TTHE TAUT POLYNOMIAL AND THE ALEXANDER POLYNOMIAL
ANNA PARLAK
Abstract.
Landry, Minsky and Taylor introduced a polynomial invariant of veeringtriangulations called the taut polynomial. We prove that when a veering triangula-tion is edge-orientable then its taut polynomial is equal to the Alexander polynomialof the underlying manifold. For triangulations which are not edge-orientable we givea sufficient condition for the equality between the support of the taut polynomialand that of the Alexander polynomial. We also consider Dehn fillings of 3-manifoldsequipped with a veering triangulation. In this case we compare the image of thetaut polynomial under a Dehn filling and the Alexander polynomial of the Dehn-filled manifold.As an application we extend the results of McMullen which relate theTeichm¨uller polynomial with the Alexander polynomial. Using this we find firstcohomology classes corresponding to fibrations whose monodromy determines ori-entable invariant laminations in the fibre. Introduction
In this paper we consider 3-manifolds which admit a (transverse taut) veeringtriangulation. Such triangulations have been introduced by Agol as canonical trian-gulations of pseudo-Anosov mapping tori [2, Section 4]. In an unpublished work Agoland Gu´eritaud generalised this construction and showed that a veering triangulationcan be derived from any pseudo-Anosov flow without perfect fits. Work is underwayto prove the converse [19], that a 3-manifold M with a veering triangulation admits aflow which can be extended to pseudo-Anosov flows on certain Dehn fillings of M .The connection between veering triangulations and pseudo-Anosov flows inspires anovel direction of research: studying pseudo-Anosov flows via veering triangulations.This idea is exploited by Landry, Minsky and Taylor in [10, 11]. In [11] they introducedtwo polynomial invariants of veering triangulations — the taut polynomial and theveering polynomial. In [10] they showed that these polynomials carry informationabout the exponential growth rate of the closed orbits of the associated pseudo-Anosovflows [10].Our goal is to study the relationship between the taut polynomial of a veeringtriangulation and the Alexander polynomial of the underlying manifold. Throughoutthe paper we assume that M is a finite volume 3-manifold equipped with a veeringtriangulation V . Let H M = H ( M ; Z ) /torsion . Denote by M ab the maximal free abelian cover of M . It admits a veering triangula-tion V ab . The taut polynomial Θ V ∈ Z [ H M ] of V is an invariant derived from a certain Z [ H M ]-module associated to V ab . The Alexander polynomial ∆ M ∈ Z [ H M ] of M canbe computed from a presentation for H ( M ab ; Z ). Mathematics Subject Classification.
Primary 57M27; Secondary 57Q15.
Key words and phrases. a r X i v : . [ m a t h . G T ] J a n ANNA PARLAK
We distinguish two types of veering triangulations which have to be treated differ-ently: edge-orientable ones and not edge-orientable ones. After defining edge-orientableveering triangulations we introduce the edge-orientation double cover V or of a veeringtriangulation V . The covering V or → V determines the edge-orientation homomor-phism ω : π ( M ) → {− , } . In Lemma 4.9 we prove that if the triangulation V ab of the maximal free abelian cover is edge-orientable then ω factors through H M . Theobtained factor σ : H M → {− , } is used to relate the taut polynomial Θ V with theAlexander polynomial ∆ M . Proposition 5.7.
Let V = (( T, F, E ) , α, ν ) be a veering triangulation of a 3-manifold M .Set r = rank H M and let ( h , . . . , h r ) be any basis of H M . If V ab is edge-orientablethen Θ V ( h , . . . , h r ) = ∆ M ( σ ( h ) · h , . . . , σ ( h r ) · h r ) where σ : H M → {− , } is the factor of the edge-orientation homomorphism of V . We then generalise our findings by considering a (not necessarily closed) Dehn fill-ing N of the compact core of M . We set H N = H ( N ; Z ) /torsion . The inclusion of M into N determines an epimorphism i ∗ : H M → H N . We comparethe image i ∗ (Θ V ) of the taut polynomial of V under the Dehn filling and the Alexan-der polynomial of N . In order to do that we need to consider an intermediate freeabelian cover M N of M with the deck group isomorphic to H N . It admits a veeringtriangulation V N . If this triangulation is edge-orientable, then the edge-orientationhomomorphism factors through σ N : H N → {− , } . Using Proposition 5.7 we provethe main theorem of this paper. Theorem 6.3.
Let V be a veering triangulation of a finite volume 3-manifold M . Let N be a Dehn filling of the compact core of M such that the rank of H N is positive.Denote by (cid:96) , . . . , (cid:96) k the core curves of the filling solid tori in N and by i ∗ : H M → H N the epimorphism induced by the inclusion of M into N . Assume that the veeringtriangulation V N is edge-orientable and that for every j ∈ { , . . . , k } the class [ (cid:96) j ] ∈ H N is nontrivial. Set s = rank H N and let ( h , . . . , h s ) denote any basis of H N .I. rank H M ≥ and(a) s ≥ . Then i ∗ (Θ V )( h , . . . , h s ) = ∆ N ( σ N ( h ) · h , . . . , σ N ( h s ) · h s ) · k (cid:89) j =1 ([ (cid:96) j ] − . (b) s = 1 . Let h denote the generator of H N . • If ∂N (cid:54) = ∅ then i ∗ (Θ V )( h ) = ( h − − · ∆ N ( σ N ( h ) · h ) k (cid:89) j =1 ([ (cid:96) j ] − . • If N is closed then i ∗ (Θ V )( h ) = ( h − − · ∆ N ( σ N ( h ) · h ) · k (cid:89) j =1 ([ (cid:96) j ] − . II. rank H M = 1 . Let h denote the generator of H N ∼ = H M .(a) If ∂N (cid:54) = ∅ then i ∗ (Θ V )( h ) = ∆ N ( σ N ( h ) · h ) . HE TAUT POLYNOMIAL AND THE ALEXANDER POLYNOMIAL 3 (b) If N is closed set (cid:96) = (cid:96) and then i ∗ (Θ V )( h ) = ( h − − ([ (cid:96) ] − · ∆ N ( σ N ( h ) · h ) . The motivation for studying the image of the taut polynomial under a Dehn fillingcomes from its connection with the
Teichm¨uller polynomial . This invariant has beenintroduced by McMullen in [13, Section 3] and is associated to a fibred face of theThurston norm ball. As explained in [11, Proposition 7.2] every Teichm¨uller polynomialcan be expressed as the image of the taut polynomial of a layered veering triangulationunder a fibre-preserving Dehn filling. Theorem 6.3 can therefore be interpreted as atheorem which relates the Teichm¨uller polynomial Θ F of a fibred face F in H ( N ; R )with the Alexander polynomial ∆ N of N . In [13, Theorem 7.1] McMullen proved Theorem (McMullen).
Let F be a fibred face in H ( N ; R ) , where b ( N ) ≥ . Then F ⊂ A for a unique face A of the Alexander norm ball in H ( N ; R ) . If moreover thelamination L associated to F is transversely orientable, then F = A and ∆ N divides Θ F . Theorem 6.3 extends this result in two ways.(1) When the veering triangulation is layered and the Dehn filling preserves a car-ried fibration it gives a stronger relation between the Teichm¨uller polynomial andthe Alexander polynomial; see Corollary 7.5. For a fibred face F whose associatedlamination L is transversely orientable we give sufficient conditions for the equalitybetween Θ F and ∆ N . When ∆ N only divides Θ F we give the formula for the remainingfactors of Θ F . We also give precise formulas relating Θ F and ∆ N in the case when L is not transversely orientable — we only need to assume that the lamination inducedby L in the maximal free abelian cover N ab of N is transversely orientable. Note thattransverse orientability of the induced lamination in N ab is a much weaker conditionthan transverse orientability of L itself. For instance, if the fibred face F is fully-punctured the induced lamination in N ab can fail to be transversely orientable only ifthe torsion subgroup of H ( N ; Z ) is of even order.(2) Theorem 6.3 does not require a veering triangulation to be layered. In particular,when a veering triangulation V of M is not layered, but does carry a surface, it givesa relation between an invariant of a non-fibred face of the Thurston norm ball in H ( M ; R ) (see [11, Theorem 5.12]) and the Alexander polynomial of M .In Subsection 7.4 we give a dynamical explanation of the relation between theAlexander polynomial and the Teichm¨uller polynomial obtained in Corollary 7.5. Weuse this in Corollary 7.7 to find cohomology classes φ ∈ H ( N ; R ) corresponding tofibrations of N over the circle whose monodromy determines orientable invariant lam-inations in the fibre. Acknowledgements.
I am grateful to Samuel Taylor for explaining to me his work onthe taut and veering polynomials during my visit at Temple University in July 2019and subsequent conversations. I thank Saul Schleimer and Henry Segerman for theirgenerous help in implementing the computation of the taut polynomial. I also thankJoe Scull for discussions on various topology-related topics.This work has been written during PhD studies of the author at the University ofWarwick, funded by the EPSRC and supervised by Saul Schleimer.2.
Transverse taut veering triangulations
Ideal triangulations of 3-manifolds. An ideal triangulation of a 3-manifold M withtorus cusps is a decomposition of M into ideal tetrahedra. We denote an ideal tri-angulation by T = ( T, F, E ), where
T, F, E denote the set of tetrahedra, triangles(2-dimensional faces) and edges respectively.
ANNA PARLAK
Transverse taut triangulations.
Let t be an ideal tetrahedron. Assume that a coori-entation is assigned to each of its faces. We say that t is transverse taut if on twoof its faces the coorientation points into t and on the other two it points out of t [8,Definition 1.2]. We call the pair of faces whose coorientations point out of t the topfaces of t and the pair of faces whose coorientations point into t the bottom faces of t .We also say that t is immediately below its top faces and immediately above its bottomfaces. We can encode a transverse taut structure on a tetrahedron by drawing it as aquadrilateral with two diagonals — one on top of the other; see Figure 1. Then theconvention is that the coorientation on all faces points towards the reader.The top diagonal is the common edge of the two top faces of t and the bottomdiagonal is the common edge of the two bottom faces of t . The remaining four edgesof t are called its equatorial edges . Presenting a transverse tetrahedron as in Figure 1naturally endows it with an abstract assignment of angles from { , π } to its edges. Thediagonal edges have the π angle assigned, and the equatorial edges have the 0 angleassigned. Such an assignment of angles is called a taut structure on t [8, Definition 1.1].A triangulation T = ( T, F, E ) is transverse taut if • every ideal triangle f ∈ F is assigned a coorientation so that every ideal tetra-hedron is transverse taut, and • for every edge e ∈ E the sum of angles of the underlying taut structure of T ,over all embeddings of e into tetrahedra, equals 2 π [8, Definition 1.2].We denote a triangulation with a transverse taut structure by ( T , α ). Remark.
Triangulations as described above were introduced by Lackenby in [9]. Someauthors, including Lackenby, call them taut triangulations . Then triangulations en-dowed only with a { , π } angle structure are called angle taut triangulations . Veering triangulations. A veering tetrahedron is an oriented taut tetrahedron whoseequatorial edges are coloured alternatingly red and blue as shown in Figure 1. Ataut triangulation is veering if a colour (red/blue) is assigned to each edge of thetriangulation so that every tetrahedron is veering. Note that any triangle of a veeringtriangulation has two edges of the same colour and one edge of the other colour. Figure 1.
A veering tetrahedron. The angle structure assigns 0 to the equa-torial edges and π to the diagonal edges of the tetrahedron. Definition 2.1.
We say that a triangle of a veering triangulation is red (respectively blue ) if two of its edges are red (respectively blue).In this paper we consider only transverse taut veering triangulations, where we candistinguish top faces from bottom faces. We skip “transverse taut” in the rest of thepaper. We denote a veering triangulation by V = ( T , α, ν ), where ν corresponds tothe colouring on edges. HE TAUT POLYNOMIAL AND THE ALEXANDER POLYNOMIAL 5 The lower and upper tracks
Let ( T , α ) be a transverse taut triangulation of M . The taut structure on T makesits 2-skeleton T (2) of T into a branched surface . It is called the horizontal branchedsurface and denoted by B [19, Subsection 2.12]. B is ideally triangulated by thetriangular faces of T . We denote this triangulation of B by ( F, E ).In this subsection we describe a pair of canonical train tracks embedded in B , calledthe lower and upper tracks of a ( T , α ). They are key in the definition of the tautpolynomial. Moreover, we use them to divide veering triangulations into two classes: edge-orientable ones and not edge-orientable ones.3.1. Dual train tracks. A dual train track τ ⊂ B is a train track in B which restrictedto any face f ∈ F looks like in Figure 2. We denote the restriction of τ to f by τ f .The edges of τ f are called the half-branches of τ f . Two of the half-branches of τ f arecalled its small half-branches. The remaining half-branch is called large .The train track τ has two types of special points: switches (one for each f ∈ F ; thecommon point of all half-branches of τ f ) and midpoints of edges (one for each e ∈ E ).We say that a half-branch b meets e ∈ E , or vice versa, if one of the endpoints of b isthe midpoint of e . largesmall small Figure 2.
Dual train track restricted to a face.
Definition 3.1.
We say that a dual train track τ ⊂ B is transversely orientable if thereexists a nonvanishing continuous vector field on τ which is tangent to B and transverseto τ . Figure 3.
A local picture of a transversely oriented dual train track in thehorizontal branched surface. For clarity the train track in the bottom triangleis presented separately on the right.Figure 3 presents a local picture of a transversely oriented dual train track. Trans-verse orientation is indicated by arrows transverse to the branches of the track pointingin one of the two possible directions.
Remark 3.2.
Let τ be a dual train track in B and f ∈ F . If we fix a transverseorientation on one half-branch of τ f there is no obstruction to extend it over all half-branches of τ f . Hence the train track τ f is always transversely orientable. Definition 3.3.
Suppose that a dual train track τ in the horizontal branched surfaceof ( T , α ) is transversely oriented. The restriction of the transverse orientation tothe edge midpoints endows the edges of T with an orientation. We say that theseorientations on edges are determined by the transverse orientation on τ . ANNA PARLAK
Example 3.4.
Figure 4 presents a dual train track τ in the horizontal branched surfaceof the veering triangulation cPcbbbdxm 10 of the manifold m003, also known as thefigure eight knot sister. In the face labelled with f there are two half-branches of τf f f f f f f f Figure 4.
A dual train track in the 2-skeleton of the veering triangulation V = cPcbbbdxm 10 of the figure eight knot sister. The yellow shaded region ishomeomorphic to the M¨obius band.which together form the core curve of the M¨obius band (shaded in yellow). Hence thepresented train track cannot be transversely oriented.3.2. The lower and upper tracks.
A transverse taut structure α on T endows its hori-zontal branched surface B with a pair of canonical train tracks which we call, following[19, Definition 4.7], the lower and upper tracks of ( T , α ). Definition 3.5.
Let ( T , α ) be a transverse taut triangulation and let B be the corre-sponding horizontal branched surface equipped with the ideal triangulation ( F, E ).The lower track τ L of T is the dual train track in B such that for every f ∈ F thelarge-half branch of τ Lf meets the top diagonal of the tetrahedron immediately below f .The upper track τ U of T is the dual train track in B such that for every f ∈ F the large-half branch of τ Uf meets the bottom diagonal of the tetrahedron immediatelyabove f . Example 3.6.
The upper track of the veering triangulation cPcbbbdxm 10 of the figureeight knot sister is presented in Figure 4.We introduce the following names for the edges of f ∈ F which meet large half-branches of τ Lf or τ Uf . Definition 3.7.
Let ( T , α ) be a transverse taut triangulation. We say that an edge inthe boundary of f ∈ F is the lower large (respectively the upper large ) edge of f if itmeets the large half-branch of τ Lf (respectively τ Uf ).To define the lower and upper tracks we do not need a veering structure on atransverse taut triangulation. However, if the triangulation is veering the lower andupper tracks restricted to the faces of a single tetrahedron t are determined by thecolours of the diagonal edges of t ; see the lemma below. Lemma 3.8 (Lemma 3.9 in [18]) . Let V = ( T , α, ν ) be a veering triangulation. Let t beone of its tetrahedra. The lower large edges of the bottom faces of t are the equatorialedges of t which are of the same colour as the bottom diagonal of t . The upper largeedges of the top faces of t are the equatorial edges of t which are of the same colour asthe top diagonal of t . (cid:3) HE TAUT POLYNOMIAL AND THE ALEXANDER POLYNOMIAL 7
Using Lemma 3.8 we present the pictures of the lower track and upper track in aveering tetrahedron in Figure 5(a) and Figure 5(b), respectively.(a) (b)
Figure 5.
Squares in the top row represent top faces of a tetrahedron andsquares in the bottom row represent bottom faces of a tetrahedron. (a) Thelower track in a veering tetrahedron. There are two options, depending on thecolour of the bottom diagonal. (b) The upper track in a veering tetrahedron.There are two options, depending on the colour of the top diagonal.In Remark 3.2 we noted that every dual train track restricted to a single face of thetriangulation is transversely orientable. We are interested in extending this transverseorientation further, not yet over the whole 2-skeleton of the triangulation, but overfour faces of a given tetrahedron. This is not possible for all dual train tracks. Butit is possible in the case of the upper and lower tracks of a veering triangulation; seeFigure 5. Corollary 3.9.
Let V = ( T , α, ν ) be a veering triangulation. Fix t ∈ T and let f i ∈ F , i = 1 , , , be the four faces of t . Denote by τ i the restriction τ Uf i of the upper track τ U of V to f i . Then the train track τ t = (cid:91) i =1 τ i ⊂ (cid:91) i =1 f i is transversely orientable. (cid:3) Edge-orientable veering triangulations.
We divide all veering triangulations intotwo classes. The division depends on whether the lower/upper track is transverselyorientable. First we prove that it does not matter which track we consider.
Lemma 3.10.
Let V be a veering triangulation. The lower track of V is transverselyorientable if and only if the upper track of V is transversely orientable.Proof. Suppose τ L is transversely oriented. Pick the orientation on the edges of V determined by the transverse orientation on τ L .Note that for any triangle f ∈ F the lower large edge of f and the upper large edgeof f are different edges of f , but both are of the same colour as f [18, Lemma 2.1].Figure 6 presents the lower and upper tracks in a red and a blue triangle. If we reversethe orientation of all edges of one colour, say blue, then the new orientations determinea transverse orientation on τ U . (cid:3) Definition 3.11.
We say that a veering triangulation V is edge-orientable if the uppertrack of V is transversely orientable. ANNA PARLAK
Figure 6.
Using a transverse orientation on the lower track to find a trans-verse orientation on the upper track. On the left: in red triangles. On theright: in blue triangles.By Lemma 3.10 a veering triangulation V is edge-orientable if and only if the lowertrack of V is transversely orientable. For the remaining of the paper we only considerthe upper track and use a shorter notation τ = τ U . Example 3.12.
The veering triangulation cPcbbbdxm 10 of the figure eight knot sisteris not edge-orientable. Its upper track is presented in Figure 4.4.
The edge-orientation double cover
Let V = (( T, F, E ) , α, ν ) be a veering triangulation of a 3-manifold M . Let τ = τ U be the upper track of V . In this section we construct the edge-orientation double cover V or of V . Definition 4.1.
Let f ∈ F . Fix a transverse orientation on the upper track τ f ; thisis possible by Remark 3.2. We say that f is edge-oriented if the orientation of theedges in the boundary of f agrees with the orientation determined by the transverseorientation on τ f ; see Figure 7. Figure 7.
Edge-oriented triangles. There are two possibilities depending onthe chosen transverse orientation on τ f .Naturally, if V is edge-orientable, then there is a choice of orientations on the edgesof V such that every f ∈ F is edge-oriented. Definition 4.2.
Let t ∈ T . Fix a transverse orientation on the upper track τ t restrictedto the faces of t ; this is possible by Corollary 3.9. We say that t is edge-oriented if theorientation on the edges of t agrees with the orientation determined by the transverseorientation on τ t . In that case every face of t is edge-oriented.We construct the edge-orientation double cover V or of V using edge-oriented tetra-hedra — twice as many as the number of tetrahedra of V . Then we use the fact thatevery face can be edge-oriented in precisely two ways (see Figure 7) to argue that thefaces of the edge-oriented tetrahedra can be identified in pairs to give a double coverof V . Construction 4.3. (Edge-orientation double cover) Let V = (( T, F, E ) , α, ν ) be a veeringtriangulation. For t ∈ T we take two copies of t and denote them by t, t . Both t, t are HE TAUT POLYNOMIAL AND THE ALEXANDER POLYNOMIAL 9 endowed with the upper track of V restricted to the faces of t . We denote the trackin t by τ t and the track in t by τ t .We fix the orientation on the bottom diagonal of t and choose the opposite orienta-tion on the bottom diagonal of t . This determines (opposite) transverse orientationson τ t , τ t by Corollary 3.9. We orient the edges of t, t so that they are edge-oriented by τ t , τ t , respectively.Let T = { t | t ∈ T } . Every face f ∈ F appears four times as a face of an edge-oriented tetrahedron from T ∪ T : twice as a bottom face and twice as a top face.Moreover, its two bottom copies are edge-oriented in the opposite way; the same istrue for its two top copies. Hence we can identify the faces of the tetrahedra T ∪ T inpairs to obtain an edge-oriented veering triangulation which double covers V . Definition 4.4.
Let V be a veering triangulation. The edge-orientation double cover V or of V is the ideal triangulation obtained by Construction 4.3. Example 4.5.
Let V denote the veering triangulation cPcbbbdxm 10 of the the figureeight knot sister (manifold m003). It is presented in Figure 4. We follow Construc-tion 4.3 to build the edge-orientation double cover V or of V . The result is presentedin Figure 8. f f f f f f f f f f f f f f f f Figure 8.
The edge-orientation double cover of the veering triangulation V = cPcbbbdxm 10 of the the figure eight knot sister. Two lifts of an edge of V have the same colour, but the orientation on one of them is indicated by asingle arrow, while on the other — by a double arrow.Since we build the cover V or using edge-oriented tetrahedra, we immediately get thefollowing lemma. Lemma 4.6.
Let V be a veering triangulation. The edge-orientation double cover V or of V is edge-orientable. (cid:3) Lemma 4.7.
Let V be a veering triangulation. The edge-orientation double cover V or is connected if and only if V is not edge-orientable.Proof. Suppose that V is edge-orientable. Fix the orientations on the edges of V suchthat every face of V is edge-oriented. With this choice of orientations following Con-struction 4.3 gives a disjoint union of two copies of V . Conversely, if V or is disconnected,then it is a disjoint union of two copies of V . By Lemma 4.6 the triangulation V or isedge-orientable and hence so is V . (cid:3) The edge-orientation homomorphism.
The edge-orientation double cover V or → V determines a homomorphism ω : π ( M ) → {− , } γ (cid:55)→ (cid:40) γ lifts to a loop in V or − . (4.8)We call this homomorphism the edge-orientation homomorphism of V .Let H be a quotient of π ( M ). We are interested in a combinatorial conditionwhich ensures that the edge-orientation homomorphism ω : π ( M ) → {− , } fac-tors through H . The obtained factors H → {− , } are used in Proposition 5.7 andTheorem 6.3.Note that there is a regular cover M H of M with the deck group isomorphic to H and the fundamental group isomorphic to the kernel of the projection π ( M ) → H .This cover admits a veering triangulation V H induced by the veering triangulation V of M .The reasoning in the following lemma is completely analogous to the case of factoringthe orientation character; see for example [1, Lemma 1.1]. Lemma 4.9.
The veering triangulation V H is edge-orientable if and only if the edge-orientation homomorphism ω : π ( M ) → {− , } factors through H .Proof. Let q : π ( M ) → H be the natural projection. Its kernel is isomorphic to π (cid:0) M H (cid:1) . To show that ω factors through H it is enough to show that ker q ≤ ker ω .This is clearly the case when V H is edge-orientable, because then the edge-orientationhomomorphism ω H : π (cid:0) M H (cid:1) → {− , } is trivial. Therefore for every γ ∈ ker q ∼ = π (cid:0) M H (cid:1) we have ω ( γ ) = 1. π (cid:0) M H (cid:1) π ( M ) {− , } H ω H ωq Conversely, suppose that V H is not edge-orientable. Then the edge-orientation ho-momorphism ω H : π (cid:0) M H (cid:1) → {− , } is not trivial. Let γ ∈ ker q ∼ = π (cid:0) M H (cid:1) be anelement such that ω ( γ ) = − β ∈ π ( M ). Then q ( β ) = q ( βγ ) and ω ( βγ ) = ω ( β ) ω ( γ ) = − ω ( β ) (cid:54) = ω ( β ) . This implies that ω does not factor through H . (cid:3) The taut polynomial and the Alexander polynomial
Let M be a finite volume 3-manifold equipped with a veering triangulation V =(( T, F, E ) , α, ν ). Let(5.1) H M = H ( M ; Z ) /torsion . Remark 5.2.
We regularly abusively identify cusped 3-manifolds with their compactcore. Therefore M as above is often seen as a manifold with toroidal boundary com-ponents instead of torus cusps. HE TAUT POLYNOMIAL AND THE ALEXANDER POLYNOMIAL 11
The regular cover of M determined by H M is called the maximal free abelian cover of M and denoted by M ab . By V ab we denote the veering triangulation of M ab inducedby V . The sets of ideal tetrahedra, triangles and edges of V ab can be identified with H M × T, H M × F, H M × E , respectively. We denote elements of H M × T, H M × F, H M × E by h · t , h · f , h · e , where h ∈ H M , t ∈ T , f ∈ F , e ∈ E . The free abelian groupsgenerated by H M × T , H M × F and H M × E are isomorphic as Z [ H M ]-modules to thefree Z [ H M ]-modules Z [ H M ] T , Z [ H M ] F , Z [ H M ] E generated by T, F, E , respectively. Weorient ideal simplices of V ab in such a way that the restriction of M ab → M to eachideal simplex is orientation-preserving.In this section we recall the definitions of the taut polynomial Θ V of a veeringtriangulation V (introduced in [11, Section 3]) and the Alexander polynomial ∆ M ofthe manifold M . Then we give a sufficient condition on V which ensures that, up toa unit in Z [ H M ], these polynomials differ at most by changing the sign of some of itsterms.5.1. Polynomial invariants of finitely presented Z [ H ] -modules. The definitions of boththe taut and the Alexander polynomials follow the same pattern. Namely, they arederived from the Fitting ideals of certain Z [ H M ]-modules associated to the maximalfree abelian cover M ab . For that reason in this subsection we recall definitions ofFitting ideals and their invariants.Let H be a finitely generated free abelian group. Let M be a finitely presentedmodule over the integral group ring Z [ H ]. Then there exist integers k, l ∈ N and anexact sequence Z [ H ] k A −→ Z [ H ] l −→ M −→ Z [ H ]-homomorphisms called a free presentation of M . The matrix of A , writtenwith respect to any bases of Z [ H ] k and Z [ H ] l , is called a presentation matrix for M . Definition 5.3. [17, Section 3.1] Let M be a finitely presented Z [ H ]-module with apresentation matrix A of dimension l × k . We define the i -th Fitting ideal
Fit i ( M )of M to be the ideal in Z [ H ] generated by the ( l − i ) × ( l − i ) minors of A .In particular Fit i ( M ) = Z [ H ] for i ≥ l , as the determinant of the empty matrixequals 1, and Fit i ( M ) = 0 for i < i < l − k . The Fitting ideals are independentof the choice of a free presentation of M [17, p. 58]. Remark.
Fitting ideals are called determinantal ideals in [22] and elementary ideals in[4, Chapter VIII].For any finitely generated ideal I ⊂ Z [ H ] there exists a unique minimal principalideal I ⊂ Z [ H ] which contains it [4, p. 117]. The ideal I is generated by the greatestcommon divisor of the generators of I [4, p. 118]. This motivates the followingdefinition. Definition 5.4.
Let M be a finitely presented Z [ H ]-module. We define the i -th Fittinginvariant δ i ( M ) of M to be the greatest common divisor of elements of Fit i ( M ).When Fit i ( M ) = (0) we set δ i ( M ) to be equal to 0.Note that Fitting invariants are well-defined only up to a unit in Z [ H ]. In particular,all equalities proved in this paper hold only up to a unit in the appropriate integralgroup ring.5.2. The Alexander polynomial.
Since there is an action of H M on M ab , the firsthomology group H ( M ab ; Z ) admits a Z [ H M ]-module structure. Its zeroth Fittinginvariant is called the Alexander polynomial of M . We denote it by ∆ M . An efficient algorithm to compute ∆ M directly from the presentation of π ( M ) relieson Fox calculus [12, pp. 116 – 118] and has been implemented in SnapPy [5]. It is alsopossible to compute ∆ M using the 2-complex dual to the ideal triangulation of M ; see[14, Section 5].Following Remark 5.2 we identify M with its compact core and consider H ( M ab , ∂M ab ; Z ). This abelian group also admits a Z [ H M ]-module structure. Wedenote its zeroth Fitting invariant by ∆ ( M,∂M ) . From the results of Turaev on Milnortorsion we can deduce the following lemma. Lemma 5.5 (Theorem 11.10 of [23]) . Let M be a compact, connected, triangulated 3-manifold whose boundary is non-empty and consists of tori. Then ∆ ( M,∂M ) = ∆ M . (cid:3) The taut polynomial.
Recall that M denotes a finite volume 3-manifold equippedwith a veering triangulation V = (( T, F, E ) , α, ν ) and H M denotes the torsion-free partof H ( M ; Z ). The taut polynomial Θ V of V is defined in [11, Section 3] by explicitlygiving a free presentation(5.6) Z [ H M ] F D −→ Z [ H M ] E −→ E α ( V ) −→ taut module E α ( V ). The definition of D depends on the upper track of V , asexplained below.Let τ denote the upper track of V . It induces a train track τ ab in the 2-skeletonof the veering triangulation V ab of M ab . Let h · f ∈ H M × F be a triangle of thattriangulation. Consider the restriction of τ ab to h · f ; see Figure 9. This train trackdetermines a switch relation between its three half-branches: the large half-branch isequal to the sum of the two small half-branches. By identifying the half-branches withthe edges in the boundary of h · f which they meet, we obtain a switch relation betweenthe edges in the boundary of h · f . h · e h · e h · e Figure 9.
The upper track in a triangle determines a switch relation h · e = h · e + h · e between the edges in its boundary.We rearrange the switch relation determined by h · f into a linear combination ofedges from H M × E . This linear combination is the value of D at h · f . For example,the image of the triangle presented in Figure 9 under D is equal to h · e − h · e − h · e ∈ Z [ H M ] E . The taut polynomial Θ V is defined as the zeroth Fitting polynomial of E α ( V ). Inother words Θ V = gcd { maximal minors of D } . Comparison of polynomials.
In this subsection we give a sufficient condition onthe veering triangulation for its taut polynomial to be very closely related to theAlexander polynomial of the underlying manifold.Recall from Subsection 4.1 that a veering triangulation V of M determines theedge-orientation homomorphism ω : π ( M ) → {− , } . By Lemma 4.9 if V ab is edge-orientable then ω factors through σ : H M → {− , } . HE TAUT POLYNOMIAL AND THE ALEXANDER POLYNOMIAL 13
Proposition 5.7.
Let V = (( T, F, E ) , α, ν ) be a veering triangulation of a 3-manifold M .Set r = b ( M ) and let ( h , . . . , h r ) be any basis of H M . If V ab is edge-orientable then Θ V ( h , . . . , h r ) = ∆ M ( σ ( h ) · h , . . . , σ ( h r ) · h r ) where σ : H M → {− , } is the factor of the edge-orientation homomorphism of V .Proof. The relative chain complex of the pair ( M ab , ∂M ab ) can be identified with0 −→ Z [ H M ] T ∂ −→ Z [ H M ] F ∂ −→ Z [ H M ] E ∂ −→ −→ . Therefore Z [ H M ] F ∂ −→ Z [ H M ] E −→ H ( M ab , ∂M ab ; Z ) −→ , is a free presentation of H ( M ab , ∂M ab ; Z ). Observe that it resembles the presentation(5.6) for the taut module. The difference is that the presentation matrix D of the tautmodule does not depend on the orientations of the triangles and edges of V ab — it onlydepends on the transverse taut structure α on the triangulation.If V is edge-orientable choose an orientation on its edges so that it is edge-oriented.Otherwise, pick any orientation on the edges of V . Recall that we orient the edges of V ab so that the covering map M ab → M restricted to any edge is orientation-preserving.Therefore clearly when V is edge-orientable then for any h · f ∈ H M × F we have D ( h · f ) = ± ∂ ( h · f ) and thus Θ V = ∆ ( M,∂M ) .More generally, the group homomorphism σ induces a ring homomorphism σ : Z [ H M ] → Z [ H M ] defined by σ (cid:88) h ∈ H M a h · h = (cid:88) h ∈ H M σ ( h ) · a h · h. Consider the following diagram. Z [ H M ] F Z [ H M ] E H ( M ab , ∂M ab ; Z ) 0 Z [ H M ] F Z [ H M ] E E α ( V ) 0 ∂ id σ ⊕ E D The diagram commutes (after reversing the orientation of some triangular faces of V if necessary) because reversing the orientation of the edge h · e of V ab if and only if σ ( h ) = − V ab into an edge-oriented triangulation. It follows that the presentation matricesfor H ( M ab , ∂M ab ; Z ) and E α ( V ) differ by reversing the sign of variables correspondingto the basis elements of H M with a nontrivial image under σ . And so do their Fittinginvariants. ThereforeΘ V ( h , . . . , h r ) = ∆ ( M,∂M ) ( σ ( h ) · h , . . . , σ ( h r ) · h r ) . The claim now follows from Lemma 5.5. (cid:3)
Remark. If V is not edge-orientable then E α ( V ) and H ( M ab , ∂M ab ; Z ) are not iso-morphic as Z [ H M ]-modules (otherwise their Fitting invariants would be the same).When σ is nontrivial the induced ring homomorphism σ : Z [ H M ] → Z [ H M ] is not a Z [ H M ]-module homomorphism. For h ∈ H M such that σ ( h ) = − h · σ ( h ) = − h , but σ ( h · h ) = h . Proposition 5.7 allows us to easily identify veering triangulations for which theequality Θ V ( g , . . . , g r ) = ∆ M ( ± g , . . . , ± g r ) might fail. Corollary 5.8.
Let V be a veering triangulation of a 3-manifold M . Set r = b ( M ) . If Θ V ( g , . . . , g r ) (cid:54) = ∆ M ( ± g , . . . , ± g r ) up to a unit in Z [ H M ] then the torsion subgroup of H ( M ; Z ) has even order.Proof. By Proposition 5.7 ifΘ V ( g , . . . , g r ) (cid:54) = ∆ M ( ± g , . . . , ± g r )then V ab is not edge-orientable. If follows that the deck group of V or → V is a quotientof the torsion subgroup of H ( M ; Z ). (cid:3) Dehn filling veering triangulations
Let M be a 3-manifold equipped with a veering triangulation V = (( T, F, E ) , α, ν ).Let N be a Dehn filling of the compact core of M . We do not assume that N is closed.Following (5.1) we set H N = H ( N ; Z ) /torsion . We denote the ranks of H M , H N by b ( M ), b ( N ), respectively. The inclusion of M into N induces an epimorphism i ∗ : H M → H N . In this section we compare the image i ∗ (Θ V ) of the taut polynomial under the Dehn filling and the Alexander polynomial∆ N of N . Remark.
We treat i ∗ (Θ V ) ∈ Z [ H N ] as an invariant of the Dehn filling of a veeringtriangulation. There is another natural invariant corresponding to this Dehn filling.Namely, instead of computing i ∗ (Θ V ) = i ∗ (gcd(minors of D ))we can compute Θ i ∗ V = gcd( i ∗ (minors of D )) . The latter is the zeroth Fitting invariant of the module obtained from the taut moduleby the extension of scalars through i ∗ . Both these polynomials are elements of Z [ H N ].Clearly, i ∗ (Θ V ) divides Θ i ∗ V , but equality does not always hold.In this paper we consider only i ∗ (Θ V ), because it is a generalisation of the Te-ichm¨uller polynomial; see Lemma 7.1. Ultimately we are interested in the relationbetween the Teichm¨uller polynomial of a fibred face of the Thurston norm ball in H ( N ; R ) and the Alexander polynomial of N ; see Section 7.6.1. The image i ∗ (∆ M ) ∈ Z [ H N ] . The first step in finding a relation between i ∗ (Θ V )and ∆ N is to compare i ∗ (∆ M ) with ∆ N . This has been previously studied by Turaev.By first recursively applying the formulas given in Corollary 4.2 of [24] and then usingCorollary 4.1 of [24] we obtain the following lemma. Lemma 6.1.
Let M be a compact, orientable, connected 3-manifold with nonemptyboundary consisting of tori. Let N be any Dehn filling of M such that b ( N ) ispositive. Denote by (cid:96) , . . . , (cid:96) k the core curves of the filling solid tori in N and by i ∗ : H M → H N the epimorphism induced by the inclusion of M into N . Assume thatfor every j ∈ { , . . . , k } the class [ (cid:96) j ] ∈ H N is nontrivial.I. b ( M ) ≥ and(a) b ( N ) ≥ . Then i ∗ (∆ M ) = ∆ N · k (cid:89) j =1 ([ (cid:96) j ] − . HE TAUT POLYNOMIAL AND THE ALEXANDER POLYNOMIAL 15 (b) b ( N ) = 1 . Let h denote the generator of H N . • If ∂N (cid:54) = ∅ then i ∗ (∆ M ) = ( h − − · ∆ N · k (cid:89) j =1 ([ (cid:96) j ] − . • If N is closed i ∗ (∆ M ) = ( h − − · ∆ N · k (cid:89) j =1 ([ (cid:96) j ] − . II. b ( M ) = 1 . Let h denote the generator of H N ∼ = H M .(a) If ∂N (cid:54) = ∅ then i ∗ (∆ M ) = ∆ N . (b) If N is closed set (cid:96) = (cid:96) and then i ∗ (∆ M ) = ( h − − ([ (cid:96) ] − · ∆ N . (cid:3) The image i ∗ (Θ V ) ∈ Z [ H N ] . Let N ab denote the maximal free abelian coverof N . By removing the preimages of the filling solid tori under N ab → N we obtain afree abelian cover M N of M . The deck group of the covering M N → M is isomorphicto H N . This situation is illustrated in the following commutative diagram.(6.2) M ab M N MN ab N. i The veering triangulation V of M induces a veering triangulation V N of M N . ByLemma 4.9 if V N is edge-orientable, then the edge-orientation homomorphism of V factors through H N . We denote this factor by σ N : H N → {− , } . Theorem 6.3.
Let V be a veering triangulation of a finite volume 3-manifold M . Let N be a Dehn filling of the compact core of M such that s = b ( N ) is positive. Denoteby (cid:96) , . . . , (cid:96) k the core curves of the filling solid tori in N and by i ∗ : H M → H N the epimorphism induced by the inclusion of M into N . Assume that the veeringtriangulation V N is edge-orientable and that for every j ∈ { , . . . , k } the class [ (cid:96) j ] ∈ H N is nontrivial. Let σ N : H N → {− , } be the homomorphism through which the edge-orientation homomorphism of V factors.I. b ( M ) ≥ and(a) s ≥ . Then i ∗ (Θ V )( h , . . . , h s ) = ∆ N ( σ N ( h ) · h , . . . , σ N ( h s ) · h s ) · k (cid:89) j =1 ([ (cid:96) j ] − . (b) s = 1 . Let h denote the generator of H N . • If ∂N (cid:54) = ∅ then i ∗ (Θ V )( h ) = ( h − − · ∆ N ( σ N ( h ) · h ) k (cid:89) j =1 ([ (cid:96) j ] − . • If N is closed then i ∗ (Θ V )( h ) = ( h − − · ∆ N ( σ N ( h ) · h ) · k (cid:89) j =1 ([ (cid:96) j ] − . II. b ( M ) = 1 . Let h denote the generator of H N ∼ = H M .(a) If ∂N (cid:54) = ∅ then i ∗ (Θ V )( h ) = ∆ N ( σ N ( h ) · h ) . (b) If N is closed set (cid:96) = (cid:96) and then i ∗ (Θ V )( h ) = ( h − − ([ (cid:96) ] − · ∆ N ( σ N ( h ) · h ) . In particular with the above assumptions we have i ∗ (Θ V )( h , h , . . . , h s ) = ∆ N ( ± h , . . . , ± h s ) if and only if one of the following four conditions holds • N = M , or • b ( N ) = 1 , ∂N (cid:54) = ∅ , k = 1 , and [ (cid:96) ] generates H N , or • b ( N ) = 1 , N is closed, k = 2 and [ (cid:96) ] = [ (cid:96) ] generates H N , or • b ( M ) = 1 , N is closed and [ (cid:96) ] generates H N . (cid:3) Proof.
Set r = b ( M ). First note that if V N is edge-orientable, then V ab is edge-orientable as well. Therefore there is a homomorphism σ : H M → {− , } throughwhich the edge-orientation homomorphism of V factors. By Proposition 5.7 if V ab isedge-orientable, thenΘ V ( g , . . . , g r ) = ∆ M ( σ ( g ) · g , . . . , σ ( g r ) · g r ) . Since V N is edge-orientable the kernel of i ∗ is contained in the kernel of σ and σ ( g ) = σ N ( i ∗ ( g )) for every g ∈ H M . Therefore i ∗ (Θ V )( h , . . . , h s ) = i ∗ (∆ M )( σ N ( h ) · h , . . . , σ N ( h s ) · h s ) , where s = b ( N ). Now the claim follows from Lemma 6.1. (cid:3) Edge-orientability of V N . We already noted in the proof of Corollary 5.8 that theveering triangulation V ab of the maximal free abelian cover of M can fail to be edge-orientable only if the torsion subgroup of H ( M ; Z ) is of even order. Of course, edge-orientability of the veering triangulation of an intermediate free abelian cover of M isa stronger condition: for many Dehn fillings N of M the veering triangulation V N isnot edge-orientable.Suppose that the compact core of M has b boundary components T , . . . , T b . Let γ i be a Dehn filling slope on T i . Consider the 3-manifold N = M ( γ , . . . , γ b )obtained from M by Dehn filling T i along γ i . A necessary condition for V N to beedge-orientable is that ω ( γ i ) = 1 for every i ∈ { , . . . , b } . In particular, if • V ab is edge-orientable, and • the preimage of every torus cusp of M under the covering map V or → V is connected,then for each i ∈ { , . . . , b } there is a unique slope γ i on T i such that V N is edge-orientable. In contrast, if • V ab is edge-orientable, and • the preimage of every torus cusp of M under the covering map V or → V is discon-nected,then any Dehn filling N of M determines an edge-orientable veering triangulation V N .There are 87047 veering triangulations in the Veering Census [7]. 62536 (71.8%) ofthem are not edge-orientable. Out of 62536 not edge-orientable veering triangulationsthere are 49637 (79.4%) whose edge-orientable double cover V or has the same numberof cusps as V . There are only 5854 (9.4%) whose edge-orientable double cover V or hastwice as many cusps as V . HE TAUT POLYNOMIAL AND THE ALEXANDER POLYNOMIAL 17 Consequences for the Teichm¨uller polynomial and fibred faces
The taut polynomial of a veering triangulation is related to an older polynomialinvariant of 3-manifolds called the
Teichm¨uller polynomial . It was introduced by Mc-Mullen in [13] and is associated to a fibred face of the Thurston norm ball . In thissection we interpret Theorem 6.3 as a result relating the Teichm¨uller polynomial of afibred face with the Alexander polynomial of the manifold. Then we discuss implica-tions of this theorem for fibred faces.7.1.
Fibred faces of the Thurston norm ball and the Teichm¨uller polynomial. If N is acompact, oriented, hyperbolic 3-manifold then H ( N ; R ) ∼ = H ( N, ∂N ; R ) is equippedwith the Thurston norm [20]. Its unit ball is a polytope with rational vertices [20,Theorem 2]. It may admit finitely many fibred faces . These are top dimensional facesof the Thurston norm unit ball which organise the ways in which N fibres over thecircle [20, Theorem 3].Let N be as above. Let F be a fibred face of the Thurston norm ball in H ( N ; R ).An integral primitive class from the interior of the cone R + · F determines a fibration S → N → S of N over the circle. N can be expressed as the mapping torus N = ( S × [0 , (cid:46) { ( x, ∼ ( ψ ( x ) , } of a pseudo-Anosov homeomorphism ψ : S → S [21, Proposition 2.6]. This homeo-morphism is also called the monodromy of the fibration S → N → S .The monodromy ψ determines a pair of 1-dimensional laminations in S which areinvariant under ψ . Let λ denote the stable lamination for ψ . The mapping torus of λ is a 2-dimensional lamination in N . We denote it by L . McMullen proved that upto isotopy the lamination L does not depend on the chosen primitive integral classin the interior of R + · F [13, Corollary 3.2]. Using this he defined the Teichm¨ullerpolynomial Θ F of F as the zeroth Fitting invariant of the module of transversals to thelamination induced by L in the maximal free abelian cover of N [13, Section 3]. Themain feature of this polynomial is that it encodes information on the stretch factors ofmonodromies of all fibrations lying in the fibred cone R + · F [13, Theorem 5.1].7.2. The Teichm¨uller polynomial and the taut polynomial.
Let N be a compact, ori-ented, hyperbolic 3-manifold which is fibred over the circle. Let F be a fibred faceof the Thurston norm ball in H ( N ; R ). By sing( F ) we denote the set { (cid:96) , . . . , (cid:96) k } ofclosed curves in N corresponding to the singular orbits of the suspension flow of themonodromy of any fibration in R + · F . This is well-defined by [6, Theorem 14.11].If we pick a fibration lying over F we can follow Agol’s algorithm [2, Section 4] toconstruct a layered veering triangulation V of M = N \ sing( F ). The fact that V does not depend on the chosen fibration from R + · F is proven in [16, Proposition 2.7].Landry, Minsky and Taylor observed that we can compute the Teichm¨uller polynomialof F using the taut polynomial of V . Lemma 7.1 (Proposition 7.2 of [11]) . Let N be a compact, oriented, hyperbolic 3-manifold. Let F be a fibred face of the Thurston norm ball in H ( N ; R ) . Denoteby V the veering triangulation of M = N \ sing( F ) associated to F . Let i ∗ : H M → H N be the epimorphism induced by the inclusion of M into N . Then Θ F = i ∗ (Θ V ) . (cid:3) Lemma 7.1 allows us to interpret Theorem 6.3 as a theorem which relates the Te-ichm¨uller polynomial with the Alexander polynomial.
Remark.
An algorithm to compute the Teichm¨uller polynomial using Lemma 7.1 isgiven in [18, Section 8].7.3.
The Teichm¨uller polynomial and the Alexander polynomial.
In this subsection weprove lemmas that allow us to interpret Theorem 6.3 as a strenghtening of Theorem 7.1of [13]. We summarise the findings in Corollary 7.5.Recall that given a fibred face F of the Thurston norm ball in H ( N ; R ) there is • a unique (up to isotopy) 2-dimensional lamination L in N associated to F [13,Corollary 3.2], • a unique veering triangulation V of M = N \ sing( F ) carrying fibres from R + · F punctured at the singularities of the monodromies [16, Proposition 2.7].The lamination L misses sing( F ) and therefore can also be seen as a lamination in M .First we establish a relation between edge-orientability of V and transverse orientabilityof L . Lemma 7.2.
Let F be a fibred face of the Thurston norm ball in H ( N ; R ) . Let • L be the 2-dimensional lamination in N associated to F , • V be the veering triangulation of M = N \ sing( F ) associated to F .Then V is edge-orientable if and only if L is transversely orientable.Proof. Let t be a tetrahedron of V . First note that the upper track in the two bottomfaces of t and in its two top faces differ by a splitting; see Figure 5. Performing thissplit continuously sweeps out a 2-dimensional complex in t . In Subsection 6.1 of [19]it is explained how to transform this complex into a branched surface B Ut . The union B U = (cid:83) t ∈ T B Ut forms a branched surface in M .Edge-orientability of V is equivalent to transverse orientability of B U . Moreover, thelamination L is fully carried by B U [19, Corollary 8.34]. Therefore L is transverselyorientable if and only if B U is transversely orientable. (cid:3) Consistently with the diagram (6.2) we denote the laminations in M N , M ab inducedby L ⊂ M by L N , L ab , respectively. The lamination L N can also be seen inside N ab . Corollary 7.3.
Let L and V be as in Lemma 7.2. The lamination L N is transverselyorientable if and only if V N is edge-orientable. (cid:3) Recall that by sing( F ) = { (cid:96) , . . . , (cid:96) k } we denote the singular orbits of the flowcanonically associated to F . In the language of Theorem 6.3 they correspond to thecore curves of the filling solid tori in N . In Theorem 6.3 we needed to assume thattheir classes in H N = H ( N ; Z ) /torsion are nontrivial. In the fibred setting we knowthat the classes [ (cid:96) j ] have nonzero algebraic intersection with every [ S ] ∈ int( R + · F ) ∩ H ( N, ∂N ; Z ). Therefore we get the following lemma. Lemma 7.4.
Let F be a fibred face of the Thurston norm ball in H ( N ; R ) . Let sing( F ) = { (cid:96) , . . . , (cid:96) k } be the singular orbits of the flow associated to F . Then for every j ∈ { , , . . . , k } the class [ (cid:96) j ] in H N is nontrivial. (cid:3) Using Lemma 7.1, Corollary 7.3, Lemma 7.4 and Theorem 6.3 we can derive relationsbetween the Teichm¨uller polynomial of F and the Alexander polynomial of N . Corollary 7.5.
Let N be a compact, oriented, hyperbolic 3-manifold with a fibred face F ⊂ H ( N, ∂N ; R ) of the Thurston norm ball. Let sing( F ) = { (cid:96) , . . . , (cid:96) k } . Denote by L the 2-dimensional lamination in N associated to F . Assume that the lamination in N ab induced by L is transversely orientable. Set s = b ( N ) and M = N \ sing( F ) .I. b ( M ) ≥ and HE TAUT POLYNOMIAL AND THE ALEXANDER POLYNOMIAL 19 (a) s ≥ . Then Θ F ( h , . . . , h s ) = ∆ N ( σ N ( h ) · h , . . . , σ N ( h s ) · h s ) · k (cid:89) j =1 ([ (cid:96) j ] − . (b) s = 1 . Let h denote the generator of H N . • If ∂N (cid:54) = ∅ then Θ F ( h ) = ( h − − · ∆ N ( σ N ( h ) · h ) k (cid:89) j =1 ([ (cid:96) j ] − . • If N is closed then Θ F ( h ) = ( h − − · ∆ N ( σ N ( h ) · h ) · k (cid:89) j =1 ([ (cid:96) j ] − . II. b ( M ) = 1 . Let h denote the generator of H N ∼ = H M .(a) If ∂N (cid:54) = ∅ then Θ F ( h ) = ∆ N ( σ N ( h ) · h ) . (b) If N is closed set (cid:96) = (cid:96) and then Θ F ( h ) = ( h − − ([ (cid:96) ] − · ∆ N ( σ N ( h ) · h ) . (cid:3) Corollaries of the main theorem in the fibred case.
Corollary 7.5 has a coupleof interesting consequences for the Teichm¨uller polynomials and fibred faces of theThurston norm ball.
Corollary 7.6.
Let N be a compact, oriented, hyperbolic and fibred 3-manifold. Let F , F be two fibred faces of the Thurston norm ball in H ( N ; R ) . For i = 1 , denoteby L i the lamination associated to F i and by Θ i the Teichm¨uller polynomial of F i . Ifthe induced laminations L N , L N in N ab are transversely orientable then Θ ( h , . . . , h s ) = P ( h , . . . , h s ) · Θ ( ± h , . . . , ± h s ) where P is a product of factors of the form ( h − and ( h − − , h ∈ H N . (cid:3) Corollary 7.6 says that when a 3-manifold N has two fibred faces which determinetransversely orientable laminations in the maximal free abelian cover N ab , then theirTeichm¨uller polynomials are almost the same. Below we explain why this is not asurprising behaviour.First note that by fixing a cohomology class φ ∈ H ( N ; Z ) we can transform ev-ery Laurent polynomial Q ∈ Z [ H N ] into a Laurent polynomial in one variable via specialisation . More precisely, for Q = (cid:88) h ∈ H N a h · h, we define the specialisation of Q at φ by setting Q ( z φ ) = (cid:88) h ∈ H N a h · z φ ( h ) . A cohomology class φ ∈ H ( N ; Z ) from the interior of a cone over a fibred face F determines a fibration of N over the circle. Let ψ denotes the monodromy of thisfibration. McMullen showed that the largest real root of Θ F ( z φ ) is equal to the stretchfactor of ψ [13, Theorem 5.1]. On the other hand, the largest in the absolute value realroot of ∆ N ( z φ ) is equal to the homological stretch factor of ψ [15, Assertion 4]. It isknown that these two numbers are equal if and only if the 1-dimensional laminations in S which are invariant under ψ are orientable [3, Lemma 4.3]. In particular, it isclear that when the lamination L associated to a fibred face F is transversely orientable,then the Teichm¨uller polynomial of F and the Alexander polynomial of N have to bevery tightly related; for all cohomology classes φ ∈ int( R + · F ) ∩ H ( N ; Z ) the largestreal roots of their specialisations at φ have to be equal up to the sign.When the lamination L is not transversely orientable it is still possible that itsintersection λ with a given fibre is an orientable 1-dimensional lamination. In this casethe monodromy ψ must reverse the orientation of λ . In fact, Corollary 7.5 allows usto identify cohomology classes determining fibrations with this property. Corollary 7.7.
Let N be a compact, oriented, hyperbolic, fibred 3-manifold with b ( N ) > .Let F be a fibred face of the Thurston norm ball in H ( N ; R ) . Let M = N \ sing( F ) .Assume that the lamination L associated with F induces a transversely orientable lami-nation in M ab . Then in the interior of the cone R + · F there are rational rays determiningfibrations for which the intersection of the fibre with L is an orientable 1-dimensionallamination.Proof. Let i ∗ : H ( N ; R ) → H ( M ; R ) be the homomorphism induced by the inclusionof M into N . There is a fibred face ˚ F in H ( M ; R ) such that i ∗ ( R + · F ) ⊂ R + · ˚ F . The fibres dual to cohomology classes from i ∗ ( R + · F ) are obtained from the fibres dualto cohomology classes from R + · F by puncturing them at the singularities of the mon-odromy. Therefore it is enough to find cohomology classes in i ∗ ( R + · F ) correspondingto fibrations of M for which the intersection of L with the fibre is orientable.Let r = b ( M ). Corollary 7.5 and Lemma 7.1 imply thatΘ ˚ F ( g , . . . , g r ) = ∆ M ( σ ( g ) · g , . . . , σ ( g r ) · g r ) . We can reorder the basis elements for H M = H ( M ; Z ) /torsion so thatΘ ˚ F ( g , . . . , g r ) = ∆ M ( − g , − g , . . . , − g j , g j +1 , g j +2 , . . . , g r ) , for some j ∈ { , , . . . , r − } . Let ( η , . . . , η r ) be the basis of H ( M ; R ) dual to thebasis ( g , . . . , g r ) of H ( M ; R ). If j (cid:54) = 0 consider a primitive class φ ∈ int( R + · F ) suchthat i ∗ φ = a η + . . . + a r η r , satisfies a i / ∈ Z for i = 1 , , . . . , j and a i ∈ Z for i = j + 1 , j + 2 , . . . , r . We can choosesuch because we assumed that b ( N ) >
1. ThenΘ ˚ F ( z i ∗ φ ) = ∆ M ( − z i ∗ φ ) . If j = 0 then for any primitive integral class φ ∈ int( R + · F ) we haveΘ ˚ F ( z i ∗ φ ) = ∆ M ( z i ∗ φ ) . In both cases we get that the largest in the absolute value real roots of Θ ˚ F ( z i ∗ φ )and ∆ M ( z i ∗ φ ) are equal up to the sign. Therefore the invariant laminations of themonodromy of the fibration of M determined by i ∗ φ are orientable by Lemma 4.3of [3]. (cid:3) HE TAUT POLYNOMIAL AND THE ALEXANDER POLYNOMIAL 21
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