. It also provides a sequence of Pachner moves from T ′ to T ′′ .5. For each order 2 symmetry η of M − int( N ( K )), determine whether it preserves ∂N ( K ) anddetermine whether it acts by − id on it. If this is not the case, ignore it and move on.6. If η does act in this way on ∂N ( K ), then it extends to an order two symmetry of M , which wewill also call η . Extend T ′′ to a triangulation T ′′′ of M that is invariant under η and build asequence of Pachner moves from T to T ′′′ .7. Using Theorem 8.2, determine whether η is isotopic to the involution τ of M that is given bythe construction of M as a branched double cover. If it is not, then discard it.8. Assuming that η is isotopic to τ , Theorem 8.3 provides a piecewise-linear homeomorphism φ of M isotopic to the identity such that η = φ − τ φ . The arc φ ( K ) is invariant under τ and itsimage in S − int( N ( L − L ′ )) is an arc β with endpoints on L ′ .9. Lemma 9.1 only provides the arc β up to homeomorphism of S − int( N ( L )), whereas we wantall possibilities for β up to isotopy. So, for each arc β constructed as above, consider all itsimages under the mapping class group of S − int( N ( L )), which is determined using Theorem8.1.10. Feed all these arcs β into the subroutine below that constructs arcs α from them.22 he Seifert fibred case
1. Use Theorem 6.6 to produce a Seifert fibration of M that is invariant under the involution.More specifically, this produces a union of disjoint simple closed curves K in M that are thesingular fibres. Also, for each such simple closed curve, it produces the slope σ on ∂N ( K ) thatis slope of the regular fibres.2. For each possibility for K , observe whether it is invariant under the involution. If it is not,discard it. If it is, its image in S − int( N ( L − L ′ )) is an arc β . Feed β into the subroutinebelow that constructs arcs α from it. Constructing the arcs α
1. Using β , we determine one or two possibilities for α , corresponding to conclusions (i) and (ii)of Theorem 2.1. In case (i), we set α to be equal to β . We check that the distance of the tanglereplacement is two. If it is not, we discard this possibility. In case (ii), we parametrise theslopes on ∂N ( K ) by Q ∪ {∞} where ∞ is the meridional slope giving M . We let the slope of σ be p/q . If q is of the form 2 a , then we write p/q as (1 ± ab ) / ± a , and then set α as in(ii) of Theorem 2.1.2. If we are determining whether s d ( L ) = 1 or ts d ( L ) = 1, then we only consider arcs α that haveendpoints on distinct components of L . So under these circumstances, if the endpoints of α lieon the same component of L , then we discard it.3. For each of these possibilities for α , we perform this tangle replacement. The image of L is alink L ◦ . Determine whether L ◦ is the unlink, a split link or totally split, as appropriate. Forexample, one can use Theorem 5.2 in [17] to determine whether S − int( N ( L ◦ )) is reducibleand if it is, to find a reducing sphere. Then one can decompose along it, fill in with 3-balls andrepeat. In this way, we can determine whether L ◦ is split and whether it is totally split. In thelatter case, we can also determine whether the components are unknots, and so whether L ◦ isthe unlink.4. For any relevant crossing arc α , we can then construct the associated crossing circles and ± α equal to β , but we do not check that the distance of thetangle replacement is two. This was done solely to ensure that the tangle replacement correspondedto a crossing change. In case (ii) of Theorem 2.1, we consider all possible ways of writing p/q as(1 ± dab ) / ± da , where a and b are integers and d is an integer at least two. For each such possibility,we get an arc α as in (ii) of Theorem 2.1. Thus, checking each of these tangle replacements in turn,we obtain the following result. Theorem 9.2.
Let L be a link in S with at least two components. If L has exactly two components,suppose that these have zero linking number. Suppose that L is hyperbolic and 2-string prime. Then here is an algorithm to find all possible trivial tangle replacements that can be made to L withdistance at least two that turn it into a split link. Remark 9.3.
The algorithm given above required us to consider all sublinks L ′ of L with one or twocomponents. However, in the case where | L | ≥
3, we can focus just on the case where | L ′ | = 2. Thereason for this is as follows. The manifold M is the double cover of S − int( N ( L − L ′ )) branchedover L ′ . It was important for the endpoints of α to lie in L ′ , so that the inverse image of α in M is a knot K . But if α has endpoints in the same component of L , then we can choose L ′ to be theunion of this component plus one other chosen arbitrarily. The conditions on M − int( N ( K )) areall easily verified, as in Section 7. In particular, the condition H ( M − int( N ( K )) , ∂M ) = 0 holds.
10. The algorithm to enumerate knots with exceptional surgeries
In the previous section, an algorithm that solves the decision problems in Theorem 1.1 was given.The crucial ingredient was Theorem 5.1, which provided an algorithm to enumerate the exceptionalsurgery curves within a 3-manifold satisfying certain conditions. This algorithm is difficult to workwith in practice, and so in many concrete examples, it is better to use the techniques behind Theorem5.1. Therefore in this section, we give an overview of the proof of Theorem 5.1.The proof relied heavily on sutured manifold theory. An excellent reference for this is [41]. Thefirst part of the argument closely follows Section 5 of [41].Suppose that M is a compact orientable 3-manifold with ∂M a (possibly empty) union of tori.Let K be a knot in M and let σ be a slope on ∂N ( K ) satisfying (i) and (ii) of Theorem 5.1. Give M − int( N ( K )) the structure of a sutured manifold with R − = ∅ and R + = ∂M ∪ ∂N ( K ) andtherefore with sutures γ = ∅ . We may find a taut sutured manifold hierarchy( M − int( N ( K )) , γ ) S −→ ( M − int( N ( K )) , γ ) S −→ . . . S n − −→ ( M n − int( N ( K )) , γ n )such that the following hold:(i) Each surface S i is disjoint from ∂N ( K ).(ii) Each surface S i contains no closed separating components.(iii) No surface S i has a boundary component that bounds a disc in ∂M i disjoint from γ i .(iv) H ( M n − int( N ( K )) , ∂M n ) = 0.(v) In the case where σ is norm-exceptional there is some z ∈ H ( M − int( N ( K )) , ∂M ) that mapsto an element z σ ∈ H ( M K ( σ ) , ∂M K ( σ )), such that the Thurston norm of z σ is less than theThurston norm of z . We require that [ S ] = z .Condition (iv) implies that ∂M n consists of a collection of spheres and at most one torus. Bythe tautness of ( M n − int( N ( K )) , γ n ), each sphere bounds a ball in M n . Hence, there is exactlyone component Y of M n − int( N ( K )) that is not a ball. This forms a rational homology cobordismbetween a toral component T of ∂M n and ∂N ( K ). This torus T is incompressible in M − int( N ( K )).24his is because a compressible torus in an irreducible 3-manifold bounds a solid torus or lies withina 3-ball. If T bounds a solid torus in M − int( N ( K )), then some surface S i must intersect this solidtorus, and this gives rise to a closed separating component of S i , contradicting (ii). If T lies withina 3-ball in M − int( N ( K )), then again some S i must intersect this 3-ball and again this gives aclosed separating component. Now we are assuming that M − int( N ( K )) is atoroidal, and so theincompressible torus T is boundary parallel in M − int( N ( K )). It cannot be parallel to componentof ∂M , because Y would then be the region between T and ∂N ( K ) and hence would be a copyof M − int( N ( K )). However, M − int( N ( K )) is not a rational homology cobordism between ∂M and ∂N ( K ), since H ( M − int( N ( K )) , ∂M ) is non-trivial. We therefore deduce that T is parallel to ∂N ( K ), and hence Y is a copy of T × [0 ,
1] with T × { } = T and T × { } = ∂N ( K ).Consider any slope ρ on ∂N ( K ) other than the one that is parallel to the sutures γ n ∩ Y . Ifwe Dehn fill each of the manifolds M i − int( N ( K )) along this slope, we obtain a sutured manifoldhierarchy ( M ( ρ ) , γ ) S −→ ( M ( ρ ) , γ ) S −→ . . . S n − −→ ( M n ( ρ ) , γ n ) . Here, M i ( ρ ) denotes the result of performing Dehn surgery along K in M i with slope ρ . Since weassumed that ρ is not parallel to the sutures γ n ∩ Y , the sutured solid torus ( M n ( ρ ) , γ n ) is taut. Usingthe theorem that tautness pulls back (Theorem 3.6 of [41]), we deduce that each of the manifoldsin the hierarchy is taut and each of the decomposing surfaces is taut. Because σ is exceptional ornorm-exceptional, ( M ( σ ) , γ ) is not taut or S is not taut in M ( σ ). Hence, σ must be the slope on ∂N ( K ) parallel to the sutures Y ∩ γ n . So, if we re-attach the solid torus N ( K ) using the meridionalslope, we deduce that ( M, γ ) S −→ ( M , γ ) S −→ . . . S n − −→ ( M n , γ n )is taut.The key part of the proof of Theorem 5.1 is to place this hierarchy into some sort of ‘normalform’ with respect to a given triangulation T of M . In fact, we first dualise T to form a handlestructure H for M , and we make the hierarchy ‘normal’ with respect to H . We will shortly make thisstatement a little more precise, but the idea is roughly that there should be only finitely possibilitiesfor M i ∩ H and γ i ∩ H , for each handle H of H . In fact, this statement is not quite correct, butwe will make it accurate shortly. But if there were only finitely many possibilities for M n ∩ H and γ n ∩ H , and we could enumerate them, then we could reconstruct the way that ( M n , γ n ) lies within M . Since M n is a regular neighbourhood of N ( K ), this would imply that there are only finitelymany possibilities for K and σ , and we could enumerate them all.To make the above discussion more precise, it is helpful to consider just the first decompositionalong S . Since S is taut in M , it is incompressible. Suppose also that S is also boundary-incompressible. Therefore, we may place S into normal form with respect to the handle structure H . (A notion of normal surfaces in a handle structure was first defined by Haken [12]; see also [15]and [28]. A variant of this notion was in fact used in [22], partly to take account of the possibilitythat S may be boundary-compressible.) Then when we cut along S , we obtain a handle structure H for M . It would be convenient if, within each handle H of H , there are only finitely many25ossibilities for M ∩ H . However, this need not be the case. The surface S may have manycomponents of intersection with H , and thereby give rise to many handles of H within H . To getaround this problem, we use the notion of the parallelity bundle of H . By definition, a handle of H is a parallelity handle if it lies between two normally parallel discs of S . This is an I -bundle,with ∂I -bundle lying in the copies of S in ∂M . The union of the parallelity handles is an I -bundle B over a surface F called the parallelity bundle for H . It is clear that, within each handle H of H ,there are only finitely many possibilities for H ∩ ( M −B ) and these are algorithmically constructible.This list is universal, in the sense that it does not depend on M or any other data. It only dependson the way that H intersects the neighbouring handles of higher index, and there are only finitelymany possibilities for this because H is dual to a triangulation.One of the key parts of the proof of Theorem 5.1 is therefore to remove the parallelity bundle B . The procedure is given in detail in Section 8 of [22]. It involves making changes to the handlestructure H . We now explain the most important of these changes now. The horizontal boundary ∂ h B is the ( ∂I )-bundle and lies in the copies of S . The vertical boundary ∂ v B is the I -bundle over ∂F . The surface P = cl( ∂ v B − ∂M ) is also an I -bundle, and hence it is a collection of discs andannuli. It is a properly embedded surface in M . A key modification that is made is to decompose M along some of the components of P . Each such component is either an annulus disjoint from thesutures γ or a product disc , which is a disc intersecting γ twice. This surface is properly embeddedin M , but there is no a priori reason why it should be disjoint from K . This is a consequence ofthe following result, which was the central result of [21]. It was here that the hypothesis that thedistance ∆( µ, σ ) > Theorem 10.1.
Let ( M, γ ) be a taut sutured manifold. Let K be a knot in M such that M − int( N ( K )) is irreducible and atoroidal. Let σ be a slope on ∂N ( K ) such that ∆( σ, µ ) > , where µ is the meridional slope. Let M K ( σ ) be the result of performing Dehn surgery along K with slope σ .Suppose that ( M K ( σ ) , γ ) is not taut. Let G be a surface properly embedded in M with components G , . . . , G | G | , none of which is a sphere or disc disjoint from γ . Then there is an ambient isotopy of K in M after which, for each integer i between and | G | , we have | K ∩ G i | ≤ − χ ( G i ) + | G i ∩ γ | µ, σ ) − . Note that when G is a union of annuli disjoint from γ and product discs, then Theorem 10.1implies that the knot K may be ambient isotoped off F . The condition that M − int( N ( K )) isatoroidal in fact can be weakened somewhat (see Theorem 1.4 in [21]).Thus, the proof of Theorem 5.1 proceeds as follows. We start with a taut decomposition( M − int( N ( K )) , γ ) S −→ ( M − int( N ( K )) , γ )satisfying (i)-(v) above. As argued above, when we attach the solid torus using the meridional Dehnfilling, we get a taut decomposition ( M, γ ) S −→ ( M , γ ) . S can be placed in a position rather similar to normal form with respect to H (specifically, it satisfies Conditions 1-5 of Section 9 in [22]). In fact, one may need to modify S toplace it in this form; we will discuss this below. Let B be the parallelity bundle in M associatedwith S . As discussed above, the surgery curve K can be isotoped off its vertical boundary, andhence off B altogether. We can then apply the procedure given in Section 8 of [22] to modify B , togive a new 3-manifold ( M ′ , γ ′ ). This modification has the effect of removing all components of B that are not I -bundles over discs and all components for which the interior of the vertical boundaryintersects ∂M . The remaining components of B , which are therefore I -bundles over discs, become2-handles of ( M ′ , γ ′ ). We have little control over the location of these 2-handles, but we do havecontrol over their attaching locus onto the 0-handles and 1-handles. This manifold ( M ′ , γ ′ ) contains K , and it has the property that when surgery along K is performed, the resulting sutured manifoldis not taut. The advantage of working with this new manifold M ′ is that, for each handle H of H , there are only finitely many possibilities for the intersection between H and the 0-handles and1-handles of M ′ , the attaching locus of the 2-handles and the sutures γ ′ .As mentioned above, before S satisfies Conditions 1-5 of Section 9 in [22], it may be necessaryto make some modifications to it. These are given in Section 9 of [22]. At each stage, it is ensuredthat K remains disjoint from S . For example, S may be boundary-compressed along a productdisc. Using Theorem 10.1, we can ensure that K avoids this product disc and so remains disjointfrom the new surface.One can then repeat this procedure. We find a taut decomposition( M ′ − int( N ( K )) , γ ′ ) S −→ ( M − int( N ( K )) , γ )satisying (i)-(iv) above. Since surgery along K with slope σ gives a sutured manifold that is nottaut, when we Dehn fill along the meridional slope, we get a taut decomposition( M ′ , γ ′ ) S −→ ( M , γ ) . Now isotope K off the parallelity bundle in ( M , γ ), and then remove this bundle to get a newsutured manifold ( M ′ , γ ′ ) containing K .We end with a manifold ( M ′ n , γ ′ n ) containing K such that H ( M ′ n − int( N ( K )) , ∂M ′ n ) is trivial.Hence, as argued above, M ′ n is some 3-balls plus a solid torus with K as its core curve, and theexceptional slope σ on ∂N ( K ) is parallel to the sutures on ∂M ′ n . For each handle H of H , itsintersection with the 0-handles of M ′ n , the 1-handles of M ′ n , the attaching locus of the 2-handles andthe sutures γ ′ n takes one of only finitely many possibilities. The algorithm proceeds by inserting allsuch possibilities into each handle of H so that they patch together correctly along adjacent handles.In this way, we can build all possibilies for the sutured manifold ( M ′ n , γ ′ n ). The algorithm checks,for each possible ( M ′ n , γ ′ n ) whether it is a taut solid torus plus possibly some taut 3-balls. If it is,the core curve of the solid torus is a possibility for K and the slope of the sutures on the solid toralcomponent of ( M ′ n , γ ′ n ) is a possibility for the slope σ . Once one has this list, one can determineeasily whether M and K really do satisfy (i) and (ii) of Theorem 5.1.27e summarise the above discussion in the following theorem. Theorem 10.2.
There is a finite computable list of 4-tuples ( H i , F i , F i , γ i ) where(i) each H i is a collection of balls embedded within a tetrahedron ∆ ;(ii) each F i is the intersection between H i and ∂ ∆ ; it a collection of discs lying in the interior ofthe faces of ∆ ;(iii) each F i is a collection of disjoint rectangles lying in ∂H i ; two opposite sides of each rectanglelie in ∂ F and the remainder of the rectangle is disjoint from F ;(iv) each γ i is a collection of disjoint arcs properly embedded in cl( ∂H i − ( F i ∪ F i )) .These have the following property. Suppose that M is a compact orientable 3-manifold with bound-ary a (possibly empty) union of tori, and that K is knot in M with an exceptional or norm-exceptionalslope σ on ∂N ( K ) , satisfying the hypotheses of Theorem 5.1. Then for any triangulation of M , onemay form a handle structure on N ( K ) as follows. Its 0-handles are obtained by inserting some H i into each tetrahedron of the triangulation. The 1-handles are dual to the discs F i . The rectangles F i patch up to form annuli, which are the attaching locus of the 2-handles. The arcs γ i patchtogether to form curves with slope σ on ∂N ( K ) . The finite list of 4-tuples in the above theorem is universal, in the sense that it does not dependon M or K . An algorithm to construct this list is given in Section 11 of [22].Thus, these 4-tuples patch together to form a handle structure on N ( K ). If one wanted to, onecould then realise K in M by picking a curve on ∂N ( K ) with winding number 1 in N ( K ). This couldthen be realised as a simplicial curve in some iterated barycentri subdivision of the triangulation of M . The above techniques also provide a proof of Theorem 6.1, which gives that certain exceptionalsurgery curves in a Seifert fibred space must be isotopic to an exceptional fibre. Proof of Theorem 6.1.
Let M be a Seifert fibre space with non-empty boundary. Let K be a knot in M and let σ be a slope on ∂N ( K ) satisfying the hypotheses Theorem 6.1. Give M − int( N ( K )) thestructure of a sutured manifold with R − = ∅ and R + = ∂M ∪ ∂N ( K ) and therefore with sutures γ = ∅ .We are assuming that M has non-empty boundary. Therefore, there is a (possibly empty)union of disjoint properly embedded arcs in its base space that avoid the exceptional points andthat decompose the base space either into a collection of regular neighbourhoods of the exceptionalpoints or, in the case where there are no singular points, into a single disc. The inverse image ofthese arcs is a union of disjoint properly embedded annuli A in M , such that M − int( N ( A )) iseither a regular neighbourhood of the singular fibres or, in the case where M has no singular fibres,a fibred solid torus. By Theorem 10.1, there is an ambient isotopy taking K off A . Hence, K lies in M − int( N ( A )). By the irreducibility and atoroidality of M − int( N ( K )), K must be a core curveof one of the components of M − int( N ( A )). Hence, K is isotopic to a fibre of M and therefore28 − int( N ( K )) is Seifert fibred. The exceptional slope σ on ∂N ( K ) must be the slope of the regularfibres. Since we are assuming that the distance between K and the meridian is more than 1, wededuce that K must be an exceptional fibre of M .
11. Finiteness of the number of splitting crossing changes
In this section, we prove Theorem 1.2.
Proof.
The algorithmic part of Theorem 1.2 was dealt with in Section 9. So we need only establishthe required upper bound on the number of splitting crossing changes that can be applied to thegiven link L .We are given a triangulation T of S with t tetrahedra in which L is simplicial. Note that ifwe are given, alternatively, a diagram of L with c crossings, then we can easily construct such atriangulation where t ≤ c . One way of doing this is as follows.First apply type 1 Reidemeister moves to remove any edges in the diagram that start and endat the same crossing. Then place an octahedron at each crossing of the diagram. The over-arc andthe under-arc at the crossing will be subcomplexes of these octahedra. Lying above the plane of thediagram and all these octahedra is a 3-ball. Its boundary has a cell structure. Any 2-cell of thiscell structure that is not already a triangle may be subdivided into triangles. We then triangulatethe ball by placing a vertex in its interior and coning off. We triangulate each octahedron using 4tetrahedra. The 3-ball lying below the plane of the diagram and the octahedra is triangulated alsoby coning off its boundary. The result is a triangulation of the 3-sphere with L has a subcomplex.It is easy to check that at most 24 c tetrahedra have been used.We apply the algorithm given in Section 9 to this triangulation, but skipping some steps thatare not relevant to the counting argument. Step 1 has already been completed.In Step 2, one considers all sublinks L ′ of L consisting of one or two components. If | L | = 2,then we require that | L ′ | = 1. The number of such sublinks is at most | L | ( | L | + 1) /
2. The numberof components of L is at most the number of 1-simplices of T , which is at most 6 t . So, the numberof relevant sublinks is at most a quadratic function of t .In Step 3, the double cover M of S − int( N ( L − L ′ )) branched over L ′ is constructed. It isstraightforward to build a triangulation of M , starting from the triangulation of S with L as asubcomplex. The number t ′ of tetrahedra in this triangulation can easily be arranged to be at mosta linear function of t .The algorithm now divides into the cases where M is hyperbolic or Seifert fibred. We considerthe hyperbolic case first.The construction of the knots K in M provided by Theorem 5.1 produces at most ( k ) t ′ possibilities for K and σ , where k is a universal computable constant. Specifically, suppose thatTheorem 10.2 provides a list of k N ( K ) is obtained by inserting29he 0-handles in one of the 4-tuples into each tetrahedron of the triangulation, in such a way theypatch together correctly along the faces. Moreover, the arcs in the 4-tuples patch together to forma representative for σ . Thus, there are at most ( k ) t ′ possibilities for N ( K ) and σ .For each possible K , one can build a triangulation T ′ for M − int( N ( K )). This is Step 2 ofthe hyperbolic case. Rather than using iterated barycentric subdivisions, we do this as follows. Wesubdivide the 0-handles from Theorem 10.2 into tetrahedra and then extend this triangulation overthe 1-handles and 2-handles of N ( K ). Thus, the number of tetrahedra is at most k t for a universalcomputable k .In Step 3 of the hyperbolic case, M − int( N ( K ) is discarded if it is not irreducible and atoroidal.In Step 4 of the hyperbolic case, the symmetry group of M − int( N ( K )) is computed. The sizeof the symmetry group for a hyperbolic 3-manifold X is at most a linear function k vol( X ) for thefollowing reason. The quotient of X by its symmetry group is a finite-volume hyperbolic orbifoldand there is a universal lower bound v on the volume of such an orbifold [18]. Thus, the order ofthe symmetry group of X is at most vol( X ) /v . Setting k = 1 /v establishes the claim.Note that k vol( M − int( N ( K ))) ≤ k v k t , where v is the volume of a regular ideal hyperbolic3-simplex. This follows from the general result [46] that the volume of hyperbolic 3-manifold with(possibly empty) toroidal boundary is at most v times the number of tetrahedra in any triangulationof the manifold. Thus, the symmetry group of M − int( N ( K )) has order at most k v k t . Eachorder two symmetry produces at most one possibility for the arc β .In Step 9 of the hyperbolic case, we consider all the images of these arcs β under the mappingclass group of S − int( N ( L )). This mapping class group has order at most k vol( S − int( N ( L ))) ≤ k v t .We now consider the case where M is Seifert fibred. By Theorem 6.1, there is a Seifert fibrationof M in which K is a singular fibre and by Addendum 6.2, this Seifert fibration can be chosen to beinvariant under τ . Since M is atoroidal and has non-empty boundary, it has at most two singularfibres. Thus, in the Seifert fibred case, there are at most two possibilities for the image β of K .Thus, in both the case where M is hyperbolic and where it is Seifert fibred, we have a boundon the number of possibilities for the arc β and the associated tangle replacement slope. For each β and associated slope, there are at most two possible arcs α . For each tangle replacement along α ,there are two associated crossing circles, by Lemma 3.1. The number of possible crossing circles istherefore at most 12 t (6 t + 1) k t ′ k v k t which is at most k t for some universal computable constant k .
12. The Whitehead link
In this section, we examine an example, the Whitehead link. We determine the complete setof crossing changes that turn the link into a split link.30 heorem 12.1.
Any crossing change that turns the Whitehead link into a split link is equivalentto changing one of the specified crossings in Figure 8. In particular, there are crossing circles upto equivalence, and crossings arcs up to equivalence, that yield splitting crossing changes. Figure 8The diagram shown in Figure 8 is rather undistinguished. The Whitehead link has an alter-nating diagram, with fewer crossings, shown in Figure 9. The crossing arcs associated with thetwo crossing changes are shown in Figure 9. One of these is isotopic to a vertical arc at one ofthe crossings in Figure 9. The other one can be made vertical, if one first performs a flype on thediagram, taking it to another alternating diagram. Thus, we obtain the following corollary.
Theorem 1.3
Any crossing change that turns the Whitehead link into a split link is equivalent tochanging some crossing in some alternating diagram.
Figure 9Associated to each of the two crossing changes in Figure 8, there are two crossing circles. Twoare of these are isotopic to each other. Thus, we get at most three inequivalent crossing circles intotal. In fact, these are readily seen to be inequivalent, for example, by examining their linkingnumber with the components of the Whitehead link.
Proof of Theorem 12.1.
We follow the procedure given in Section 10. Note first that the Whiteheadlink is hyperbolic and 2-string prime. The latter fact can be proved by observing that it is a 2-bridgelink and hence its double branched cover is a lens space, and then using Theorem 4.1.We consider all sublinks L ′ consisting of just one component. Since there is an ambient isotopythat swaps the two components of the Whitehead link L , we may fix L ′ to be one specific component.The double cover of S − int( N ( L − L ′ )) branched over L ′ is shown in Figure 10. It is the exterior M of the (4 ,
2) torus link. 31 ´ DBC over L´ Figure 10We wish to produce a finite list of knots K in M with slopes σ that satisfy the hypotheses ofTheorem 5.1. Note that the arguments in Section 7 apply and so we may assume that M − int( N ( K ))is irreducible and atoroidal.Since M is Seifert fibred, we could use Theorem 6.1. Instead, we consider the method discussedin Section 10. The relevant knots K and slopes σ arise via taut sutured manifold decompositions( M, ∅ ) S −→ . . . S n − −→ ( M n , γ n ) , where ( M n , γ n ) is a solid torus regular neighbourhood of K with sutures of slope σ , plus possiblysome taut 3-balls. We may take the homology class of the surface S to be any non-trivial class in H ( M, ∂M ) that has zero intersection number with K . This is because such classes in H ( M, ∂M )are precisely those in the image of the non-trivial classes in H ( M − int( N ( K )) , ∂M ). We will showthat we can in fact take S to be the annulus A shown in Figure 11. A Figure 11Note first that, by the Theorem 10.1, there is an isotopy of K taking it off the annulus A . Hence, K does have zero algebraic intersection number with A . So we may take S to be homologous to A .Furthermore, S is incompressible, since it is the first surface in a taut sutured manifold hierarchy.We will in fact show that any connected orientable incompressible surface properly embedded in M that is homologous to A is isotopic to A .The manifold M is Seifert fibred with base space an annulus and with a single exceptional fibre32f order 2. Any essential properly embedded surface in M is isotopic to one that is horizontal (thatis, it is transverse to the fibres) or vertical (that is, it is a union of fibres). The annulus A is vertical.Horizontal and vertical surfaces are not homologous, as can be seen for example by considering theiralgebraic intersection number with a regular fibre. Thus, any incompressible surface homologous to A is also vertical. But, because the base space of M is an annulus with a single exceptional fibre,the unique connected orientable vertical surface that is homologically non-trivial is isotopic to A .Thus, we may assume that S is A . The second manifold ( M , γ ) is therefore a solid torus. Itsboundary T lies in M − int( N ( K )). Since M − int( N ( K )) is irreducible and atoroidal, T is boundaryparallel in M − int( N ( K )). It is not parallel to ∂M , and hence it is parallel to ∂N ( K ). Hence, M − int( N ( K )) is a copy of T × I . Therefore, H ( M − int( N ( K )) , ∂M ) is trivial, and therefore M is the final manifold in the hierarchy. Hence, the only possibility, up to ambient isotopy, for K is as shown in Figure 12. K Figure 12Note that K , as shown in Figure 12 is invariant under the involution of M . The quotient arc β is shown in Figure 13. β Figure 13The final subroutine of the algorithm in Section 9 now produces two possibilities for the arc α .One of these is β . To produce the other one, we note that the slope of γ n is 1 /
2. Applying Theorem33.1, we see that conclusion (ii) there again gives α = β .Finally observe that tangle replacement along β does change L into a split link. Thus, wededuce that this is the only possibility for the arc α with endpoints in L ′ . An isotopy takes L ∪ β to the link and one of the crossing arcs shown in Figure 9.Theorem 1.3 provides some evidence for the following conjecture. Conjecture 12.2.
Any crossing change that turns an alternating link into a split link is equivalentto changing some crossing in some alternating diagram.
It is a theorem of McCoy [29] that an alternating knot has unknotting number one if and only ifone can change a crossing in some alternating diagram of the knot and obtain the unknot. However,this does not imply, of course, that every crossing change that turns an alternating knot into theunknot is equivalent to changing some crossing in some alternating diagram. Indeed it seems unlikelythat the methods developed by McCoy, which use Heegaard Floer homology, would lead to a proofof Conjecture 12.2. But McCoy’s theorem does lend weight to the conjecture.
References K. Baker, D. Buck,
The classification of rational subtangle replacements between rationaltangles,
Algebraic and Geometric Topology 13 (2013)1413–1463.2.
M. Boileau, B. Leeb, J. Porti,
Geometrization of 3-dimensional orbifolds.
Ann. of Math.(2) 162 (2005), no. 1, 195–290.3.
F. Bonahon, L. Siebenmann , The characteristic toric splitting of irreducible compact 3-orbifolds , Math. Ann. 278 (1987) 441–479.4.
F. Bonahon, L. Siebenmann , New Geometric Splittings of Classical Knots and the Classifi-cation and Symmetries of Arborescent Knots,
Preprint.5.
J. Cha, S. Friedl, M. Powell,
Splitting numbers of links , Proc. Edinb. Math. Soc. (2) 60(2017), no. 3, 587–614.6.
D. Cooper, C. Hodgson, S. Kerckhoff,
Three-dimensional orbifolds and cone-manifolds.
With a postface by Sadayoshi Kojima. MSJ Memoirs, 5. Mathematical Society of Japan,Tokyo, 2000.7.
M. Culler, C. McA. Gordon, J. Luecke, P. Shalen,
Dehn surgery on knots.
Ann. ofMath. (2) 125 (1987), no. 2, 237–300.8.
D. Gabai , Foliations and the topology of 3-manifolds. III.
J. Differential Geom. 26 (1987),no. 3, 479–536.9.
D. Gabai , Surgery on knots in solid tori.
Topology 28 (1989), no. 1, 1–6.340.
D. Gabai , The Smale conjecture for hyperbolic 3-manifolds:
Isom( M ) ≃ Diff( M ) . J. Differ-ential Geom. 58 (2001), no. 1, 113–149.11.
C. Gordon, J. Luecke,
Reducible manifolds and Dehn surgery.
Topology 35 (1996), no. 2,385–409.12.
W. Haken,
Theorie der Normal߬achen.
Acta Math. 105 (1961) 245–375.13.
W. H. Holzmann,
An equivariant torus theorem for involutions , Trans. Amer. Math. Soc326 (1991) 887–906.14.
W. Jaco,
Lectures on three-manifold topology.
CBMS Regional Conference Series in Mathe-matics, 43. American Mathematical Society, Providence, R.I., 1980.15.
W. Jaco, U. Oertel,
An algorithm to decide if a 3-manifold is a Haken manifold.
Topology23 (1984), no. 2, 195–209.16.
W. Jaco, J. H. Rubinstein,
PL minimal surfaces in 3-manifolds.
J. Differential Geom. 27(1988), no. 3, 493–524.17.
W. Jaco, J. Tollefson,
Algorithms for the complete decomposition of a closed 3-manifold ,Illinois J. Math. 39 (1995) 358–406.18.
D. A. Ka˘zdan, G. A. Margulis,
A proof of Selbergs hypothesis,
Mat. Sb. (N.S.) 75 (117)(1968), 163–168.19.
T. Kobayashi,
Equivariant annulus theorem for 3-manifolds.
Proc. Japan Acad. Ser. AMath. Sci. 59 (1983), no. 8, 403–406.20.
G. Kuperberg , Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization ,arXiv:1508.0672021.
M. Lackenby,
Surfaces, surgery and unknotting operations,
Math. Ann. 308 (1997) 615–632.22.
M. Lackenby,
Exceptional surgery curves in triangulated 3-manifolds,
Pacific J. Math. 210(2003), 101–163.23.
W. B. R. Lickorish,
The unknotting number of a classical knot.
Contemp. Math. 44 (1985)117–121.24.
C. Livingston,
KnotInfo ∼ knotinfo25. C. Livingston,
Splitting numbers of links and the four-genus.
Proc. Amer. Math. Soc. 146(2018), no. 1, 421–427.26.
J. Manning,
Algorithmic detection and description of hyperbolic structures on closed 3-manifolds with solvable word problem.
Geom. Topol. 6 (2002), 1–25.357.
S. Matveev,
Transformations of special spines, and the Zeeman conjecture.
Izv. Akad. NaukSSSR Ser. Mat. 51 (1987), no. 5, 1104–1116, 1119; translation in Math. USSR-Izv. 31 (1988),no. 2, 42343428.
S. Matveev , Algorithmic topology and classification of 3-manifolds.
Algorithms and Compu-tation in Mathematics, 9. Springer, Berlin, 2007.29.
D. McCoy,
Alternating knots with unknotting number one.
Adv. Math. 305 (2017), 757–802.30.
W. H. Meeks III, L. Simon, S.-T. Yau,
Embedded Minimal Surfaces, Exotic Spheres, andManifolds with Positive Ricci Curvature , Ann. Math. 116, No. 3 (1982) 621–659.31.
W. H. Meeks III, S.-T. Yau,
The equivariant Dehn’s lemma and the loop theorem,
Comment.Math. Helv. 56 (1981), 225–239.32.
J. Montesinos,
Three manifolds as 3-fold branched covers of S , Quart. J. Math. Oxford(2), 27 (1976), 85–94.33. J. Morgan,
The Smith conjecture. (New York, 1979) Pure Appl. Math., 112.34.
Y. Nakanishi,
A note on unknotting number.
Math. Sem. Notes Kobe Univ. 9 (1981)99–108.35.
B. Owens,
Unknotting information from Heegaard Floer homology.
Adv. Math. 217 (2008),no. 5, 2353–2376.36.
P. Ozsv´ath, Z. Szab´o , Knots with unknotting number one and Heegaard Floer homology,
Topology 44 (2005), no. 4, 705–745.37.
G. Perelman,
The entropy formula for the Ricci flow and its geometric applications,
Preprint,arxiv:math.DG/021115938.
G. Perelman,
Ricci flow with surgery on three-manifolds,
Preprint, arxiv:math.DG/030310939.
G. Perelman,
Finite extinction time for the solutions to the Ricci flow on certain three-manifolds,
Preprint, arxiv:math.DG/030724540.
C. Petronio, J. Weeks,
Partially flat ideal triangulations of cusped hyperbolic 3-manifolds ,Osaka J. Math. 37 (2000), 453–466.41.
M. Scharlemann,
Sutured manifolds and generalized Thurston norms.
J. Differential Geom.29 (1989), no. 3, 557–614.42.
M. Scharlemann,
Producing reducible 3-manifolds by surgery on a knot.
Topology 29 (1990),no. 4, 481–500.43.
P. Scott, H. Short,
The homeomorphism problem for closed 3-manifolds,
Algebr. Geom.Topol. 14 (2014), no. 4, 2431–2444 364.
A. Stoimenow,
Polynomial values, the linking form and unknotting numbers.
Math. Res.Lett. 11 (2004), no. 5-6, 755–769.45.
T. Tanaka,
Unknotting numbers of quasipositive knots,
Topology Appl. 88 (1998), no. 3,239–246.46.
W. Thurston , The geometry and topology of three-manifolds , library.msri.org/books/gt3m47.
J. Tollefson,
Involutions of Seifert fiber spaces,
Pacific J. Math. 74 (1978), no. 2, 519–529.48.
J. Weeks,
Snappea,
J. Weeks,