Dynamics of Twisted Alexander Invariants
aa r X i v : . [ m a t h . G T ] A p r Dynamics of Twisted Alexander Invariants
Daniel S. SilverSusan G. Williams ∗ Department of Mathematics and Statistics, University of South Alabama
November 19, 2018
Abstract
The Pontryagin dual of the based Alexander module of a linktwisted by a GL N Z representation is an algebraic dynamical systemwith an elementary description in terms of colorings of a diagram. Itstopological entropy is the exponential growth rate of the number oftorsion elements of twisted homology groups of abelian covers of thelink exterior.Total twisted representations are introduced. The twisted Alexan-der polynomial obtained from any nonabelian parabolic SL C repre-sentation of a 2-bridge knot group is seen to be nontrivial. The zerosof any twisted Alexander polynomial of a torus knot corresponding toa parabolic SL C representation or a finite-image permutation repre-sentation are shown to be roots of unity. Keywords:
Knot, twisted Alexander polynomial, Fox coloring, Mahler mea-sure The Alexander polynomial ∆ k ( t ), the first knot polynomial, was a result ofJ.W. Alexander’s efforts during 1920–1928 to compute torsion numbers b r ,the orders of the torsion subgroups of H M r , where M r is the r -fold cycliccover of S branched over k . Here a combinatorial approach to a topologicalproblem led to new algebraic invariants such as the Alexander module andits associated polynomials. ∗ Both authors partially supported by NSF grant DMS-0706798. Mathematics Subject Classification: Primary 57M25; secondary 37B40.
1s an abelian invariant, ∆ k can miss a great deal of information. Forexample, it is well known that there exist infinitely many nontrivial knotswith trivial Alexander polynomial. In 1990 X.S. Lin introduced a moresensitive invariant using information from nonabelian representations of theknot group [22]. Later, refinements of these twisted Alexander polynomialswere described by M. Wada [39], P. Kirk and C. Livingston [19] and J. Cha[2]. We examine twisted Alexander modules from the perspective of algebraicdynamics. Pontryagin duality converts a finitely generated Z [ t ± , . . . , t ± d ]-module M into a compact abelian group ˆ M = Hom( M , T ), where T = R / Z is the additive circle group. Multiplication in M by t , . . . , t d becomes d com-muting homeomorphisms of ˆ M , giving rise to a Z d -action σ : Z d → Aut ˆ M .Dynamical invariants of σ such as periodic point counts and topologicalentropy provide invariants of the module M .In previous work [33], the authors considered the case in which M is the(untwisted) based Alexander module A of a link ℓ of d components and t , . . . , t d correspond to meridians. When the d -variable Alexander poly-nomial ∆ ℓ is nonzero, its logarithmic Mahler measure coincides with thetopological entropy h ( σ ), a measure of the complexity of σ . Furthermore, h ( σ ) is seen to be the exponential growth rate of torsion numbers b Λ as-sociated to ℓ , where Λ is a finite-index sublattice of Z d expanding in all d directions.Given a representation γ from the link group π = π ( S \ ℓ ) to GL N Z ,a based twisted Alexander module A γ and twisted Alexander polynomial∆ ℓ,γ are defined for which analogous results hold. In particular, we give ahomological interpretation for the Mahler measure of ∆ ℓ,γ .Representations to GL N Z arise naturally from parabolic SL C represen-tations as total twisted representations. Parabolic representations of 2-bridgeknot and link groups provide many examples. We show that for any 2-bridgeknot k and nonabelian parabolic representation γ , the polynomial ∆ k,γ isnontrivial.The dynamical approach here is natural for fibered knots. In such a case,ˆ A γ is a finite-dimensional torus with an automorphism σ determined by themonodromy. The homology eigenvalues of σ are the zeroes of the twistedAlexander polynomial. We prove that any parabolic SL C representationof a torus knot group yields a twisted Alexander polynomial with trivialMahler measure.We are grateful to the Institute for Mathematical Sciences at StonyBrook University and the Department of Mathematics of the George Wash-2ngton University for their hospitality and support during the fall of 2007,when much of this work was done. We thank Abhijit Champanerkar, StefanFriedl, Jonathan Hillman, Paul Kirk, Mikhail Lyubich and Kunio Murasugifor comments and suggestions. We describe twisted Alexander modules of knots and links using extendedFox colorings of diagrams. This unconventional approach is both elementaryand compatible with the dynamical point of view that we wish to promote.We begin with the untwisted case for knots. Twisted homology willrequire only a minor modification. Later, we describe the changes that areneeded for general links.Let D be a diagram of an oriented knot k with arcs indexed by { , , . . . , q } .Recall that T = R / Z is the additive circle group. Regard its elementsas “colors.” A based dynamical coloring of D is a labeling α of the dia-gram that assigns to the i th arc the color sequence α i = ( α i,n ) ∈ T Z with α = ( . . . , , , , . . . ) (see Figure 1). At each crossing, we require: α i,n + α j,n +1 = α k,n + α i,n +1 (2.1)for all n ∈ Z , where the i th arc is the overcrossing arc and the j th and k tharcs are the left and right undercrossing arcs, respectively. Equivalently: α i + σα j = α k + σα i , where σ is the shift map , the left coordinate shift on the second subscript.The collection of all based dynamical colorings is denoted by Col ( D ).It is a closed subgroup of ( T Z ) q ∼ = ( T q ) Z , invariant under σ , which acts as acontinuous automorphism. The pair (Col ( D ) , σ ) is an algebraic dynamicalsystem, a compact topological group and continuous automorphism, whichwe call the based coloring dynamical system of D .If D ′ is another diagram for k , then its associated dynamical system(Col ( D ′ ) , σ ′ ) is topologically conjugate to (Col ( D ) , σ ) via a continuousisomorphism h : Col ( D ′ ) → Col ( D ) such that σ ◦ h = h ◦ σ ′ . This isthe natural equivalence relation on algebraic dynamical systems. Hence(Col ( D ) , σ ) is an invariant of k . Definition 2.1.
The based dynamical coloring system (Col ( k ) , σ ) of k is(Col ( D ) , σ ), where D is any diagram for k .3 kj ! !! Figure 1: Color sequences at a crossing
Definition 2.2.
An element α = ( α i,n ) ∈ Col ( k ) has period r if σ r α = α ;equivalently, α i,n + r = α i,n for all i, n . The set of all period r elements isdenoted by Fix σ r . Proposition 2.3. [31]
For any knot,
Fix σ is trivial. If α = ( α i,n ) ∈ Col ( k ) has period 2, then α i + σα i is fixed by σ forevery i , and hence σα i = − α i ; that is, α i,n +1 = − α i,n , for all i, n . Each α i,n is determined by a single color α i, ∈ T . The coloring condition (2.1) thenbecomes 2 α i, = α j, + α k, . Taking each α i, to be in Z /p for some positive integer p , the above becomesthe well-known Fox p -coloring condition. Since each group Z /p embeds nat-urally in T , we define a Fox T -coloring to be any T -coloring satisfying thiscondition. The group of Fox T -colorings of a diagram modulo monochro-matic colorings is isomorphic to the group of period-2 points of Col ( k ).For knots, all Fox T -colorings are Fox p -colorings for some p , since∆ k ( −
1) is nonzero and annihilates all period 2 points (see [31]). It is wellknown that the group of Fox p -colorings of a diagram modulo monochro-matic colorings is isomorphic to the homology H ( M ; Z /p ) of the 2-foldcyclic cover with Z /p coefficients.For each r , we denote by X r the r -fold cyclic cover of the knot exterior X = S \ k . We denote its universal abelian cover by X ∞ , and regard H X ∞ as a finitely generated Z [ t ± ]-module. Since H X r ∼ = H M r ⊕ Z , the torsionsubgroups of H X r and H M r are isomorphic, and we denote their orderby b r . We denote the rank of H M r (the dimension of the Q -vector space H M r ⊗ Q ) by β r . Proposition 2.4. [32]
For any knot k , ( k ) is Pontryagin dual to H X ∞ . The shift σ is dual to the merid-ian action of t . (ii) Fix σ r consists of b r tori, each of dimension β r . Definition 2.5. (1) The logarithmic Mahler measure of a nonzero polyno-mial p ( t ) = c s t s + . . . + c t + c ∈ C [ t ] is m ( p ) = log | c s | + s X i =1 max { log | λ i | , } , where λ , . . . , λ s are the roots of p ( t ).(2) More generally, if p ( t , . . . , t d ) is a nonzero polynomial in d variables,then m ( p ) = Z · · · Z log | p ( e πiθ , . . . , e πiθ d ) | dθ · · · dθ d . (2.2) Remark 2.6.
The integral in Definition 2.5 can be singular, but neverthelessconverges. The agreement between (1) and (2) in the case that d = 1 isassured by Jensen’s formula (see [6] or [30]). Proposition 2.7. [33]
For any knot k , the topological entropy h ( σ ) is givenby h ( σ ) = lim r →∞ r log b r = m (∆ k ( t )) . Remark 2.8.
Proposition 2.7 was proved earlier for the subsequence oftorsion numbers b r for which β r = 0 by R. Riley [29] and also F. Gonz´alez-Acu˜na and H. Short [11]. The authors showed that one need not skip overtorsion numbers for which β r vanishes. They generalized the result by show-ing that the logarithmic Mahler measure of any nonvanishing multivariableAlexander polynomial ∆ ℓ ( t , . . . , t d ) of a d -component link is again a limit ofsuitably defined torsion numbers. Both generalizations require a deep the-orem from algebraic dynamics, Theorem 21.1 of K. Schmidt’s monograph[30], which extends a theorem of D. Lind, Schmidt and T. Ward [23].Consider now a linear representation γ : π = π ( S \ k ) → GL N Z . Let x i denote the meridian generator of π corresponding to the i th arc of a diagram D for k . We let X i denote the image matrix γ ( x i ). The representation γ induces an action of π on the N -torus T N . Our colors will now be elementsof T N . 5 efinition 2.9. A based γ -twisted dynamical coloring of D is a labeling α of the arcs of D by color sequences α i = ( α i,n ) ∈ ( T N ) Z , with the 0tharc labeled by the sequence of zero vectors. The crossing condition (2.1) isreplaced by: α i,n + X i α j,n +1 = α k,n + X k α i,n +1 . We will write this as α i + X i σα j = α k + X k σα i , where again σ is the shift map , the coordinate shift on the Z -coordinates,and Xα i = ( Xα i,n ). Remark 2.10.
More generally, one can consider representations γ from π to GL N R , where Z ⊆ R ⊆ Q . In this case, T N is replaced by the N -dimensional solenoid ˆ R N , where ˆ R = Hom( R, T ). We will say more aboutthis below.The generalization from knots to links is natural. Let ℓ = ℓ ∪ · · · ∪ ℓ d bean oriented link of d components. Let D be a diagram. The abelianization ǫ : π → Z d takes the meridian generator x i corresponding to the i th arc of D to a standard basis vector ǫ ( x i ) ∈ { e , . . . , e d } for Z d .A based γ -twisted dynamical coloring of a diagram D is a labeling α of arcs by α i = ( α i, n ) ∈ ( T N ) Z d , where n = n e + · · · + n d e d ∈ Z d . For m ∈ Z d , we write σ m for the coordinate shift by m such that σ m α i has n thcoordinate equal to α i, n + m . As before, the 0th arc is labeled with the zerosequence. At each crossing we require α i, n + X i α j, n + ǫ ( x i ) = α k, n + X k α i, n + ǫ ( x k ) . (2.3)The crossing condition can be written more compactly as: α i + X i σ ǫ ( x i ) α j = α k + X k σ ǫ ( x k ) α i . The collection Col γ ( D ) of all based γ -twisted dynamical colorings is again acompact space, this time invariant under the Z d -action σ = ( σ n ), generatedby commuting automorphisms σ e , . . . , σ e d . Proposition 2.11.
Let ℓ be any oriented link. Up to topological conjugacyby a continuous group isomorphism, the dynamical system (Col γ ( D ) , σ ) isindependent of the diagram D for ℓ .
6e denote this twisted coloring system by Col γ ( ℓ ), or Col γ when thelink ℓ is understood.Proposition 2.11 can be proven in a straightforward manner by checkinginvariance under Reidemeister moves. Alternatively, it follows from Theo-rem 3.5.The topological entropy h ( σ ) of a Z d -action σ : Z d → Aut Y , where Y is a compact metric space and σ is a homomorphism to its group of home-omorphisms, is a measure of its complexity. We briefly recall a definitionfrom [30], referring the reader to that source for details. If U is an opencover of Y , denote the minimum cardinality of a subcover by N ( U ). If Q = Q dj =1 { b j , . . . , b j + l j − } ⊆ Z d is a rectangle, let | Q | be the cardinalityof Q . Finally, let h Q i = min j =1 ,...,d l j . Then h ( σ ) = sup U h ( σ, U ) , where U ranges over all open covers of Y , and h ( σ, U ) = lim h Q i→∞ | Q | log N ( ∨ n ∈ Q σ − n ( U )) . Here ∨ denotes common refinement of covers. Existence of the limit isguaranteed by subadditivity of log N ; that is, log N ( U ∨ V ) ≤ log N ( U ) +log N ( V ) for all open covers U , V of Y . Let π = h x , . . . , x q | r , . . . , r q i be a Wirtinger presentation of π = π ( S \ ℓ ),where ℓ = ℓ ∪ · · · ∪ ℓ d is an oriented link, and x is a meridian of ℓ .As above, let ǫ : π → Z d be the abelianization homomorphism. Wewill make frequent use of the natural isomorphism from the free module Z d to the multiplicative free abelian group h t , . . . , t d | [ t i , t j ] i , identifying n = ( n , . . . , n d ) ∈ Z d with the monomial t n = t n · · · t n d d .Assume that γ : π → GL N R is a representation, where R is a NoetherianUFD. Then V = R N is a right R [ π ]-module. We regard R [ Z d ] as the ring ofLaurent polynomials in t , . . . , t d with coefficients in R . We give R [ Z d ] ⊗ R V the structure of a right R [ π ]-module via( p ⊗ v ) · g = ( pt ǫ ( g ) ) ⊗ ( vγ ( g )) ∀ g ∈ π. (3.1)We denote the exterior of ℓ , the closure of S minus a regular neighbor-hood of ℓ , by X . Its universal cover will be denoted by ˜ X , and its universalabelian cover by X ′ . 7et C ∗ ( ˜ X ) denote the cellular chain complex of ˜ X with coefficients in R . It is a free left R [ π ] module, with basis obtained by choosing a liftingfor each cell of X . For purposes of calculation it is convenient to collapse X to a 2-complex with a single 0-cell ∗ , 1-cells x , . . . , x q corresponding tothe generators of π , and 2-cells r , . . . , r q corresponding to the relators. Wedenote the chosen lifted cells in ˜ X by the same symbols.We consider the γ -twisted chain complex C ∗ ( X ′ ; V γ ) of R [ Z d ]-modules C ∗ ( X ′ ; V γ ) = ( R [ Z d ] ⊗ R V ) ⊗ R [ π ] C ∗ ( ˜ X ) . (3.2)When it is clear what representation is being considered, we will shorten thenotation V γ to V .The γ -twisted chain groups C ∗ ( X ′ ; V ), and hence the γ -twisted homol-ogy groups H ∗ ( X ′ ; V ), are finitely generated R [ Z d ]-modules. The homologydepends on the representation only up to conjugacy. The action of Z d isgiven by t n · (1 ⊗ v ⊗ z ) = t n ⊗ v ⊗ z. Remark 3.1.
Notation for twisted homology varies widely among authors.We have attempted to keep ours as uncluttered as possible. The notationalreference to X ′ is justified as follows. The representation γ : π → GL N R restricts to a representation of the commutator subgroup π ′ . Then V isa right R [ π ′ ]-module. Shapiro’s Lemma (see [1], for example) implies that H ∗ ( X ′ ; V ) is isomorphic to the homology groups resulting from the twistedchain complex V ⊗ R [ π ′ ] C ∗ ( X ′ ).For 0 ≤ i ≤ q , ∂ (1 ⊗ v ⊗ x i ) = 1 ⊗ v ⊗ ∂ x i = 1 ⊗ v ⊗ ( x i ∗ − ∗ )= 1 ⊗ v ⊗ x i ∗ − ⊗ v ⊗ ∗ . Using the π -action given by (3.1), we can write ∂ (1 ⊗ v ⊗ x i ) = t ǫ ( x i ) ⊗ vX i ⊗ ∗ − ⊗ v ⊗ ∗ . (We remind the reader that X i denotes γ ( x i ).)If r = x i x j x − i x − k is a Wirtinger relator, then one checks in a similarfashion that ∂ (1 ⊗ v ⊗ r ) is given by1 ⊗ v ⊗ x i + t ǫ ( x i ) ⊗ vX i ⊗ x j − t ǫ ( x i x j x − i ) ⊗ vX i X j X − i ⊗ x i − t ǫ ( x i x j x − i x − k ) ⊗ vX i X j X − i X − k ⊗ x k . x i x j x − i x − k is trivial in π , we can express the relation ∂ (1 ⊗ v ⊗ r ) = 0more simply:1 ⊗ v ⊗ x i + t ǫ ( x i ) ⊗ vX i ⊗ x j = 1 ⊗ v ⊗ x k + t ǫ ( x k ) ⊗ vX k ⊗ x i . (3.3)An elementary ideal I ℓ,γ and γ -twisted Alexander polynomial ∆ ℓ,γ aredefined for H ( X ′ ; V ). The ideal I ℓ,γ is generated by determinants of allmaximal square submatrices of any presentation matrix, while ∆ ℓ,γ is thegreatest common divisor of the determinants, well defined up to multiplica-tion by a unit in R [ Z d ]. The independence of I ℓ,γ and ∆ ℓ,γ of the choice ofpresentation matrix is well known (see [4], p. 101, for example).Following [19], we write the chain module C ( X ′ ; V ) = W ⊕ Y , where W , Y are freely generated by lifts of x and x , . . . , x q , respectively. Ourcomplex of γ -twisted chains can be written:0 → C ( X ′ ; V ) ∂ = a ⊕ b −→ W ⊕ Y ∂ = c + d −→ C ( X ′ ; V ) → . (3.4) Definition 3.2.
The γ - twisted Alexander module A γ is the cokernel of ∂ .The based γ - twisted Alexander module A γ is the cokernel of b .Let m be a meridianal link about the component ℓ . We can regard m asa subspace of the link exterior X , and the inclusion map induces an injectionon fundamental groups. The representation γ restricts, and we can considerthe relative γ -twisted homology H ( X ′ , m ′ ; V ), where m ′ is the lift of m to the infinite cyclic cover X ′ . We can assume that cell complexes for X and m both have a single, common 0-cell. From the relative chain complex C ∗ ( X ′ , m ′ ; V ) = C ∗ ( X ′ ; V ) /C ∗ ( m ′ ; V ), we obtain the following homologicalinterpretation of A γ . Proposition 3.3.
The based γ -twisted Alexander module A γ is isomorphicto H ( X ′ , m ′ ; V ) . Although A γ is an invariant of the (ordered) link by Proposition 3.3, itsdependence on the choice of component ℓ is critical. Definition 3.4.
The γ -twisted coloring polynomial D ℓ,γ is 0th elementarydivisor ∆ ( A γ ).Since A γ has a square matrix presentation, D ℓ,γ is relatively easy tocompute. The quotient D ℓ,γ / det( t ǫ ( x ) X − I ), denoted by W ℓ,γ , is an in-variant introduced by M. Wada [39]. It is well defined up to multiplicationby a unit in R [ Z d ]. Wada’s invariant is defined generally for any finitelypresented group, given a homomorphism onto the integers and a linear rep-resentation. 9 heorem 3.5. Assume that Z ⊂ R ⊂ Q . The dual group ˆ A γ is topologicallyconjugate to the based coloring dynamical system Col γ ( ℓ ) .Proof. Elements of the dual group ˆ A γ are certain assignments α : t n ⊗ v ⊗ x i v · α i, n ∈ T , where α i, n is a vector in ˆ R N for 1 ≤ i ≤ q and n ∈ Z d . Equivalently, α is afunction taking each x i to an element α i, n ∈ ( ˆ R N ) Z d . Since ˆ A γ is the dualof the cokernel of b , such an assignment α is in ˆ A γ if and only if α ◦ b = 0.In view of Equation (3.3), the condition amounts to v · α i, n + vX i · α j, n + ǫ ( x i ) = v · α k, n + vX k · α i, n + ǫ ( e k ) , for every v ∈ V . Equivalently, α i, n + X i α j, n + ǫ ( x i ) = α k, n + X k α i, n + ǫ ( e k ) , which is the condition (2.3) for γ -twisted colorings.From the chain complex (3.4), one obtains an exact sequence:0 → H ( X ′ ; V ) → A γ → coker c f → H ( X ′ ; V ) → , where the maps are natural. (This is shown in [19] in the case that d = 1and R is a field. The general statement is proven in [13].)Exactness allows us to write a short exact sequence that expresses A γ as an extension of H ( X ′ ; V )0 → H ( X ′ ; V ) → A γ → ker f → . (3.5)Pontryagin then duality induces an exact sequence0 → [ ker f → Col γ → \ H ( X ′ ; V ) → . Recall that the shift map on Col γ is denoted by σ . We denote the shiftsof \ H ( X ′ ; V ) and [ ker f by σ ′ and σ ′′ , respectively. By Yuzvinskii’s AdditionFormula (see Theorem 14.1 of [30], for example), h ( σ ) = h ( σ ′ ) + h ( σ ′′ ) . (3.6)The module ker f can be described more explicitly. While H ( X ′ ; V )consists of the cosets of C ( X ′ ; V ) modulo the entire image of ∂ , the module10oker c is generally larger. Its elements are the cosets of C ( X ′ ; V ) modulothe image of ∂ restricted to the γ -twisted 1-chains generated only by 1 ⊗ e s ⊗ x . (As above, e s ranges over a basis of Z N .) It follows that ker f ∼ =im ∂ / im c. The exact sequence (3.5) implies that D ℓ,γ = ∆ ℓ,γ · ∆ (ker f ) . (3.7)On the other hand, the short exact sequence0 → ker f → coker c f → H ( X ′ ; V ) → (coker c ) = ∆ (ker f ) · ∆ ( H ( X ′ ; V )) . (3.9)The elementary divisor ∆ (coker c ) is the determinant of t ǫ ( x ) X − I . Notethat ∆ ( H ( X ′ ; V ) is a factor.Together equations (3.7) and (3.9) imply the following relationship pre-viously proven in [19]. Proposition 3.6.
The γ -twisted Alexander polynomial and Wada invariantare related by ∆ ℓ,γ = W ℓ,γ · ∆ ( H ( X ′ ; V )) . In earlier work [33], we interpreted the Mahler measure of any nonzerountwisted Alexander polynomial of a link as an exponential growth rate oftorsion numbers associated to finite-index abelian branched covers of thelink. Generalizing such a result for twisted Alexander polynomials presentschallenges. First, the representation γ does not induce representations of thefundamental groups of branched covers of the link; we work with unbranchedcovers instead. Second, the relationship between the periodic points of thecoloring dynamical system and torsion elements of the unbranched coverhomology groups, which we exploited, becomes more subtle when twistingis introduced. Third, H ( X ′ ; V ) has no obvious square matrix presentation,making the computation of the entropy h ( σ ′ ) more difficult.Let Λ ⊆ Z d be subgroup. An element α ∈ Col γ ( ℓ ) is a Λ- periodic point if σ n α = α for every n ∈ Λ. More explicitly, if α is given by a based γ -twisteddynamical coloring α = ( α i, n ) ∈ ( ˆ R qN ) Z d of a diagram for ℓ , then α is aΛ-periodic point if α i, n + m = α i, n , for every m ∈ Λ and 1 ≤ i ≤ q . Wedenote the subgroup of Λ-periodic points by Fix Λ σ .Associated to the homomorphism π ǫ −→ Z d → Z d / Λ, where the secondmap is the natural projection, there is a regular covering space X Λ of X Z d / Λ. The representation γ restricts to π X Λ , which is a subgroup of π . (We denote the restricted representationalso by γ .) By Shapiro’s Lemma, H ∗ ( X Λ ; V ) is isomorphic to the homologygroups that result from the chain complex (3.2) when Z d is replaced by thequotient Z d / Λ. Henceforth, we assume that R is the ring of fractions S − Z of Z by a mul-tiplicative subset S generated by a finite set of primes p , . . . , p s . The ring R can be described as the set of rational numbers of the form m/p µ · · · p µ s s ,where m ∈ Z and µ , . . . , µ s are nonnegative integers. The case S = ∅ ,whereby S − Z = Z , is included.When the index [ Z d : Λ] = | Z d / Λ | is finite, H ∗ ( X Λ ; V ) is a finitelygenerated R -module. Since R is a PID (see p. 73 of [21], for example), wehave a decomposition H ( X Λ ; V ) ∼ = R β Λ ⊕ R/ ( q n ) ⊕ · · · ⊕ R/ ( q n t t ) , (3.10)where q , . . . , q t can be taken to be prime integers. We note that R/ ( q n i i ) ∼ = Z / ( q n i i ) ⊗ Z R .The following lemma is well known. (See p. 312 of [5], for example.) Lemma 3.7.
For prime q , Z / ( q n ) ⊗ Z R = ( Z / ( q n ) if q / ∈ S, if q ∈ S. In view of Lemma 3.7 we may as well assume that no prime q i in thedecomposition (3.10) is in S . Definition 3.8.
The γ - twisted Λ -torsion number b Λ ,γ is the order of the Z -torsion subgroup of H ( X Λ ; V ). When d = 1 and Λ = r Z , we write b r,γ .Since R β Λ is Z -torsion-free, we have b Λ ,γ = q n · · · q n t t . By additivity ofthe Pontryagin dual operator and the well-known fact that the dual of anyfinite abelian group is isomorphic to itself, we have \ H ( X Λ ; V ) ∼ = ˆ R β Λ ⊕ Z / ( q n ) ⊕ · · · ⊕ Z / ( q n t t ) . If R = Z , then ˆ R is the circle. Otherwise, ˆ R is a solenoid; it is connectedbut not locally connected. Corollary 3.9.
The number of connected components of \ H ( X Λ ; V ) is equalto the γ -twisted Λ -torsion number b Λ ,γ . Z d , define h Λ i to be min {| n | : 0 = n ∈ Λ } .Our first theorem on growth rates of twisted torsion numbers assumes that R = Z (that is, S is empty). The second requires that d = 1, but allows S to be nonempty. We recall that m ( p ) is the logarithmic Mahler measure of p . We extend to rational functions by defining m ( p/q ) = m ( p ) − m ( q ) . Werecall also that the shift σ ′′ was defined following (3.5). Theorem 3.10.
Let ℓ = ℓ ∪ · · · ∪ ℓ d ⊆ S be an oriented link and γ : π → GL N Z a representation. Assume that D ℓ,γ = 0 . Then (1) lim sup h Λ i→∞ | Z d / Λ | log b Λ ,γ = h ( σ ) − h ( σ ′′ );(2) m ( W ℓ,γ ) ≤ lim sup h Λ i→∞ | Z d / Λ | log b Λ ,γ ≤ m ( D ℓ,γ ) . Here the limit is taken over all finite-index subgroups Λ of Z d , and lim sup can be replaced by an ordinary limit when d = 1 . Corollary 3.11. If H ( X ′ ; V ) has a square presentation matrix or if thespectral radius of X does not exceed 1, then m (∆ ℓ,γ ) = lim sup h Λ i→∞ | Z d / Λ | log b Λ ,γ . If we restrict to the case of a knot k (that is, d = 1), then we can saymore. Corollary 3.12.
Assume that k ⊂ S is a knot, and γ : π → GL N Z is arepresentation such that ∆ k,γ = 0 . Then m (∆ k,γ ) = lim r →∞ r log b r,γ . Corollary 3.12 generalizes for the rings S − Z . First, we express D k,γ uniquely as d ˜ D k,γ , where d ∈ R = S − Z and ˜ D k,γ is a primitive polynomialin Z [ t ± ]. Express ∆ k,γ similarly as δ ˜∆ k,γ . By Gauss’s Lemma, ˜∆ k,γ divides˜ D k,γ . Moreover, δ divides d . Theorem 3.13. If d is a unit in R , then m ( ˜∆ k,γ ) = lim r →∞ r log b r,γ . roof of Theorem 3.10. We regard the basepoint ∗ as a subspace of X . Therepresentation γ : π → GL N Z restricts trivially, and hence we can considerthe γ -twisted chain complex C ∗ ( ∗ ′ ; V ) = { C ( ∗ ′ ; V ) } , defined as in (3.2),where ∗ ′ denotes the preimage of ∗ ∈ X under the projection map X ′ → X .Clearly, C ( ∗ ′ ; V ) ∼ = Z [ Z d ] ⊗ Z V . Since C ∗ ( ∗ ′ ; V ) is a sub-chain complex of C ∗ ( X ′ ; V ), the quotient chain complex C ∗ ( X ′ , ∗ ′ ; V ) = C ∗ ( X ′ ; V ) /C ∗ ( ∗ ′ ; V )is defined. From it, relative γ -twisted homology groups H ∗ ( X ′ , ∗ ′ ; V ) aredefined. It is immediate that H ( X ′ , ∗ ′ ; V ) ∼ = A γ .Let J be the ideal of Z [ Z d ] generated by all elements of Λ. Define atwisted chain complex of Z [ Z d ]-modules J C ∗ ( X ′ , ∗ ′ ; V ) = ( J ⊗ Z V ) ⊗ Z [ π ] C ∗ ( ˜ X ) . Then
J C ∗ ( X ′ , ∗ ′ ; V ) can be considered as a subcomplex of C ∗ ( X ′ , ∗ ′ ; V ).We denote the quotient chain complex by ¯ C ∗ ( X ′ , ∗ ′ ; V ). Shapiro’s Lemmaimplies that the homology groups of the latter chain complex are isomor-phic to H ∗ ( X Λ , ∗ Λ ; V ), where ∗ Λ denotes the preimage of ∗ ∈ X under theprojection map X Λ → X .The short exact sequence of chain complexes0 → J C ∗ ( X ′ , ∗ ′ ; V ) → C ∗ ( X ′ , ∗ ′ ; V ) → ¯ C ∗ ( X ′ , ∗ ′ ; V ) → · · · → H ( J C ∗ ( X ′ , ∗ ′ ; V )) i −→ A γ → H ( X Λ , ∗ Λ ; V ) → . Since every chain in
J C ( X ′ , ∗ ′ ; V ) and C ( X ′ , ∗ ′ ; V ) is a cycle, it followsthat the image of i is J A γ . Hence by exactness, H ( X Λ , ∗ Λ ; V ) ∼ = A γ /J A γ . (3.11)Now consider the short exact sequence of chain complexes0 → ¯ C ∗ ( ∗ ′ ; V ) → ¯ C ∗ ( X ′ ; V ) → ¯ C ∗ ( X ′ , ∗ ′ ; V ) → , where ¯ C ∗ ( ∗ ′ ; V ) and ¯ C ∗ ( X ′ ; V ) are defined as ¯ C ∗ ( X ′ , ∗ ′ ; V ) was, by replac-ing the Z [ Z d ] with J . The sequence gives rise to a long exact sequence ofhomology groups:0 → H ( X Λ ; V ) → H ( X Λ , ∗ Λ ; V ) ¯ ∂ −→ Im ¯ ∂ → . (3.12)The image of ¯ ∂ is an Z -submodule of ¯ C ( ∗ ′ ; V ) ∼ = J ⊗ Z V . The latter moduleis isomorphic to the direct sum of | Z d / Λ | copies of V ∼ = Z N , and therefore14s free. Since Z is a PID, the submodule Im ¯ ∂ is also free, and the sequence(3.12) splits. We have A γ /J A γ ∼ = H ( X Λ ; V ) ⊕ Im ¯ ∂. We obtain A γ /J A γ by killing the coset of 1 ⊗ v ⊗ x ∈ A γ . In view ofthe splitting, this can be done by killing its image in Im ¯ ∂ , resulting in asummand that is easily seen to be isomorphic to ker f /J ker f . We have A γ /J A γ ∼ = H ( X Λ ; V ) ⊕ ker f /J ker f. Pontraygin duality givesFix Λ σ ∼ = \ H ( X Λ ; V ) ⊕ \ ker f /J ker f . (3.13)The decomposition (3.13) implies that the number of connected compo-nents of Fix Λ σ is the product of the numbers of connected components of \ H ( X Λ ; V ) and \ ker f /J ker f .Theorem 21.1 of [30] implies that the exponential growth rate of thenumber of connected components of Fix Λ σ equal to h ( σ ), while that of \ ker f /J ker f is h ( σ ′′ ). The number of connected components of \ H ( X Λ ; V )is b Λ ,γ , by Corollary 3.9. Hence equation (3.6) completes the proof of thefirst statement of Theorem 3.10.In order to prove the second statement of Theorem 3.10, we note that A γ has a square presentation matrix with determinant D ℓ,γ . Example 18.7of [30] implies that h ( σ ) = m ( D ℓ,γ ). Furthermore, ker f is a submodule ofcoker c , which also has a square presentation matrix t ǫ ( x ) X − I . Since d ker f is a quotient of \ coker c , the topological entropy h ( σ ′′ ) of its shift does notexceed that of \ coker c . The latter is m (det( t ǫ ( x ) X − I )), again by Example18.7 of [30]. Since m ( W ℓ,γ ) = m ( D ℓ,γ ) − m (det( t ǫ ( x ) X − I )), the secondstatement is proved. Proof of Corollary 3.11. If H ( X ′ ; V ) has a square presentation matrix, thenthe desired conclusion follows immediately from Example 18.7 of [30].The logarithmic Mahler measure of det( t ǫ ( x ) X − I ) is equal to that ofdet( t ǫ ( x ) − X ), by a simple change of variable in (2.2). The latter vanishesif no eigenvalue of X has modulus greater than 1. In this case, h ( σ ′ ) = h ( σ )by (3.6). The latter is equal to m ( D ℓ,γ ) by Example 18.7 of [30]. However,in view of (3.7), we have m ( D ℓ,γ ) = m (∆ ℓ,γ ).15 roof of Corollary 3.12. By Corollary 3.11 it suffices to show that H ( X ′ ; V )has a square presentation matrix. This follows from the fact that a finitelygenerated torsion Z [ t ± ]-module has a square presentation matrix if andonly if it has no nonzero finite submodule (see page 132 of [12]). The hy-pothesis that ∆ k,t = 0 implies that D k,t = 0, and hence the based γ -twistedAlexander module A γ is a finitely generated Z [ t ± ]-torsion module. Since A γ has a square presentation matrix, it has no nonzero finite submodule.The γ -twisted homology group H ( X ′ ; V ) is a finitely generated submoduleof A γ , and so it too has no nonzero finite submodule. Hence H ( X ′ ; V ) hasa square presentation matrix. Proof of Theorem 3.13.
The R [ t ± ]-module A γ has a square presentationmatrix, and the greatest common divisor of the coefficients of its determinant D k,γ is a unit in R . By Theorem (1.3) of [3], A γ is Z -torsion free. Since H ( X ′ ; V ) is a submodule of A γ , by the exact sequence (3.5), it is Z -torsionfree as well. Hence H ( X ′ ; V ) embeds naturally in H ( X ′ ; V ) ⊗ Z Q .Since Q [ t ± ] is a PID, there exist primitive polynomials f , . . . , f k ∈ Z [ t ]such that f j divides f j +1 , for j = 1 , . . . , k −
1, and H ( X ′ ; V ) ⊗ Z Q ∼ = Q [ t ± ] / ( f ) ⊕ · · · ⊕ Q [ t ± ] / ( f k ) := M . The product f · · · f k is equal to ˜∆ k,γ , up to multiplication by ± t n . Gener-alizing the argument of Lemma 9.1 of [30], we see that H ( X ′ ; V ) embedswith finite index in N = R [ t ± ] / ( f ) ⊕ · · · ⊕ R [ t ± ] / ( f k ) . Set N ′ = Z [ t ± ] / ( f ) ⊕ · · · ⊕ Z [ t ± ] / ( f k ). Let b ′ r be the number of con-nected components of period-r points of the dual dynamical system. Lemma17.6 of [30] implies m ( f · · · f k ) = lim r →∞ r log b ′ r . Since N ′ ⊂ N ⊂ M , and since the systems dual to N and M have the sameentropy by Lemma 17.6 of [30], we have h ( σ ) = lim r →∞ r log b ′ r . Since N = N ′ ⊗ Z R , we see from Lemma 3.7 that b r,γ is the largest factor of b ′ r not divisible by any prime in S . It was shown in [29] (see also [34]) thatfor any prime p , lim r →∞ r log b ′ ( p ) r = 0 , b ′ ( p ) r is the largest power of p dividing b r,γ . Thus b r,γ has the samegrowth rate as b ′ r , and this rate is m ( ˜∆ k,γ ). Assume that k ⊂ S is a knot and γ : π → GL N R is a representationof its group. As before R = S − Z , where S is a multiplicative subset of Z generated by finitely many primes, possibly empty. As in the previoustheorem, we assume that D k,γ is primitive, and hence A γ is Z -torsion free.We show that the based γ -twisted Alexander module A γ is completelydescribed by a pair of embeddings f, g : R M → R M . This simple descriptionappears to be well known, at least in the untwisted case. However, we havenot found a reference for it.From the description, the twisted Alexander polynomial is easily found.Also, the decomposition of entropy h ( σ ) into p -adic and Euclidean parts [24]can be seen from this perspective.Recall that A γ is generated as an R [ t ± ]-module by the 1-chains t n ⊗ v ⊗ x i , where n ∈ Z , v ranges over a fixed basis for V , and i ranges over allarcs of a diagram D for k except for a fixed arc. A defining set of relations,corresponding to the crossings of D , is given as in (3.3) by t n ⊗ v ⊗ x i + t n +1 ⊗ vX i ⊗ x j = t n ⊗ v ⊗ x k + t n +1 ⊗ vX k ⊗ x i . (4.1)Define B to be the R -module with generators t n ⊗ v ⊗ x i , as above butwith n = 0 ,
1. Relations are those of (4.1) with n = 0. Let U be the R -module freely generated by 1-chains 1 ⊗ v ⊗ x i . Let f : U → B be the obvioushomomorphism. Define g : U → B to be the homomorphism sending each1 ⊗ v ⊗ x i to t ⊗ v ⊗ x i .If either f or g is not injective, then replace U by the quotient R -module U/ (ker f + ker g ). Replace B by B/ ( f (ker g ) + g (ker f )), and f, g by theunique induced homomorphisms. If again f or g fails to be injective, thenwe repeat this process. Since R is Noetherian, we obtain injective maps f, g after finitely many iterations. Lemma 4.1.
The module A γ is isomorphic to the infinite sum · · · ⊕ R M R M ⊕ R M R M ⊕ R M · · · , with identical amalgamations given by the monomorphisms R M g ← R M f → R M .Moreover, ∆ k,γ ( t ) = det( g − tf ) . roof. For n ∈ Z , let B n be the R -module generated by 1-chains t n ⊗ v ⊗ x i and t n +1 ⊗ v ⊗ x i , and relations given by (4.1). (As above, v ranges over abasis of V and i ranges over all arcs of D except the fixed arc.) We regardthe generators t n ⊗ v ⊗ x i ∈ B n − as distinct from t n ⊗ v ⊗ x i ∈ B n .Let U n be the R -module freely generated by the 1-chains t n ⊗ v ⊗ x i .Define homomorphisms f n : U n → B n by t n ⊗ v ⊗ x i t n ⊗ v ⊗ x i . Define g n : U n → B n − by t n ⊗ v ⊗ x i t n ⊗ v ⊗ x i . It is clear that A γ isisomorphic to the infinite amalgamated sum · · · ⊕ U n B n ⊕ U n +1 B n +1 ⊕ U n +2 · · · , (4.2)where the amalgamation maps B n − g n ← U n f n → B n are not necessarily injec-tive. However, the image under f n (resp. g n ) of any element that is in thekernel of g n (resp. f n ) is trivial in A γ . Hence we can apply the operationdescribed prior to the statement of Lemma 4.2 simultaneously for all n , toensure that the maps f n and g n are injective.In any amalgamated free product G = B ∗ U B ′ , with amalgamating maps B g ← U f → B , the natural maps B → G and B ′ → G are injections. Theanalogous statement holds for amalgamated sums B ⊕ U B ′ in which theamalgamating maps are injective. The proof is easy and left to the reader.Consequently, each B n is naturally embedded in A γ .Since A γ is Z -torsion free, B n are finitely generated free R -modules iso-morphic to R M for some positive integer M , independent of n .The decomposition (4.2) induces an R [ t ± ]-module structure, with gen-erators b , . . . , b M ∈ B corresponding to module generators, and the amalga-mation B g ← U f → B inducing a defining set of module relations g ( u i ) − tf ( u i ) , where u , . . . , u M generate U . Clearly ∆ k,γ = det( g − tf ). Remark 4.2.
If ∆ k,γ = 0, then after multiplication by a unit in R [ t ± ], wecan assume that ∆ k,γ is an integer polynomial with with leading and trailingcoefficients c , c not divisible by any primes in S . Then | c | = | R M : g ( R M ) | and | c | = | R M : f ( R M ) | .When c = c = 1, the maps f and g are isomorphisms. In this case, A γ is isomorphic to R M with the action of t given by g ◦ f − . This is the casewhenever k is a fibered knot (see [2], [35]). If, in addition, R = Z , then σ is a toral automorphism and its entropy is the sum of the logs of moduli ofeigenvalues outside the unit circle, corresponding to expansive directions onthe torus.For non-fibered knots k , the coefficients c , c measure how far f, g arefrom being isomorphisms, and, in this sense, provide a measure of the non-18beredness of k . The map σ is an automorpism of a solenoid, and thecontribution of the leading coefficient to topological entropy reflects expan-sion in p-adic directions [24]. Work of S. Friedl and S. Vidussi [9] suggeststhat c and c are complete fibering obstructions with R = Z . Conjecture 4.3.
A knot k is fibered iff for every representation γ : π → GL N Z , both c and c have absolute value 1. Any finite group can be regarded as a subgroup of permutation matrices inGL N Z , for some N . Hence finite-image representations of a link group π are a natural source of examples of representations γ : π → GL N Z for whichthe techniques above can be applied. Examples can be found throughoutthe literature (see [10], [13], [36], for example).Here we focus on a less obvious source of examples, coming from repre-sentations γ of π in SL C . Following [16], we say that γ is parabolic if theimage of any meridian is a matrix with trace 2. The terminology is motivatedby the fact that γ projects to a representation π → PSL C = SL C / h − I i sending each meridian to a parabolic element. A theorem of Thurston [38]ensures that if k is a hyperbolic knot, then the discrete faithful represen-tation π → PSL C describing the hyperbolic structure of S \ k lifts to aparabolic representation γ : π → SL C .Let k be a 2-bridge knot. Such knots are parameterized by pairs ofrelatively prime integers α, β such that β is odd and 0 < β < α . The knots k ( α, β ) and k ( α ′ , β ′ ) have the same type if and only if α = α ′ and β ′ = β or ββ ′ ≡ α . (See [27], for example.)The group of k ( α, β ) has a presentation of the form π = h x, y | W x = yW i , where x, y are meridians and W is a word in x ± , y ± . As Riley showedin [28], any nonabelian parabolic representation γ : π → SL C is conjugateto one that maps x (cid:18) (cid:19) , y (cid:18) w (cid:19) (5.1)for some 0 = w ∈ C . In fact w is an algebraic integer, a zero of a monicinteger polynomial Φ α,β ( w ) that is the (1 , γ ( W ). Wecall Φ α,β ( w ) a Riley polynomial of k .Similar ideas apply to 2-bridge links, which have presentations of theform h x, y | W y = yW i (see, for example, Section 4.5 of [25]).19 emark 5.1. It is clear from what has been said that a 2-bridge knot k ( α, β )has at most two Riley polynomials Φ α,β , Φ α,β ′ , where ββ ′ ≡ α . Ei-ther can be used to determine the set of conjugacy classes of nonabelianparabolic representations of the knot group.Let φ ( w ) be an irreducible factor of Φ α,β ( w ). We define the total repre-sentation γ φ : π → GL N Z to be the representation determined by x (cid:18) I I I (cid:19) , y (cid:18) I C I (cid:19) , where C is the companion matrix of φ . Note that N is 2 · deg φ . Thecorresponding twisted invariant ∆ k,γ φ is the total γ φ -twisted Alexander poly-nomial of k . Like the classical Alexander polynomial, it is well defined upto multiplication by ± t i . Our terminology is motivated by the fact, easilyproved, that ∆ k,γ φ is the product of the ∆ k,γ as we let w range over theroots of φ . (This observation, which is a consequence of [20], yields a use-ful strategy for computing total twisted Alexander polynomials when thedegree of φ , and hence the size of the companion matrix, is large.) Remark 5.2. (1) For 2-bridge knots, the algebraic integer w is in fact analgebraic unit [28]. Hence the trailing coefficient of Φ α,β ( w ) is ±
1, and sothe companion matrix C is unimodular. However, this need not be true for2-bridge links (see Example 5.7).(2) The notions of total representation and total twisted Alexander poly-nomial generalize easily. Assume that γ : π → SL ¯ Q is a representationwith image Z [ w , . . . , w n ]. (Up to conjugation, the parabolic PSL C rep-resentation describing the complete hyperbolic structure of a knot comple-ment projects to such a representation [38].) Let φ , . . . , φ n be the minimalpolynomials of the algebraic numbers w , . . . , w n with leading coefficients c , . . . , c n . Over the ring R = Z [1 /lcm ( c · · · c n )], the polynomials are monic,and we can consider their companion matrices. In order to define the totalrepresentation of γ we embed SL ( Z [ w , . . . , w n ]) in GL N R by replacing 2 × w i is replaced bya block diagonal matrix of the form I ⊕ · · · ⊕ C i ⊕ · · · ⊕ I and N = 2Σ i deg φ i .In practice, it might be possible to find a more economical embedding.For any total representation γ φ associated to a nonabelian parabolicrepresentation γ : π → SL C , the boundary homomorphism C ( X ′ ; V ) ∂ −→ C ( X ′ ; V ) sends the 1-chain 1 ⊗ v ⊗ x − ⊗ v ⊗ y to t ⊗ v ( γ φ ( x ) − γ φ ( y )) ⊗ ∗ = t ⊗ v (cid:18) I − C (cid:19) ⊗ ∗ . C ( X ′ ; V ), and hence H ( X ′ ; V ) vanishes. Since ∆ (coker c ) =det ( tX − I ) = ( t − φ , Proposition 3.6 implies that the total γ φ -twistedAlexander polynomial of k is∆ k,γ φ = ∆ ( A γ ) / ( t − φ . (5.2) Example 5.3.
The group of the trefoil knot k = 3 has presentation π = h x, y | xyx = yxy i and Riley polynomial Φ , ( w ) = w +1. Up to conjugation, π admits a single nonabelian parabolic representation, x (cid:18) (cid:19) , y (cid:18) − (cid:19) . Using Equation 5.2 the total γ Φ -twisted Alexander polynomial of k is seento be t + 1.The based γ Φ -twisted Alexander module A γ is isomorphic to h ⊗ v ⊗ y | t ⊗ vX ⊗ y = 1 ⊗ v ⊗ y + t ⊗ vY X ⊗ y i , where v ranges over a basis for Z . The relations can be replaced by t ⊗ v ⊗ y = t ⊗ v ¯ X ¯ Y X ⊗ y − ⊗ v ¯ X ¯ Y ⊗ y, where ¯ denotes inversion. As a Z [ t ± ]-module, A γ Φ is isomorphic to Z with action of t given by right multiplication by the matrix M = (cid:18) I − ¯ X ¯ Y ¯ X ¯ Y X (cid:19) = − − − . The dual group ˆ A γ Φ , which is topologically conjugate to Col γ Φ , is a 4-torus T with shift σ given by left multiplication by M , and entropy zero. Wediscuss general torus knots in Section 7.We remark that the (untwisted) coloring dynamical system Col is a2-torus T with shift (cid:18) − (cid:19) . Example 5.4.
We consider another fibered knot, the figure-eight knot k =4 (“Listing’s knot”). Its group has presentation π = h x, y | yx ¯ yxy = xy ¯ xyx i , ( w ) = w − w + 1. Up to conjugation, π admitstwo nonabelian parabolic representations x (cid:18) (cid:19) , y (cid:18) w (cid:19) , where w = (1 ± √− /
2. (The representation with w = (1 − √− / γ Φ -twisted Alexander polynomial of k is seen to be ( t − t + 1) .The dynamical system Col γ Φ is an 8-torus T . The shift σ is describedby M = − − − − − − − − − − −
11 0 − − . The entropy of the shift is 2 m ( t − t + 1), approximately 2 . Example 5.5.
The group of k = 5 has presentation h x, y | xyx ¯ y ¯ xyx = yxyx ¯ y ¯ xy i and Riley polynomial Φ , ( w ) = w + w + 2 w + 1. The based γ Φ -twisted Alexander module A γ Φ has a presentation of the form h ⊗ v ⊗ y | t ⊗ vP ⊗ y = t ⊗ vQ ⊗ y + 1 ⊗ vR ⊗ y i , where v ranges over a basis for Z , and P, Q and R are the 6 × P = XY X ¯ Y + Y X ¯ Y ¯ XY X, Q = − I − Y X ¯ Y ¯ X, R = X + XY X ¯ Y ¯ X + Y X ¯ Y .
The module can be expressed as · · · ⊕ Z Z ⊕ Z Z ⊕ Z · · · , where the amalgamating maps g and f are given by right multiplication bythe matrices G = (cid:18) IR Q (cid:19) , F = (cid:18) I P (cid:19) . f and g have index 25, and the total twisted Alexanderpolynomial is∆ k,γ Φ = det( G − tF ) = 25 t − t + 219 t − t + 219 t − t + 25 . If replace Z by Z [1 / f, g are invertible. The dynamical system(Col γ Φ , σ ) is conjugate to an automorphism of the 12-dimensional solenoid \Z [1 / with entropy m (∆ k,γ Φ ), approximately log 25+log 1 . . . Example 5.6.
Riley concludes [28] with “an account of the representationsof our favourite knot,” a knot k that “has a confluence of noteworthy prop-erties.” Riley observed, for example, that k is the equatorial cross-section ofan unknotted 2-sphere in R . His guess that k is S-equivalent to the squareknot was confirmed in [15]. Hence abelian invariants (untwisted Alexanderpolynomials, homology of cyclic and branched covers) cannot distinguish k from the square knot.A diagram of k with Wirtinger generators for its group appears in Figure2. Riley noted that, for any complex number w , the assignment x (cid:18) (cid:19) , x (cid:18) − (cid:19) x (cid:18) − w w − w + 1 (cid:19) , x (cid:18) − w (cid:19) , determines a nonabelian parabolic representation γ .For any w , the determinant of the 3 × (cid:18) ∂x i ∂x j (cid:19) ≤ i,j ≤ with entries evaluated at γ gives the same result, ( t − ( t + 1). If welet w be an integer, then we obtain a (total) representation in SL Z . The γ -twisted Alexander polynomial of k is ∆ k,γ ( t ) = ( t − ( t + 1). We showthat this polynomial does not arise from the square knot, for any nonabelianparabolic representation of its group.Consider the square knot k ′ in Figure 3, with Wirtinger generators x , x , x indicated. Its group has presentation h x , x , x | x x x = x x x , x x x = x x x i . As shown in [28], any nonabelian parabolic representation is conjugate toone of the form: x (cid:18) (cid:19) , x (cid:18) w (cid:19) , x (cid:18) uv v − u − uv (cid:19) . x x x Figure 2: Riley’s knotOne easily verifies that w = − u = ± u = 0. If u = ±
1, then v is arbitrary, and the twisted Alexander polynomial is ( t − ( t + 1) . If u = 0, then v = 1, and the polynomial is ( t − t + 1) ( t + 1). Example 5.7.
Consider the Whitehead link ℓ in Figure 4 with Wirtingergenerators indicated. Its group π has presentation h x, y | wy = yw i , where w = ¯ x ¯ yxyx ¯ y ¯ x. The associated Riley polynomial is Φ( w ) = w + 2 w + 2. Let γ Φ : π → GL Z denote the total representation. It is a straightforward matter to checkthat the based γ Φ -twisted Alexander module is generated by the 1-chains x x x Figure 3: Square knot24 y Figure 4: Whitehead link1 ⊗ e i ⊗ y (1 ≤ i ≤
4) with relators − ⊗ v ¯ X ¯ Y ⊗ y + t ⊗ v ¯ X ¯ Y X ⊗ y − t ⊗ v ¯ X ¯ Y XY X ¯ Y ⊗ y + t ⊗ v ¯ X ¯ Y XY X ¯ Y ¯ X ⊗ y − t t ⊗ v ⊗ y + t ⊗ vY ¯ X ¯ Y ⊗ y − t t ⊗ vY ¯ X ¯ Y X ⊗ y + t t ⊗ vY ¯ X ¯ Y XY X ¯ Y ⊗ y. Proposition 6.11 below implies that H ( X ′ , V ) is finite. Hence Proposi-tion 3.6 enables us to compute the γ Φ -twisted Alexander polynomial of ℓ .It is given by the following array in which the number in position i, j is thecoefficient of t i t j . − − − −
16 12 − −
16 28 −
16 6 − −
16 12 − − − t − t − − − ℓ,γ Φ is approximately 2.5. In particular,Theorem 3.10 ensures the growth of torsion of γ Φ -twisted homology in thecyclic covers of the link complement.25 Evaluating the twisted Alexander polynomial for2-bridge knots and links
The following establishes the nontriviality of twisted Alexander polynomialsfor any 2-bridge knot and nonabelian parabolic representation of that group.
Theorem 6.1.
Let k be a 2-bridge knot, and let γ φ be a total representationassociated to a nonabelian parabolic representation γ : π → SL C . Then | ∆ k,γ φ (1) | = 2 deg φ . Corollary 6.2.
Let k be a -bridge knot. For any nonabelian parabolicrepresentation γ , the γ -twisted Alexander polynomial ∆ k,γ ( t ) is nontrivial.Proof. Assume after conjugation that γ has the standard form given by (5.1)above. Let φ be the minimal polynomial of w , an irreducible factor of theRiley polynomial. Denote its roots by w = w, w , . . . , w m , where m is thedegree of φ . The Galois group of Q [ w ] over Q is transitive on the roots. Itfollows that if ∆ k,γ ( t ) is trivial, then it is trivial whenever any w i , ( i >
1) issubstituted for w in (5.1). However, the total twisted polynomial ∆ k,γ φ ( t )is the product of such polynomials. Hence ∆ k,γ ( t ) is nontrivial. Remark 6.3.
Corollary 6.2 implies that the Wada invariant of any 2-bridgeknot corresponding to a nonabelian parabolic representation is nontrivial.Whether or not the Wada invariant of a knot corresponding to any non-abelian representation is nontrivial is unknown (cf. [26], [8]). One might sus-pect a negative answer in view of M. Suzuki’s result [37] that the Lawrence-Krammer representation of the braid group B , a faithful nonabelian repre-sentation, yields a trivial Wada invariant. Conjecture 6.4.
For any -bridge knot k and total representation γ φ as inTheorem 6.1, | ∆ k,γ φ ( − | = 2 deg φ µ , for some integer µ . Remark 6.5.
Theorem 6.1 and Conjecture 6.4 have been shown by M.Hirasawa and K. Murasugi [14] in the case that the group of k maps onto atorus knot group, preserving meridian generators, and γ is the pullback ofa nonabelian parabolic representation.In order to prove Theorem 6.1, we consider an arbitrary 2-bridge knot k and total representation γ φ associated to a nonabelian parabolic represen-tation γ : π → SL C . Denote the degree of φ by m . The following lemmafollows from the proof of Corollary 3.12. However, the direct argumentbelow yields extra, useful information.26 emma 6.6. The Z [ t ± ] -module H ( X ′ ; V ) has a square matrix presenta-tion.Proof. Consider the chain complex (3.2):0 → C ( X ′ ; V ) ∂ −→ C ( X ′ ; V ) ∂ −→ C ( X ′ ; V ) → . It can be written as0 → Z [ t ± ] m ∂ −→ Z [ t ± ] m ⊕ Z [ t ± ] m ∂ −→ Z [ t ± ] m → . The module of 1-chains is freely generated by 1 ⊗ e i ⊗ x and 1 ⊗ e i ⊗ y , where e i ranges over a basis for V = Z m .The ∂ -images of the chains a i = 1 ⊗ e i ⊗ ( x − y ) generate C ( X ′ ; V )(see the discussion preceding Example 5.3). For each i = 1 , . . . , m , choosea Z [ t ± ]-linear combination u i of the chains a i such that b i = 1 ⊗ e i ⊗ y − u i is a cycle. Then a , . . . , a m , b , . . . , b m is a second basis of C ( X ′ ; V ) suchthat b , . . . , b m freely generate the submodule of 1-cycles. Hence H ( X ′ ; V )is isomorphic to a quotient Z [ t ± ] m / Z [ t ± ] m A , for some square matrix A of size 2 m . Corollary 6.7. | ∆ k,γ φ (1) | is equal to the order of the twisted homology group H ( X ; V ) = H ( X ; V ) .Proof. The argument above shows that ∂ is surjective. Hence H ( X ′ ; V ) =0. A Wang sequence shows that H ( X ; V ) ∼ = H ( X ′ ; V ) / ( t − H ( X ′ ; V ) . From Lemma 6.6, the matrix A (1) obtained from A by setting t = 1 presents H ( X ; V ). The order of H ( X ; V ) is | det A (1) | = | ∆ k,γ φ (1) | . In view of the corollary, we focus our attention on H ( X ; V ). This finitelygenerated abelian group is determined by the γ φ -twisted chain complex C ∗ ( X ; V ) = V ⊗ Z [ π ] C ∗ ( ˜ X ) , (6.1)just as in Section 3 but with trivial t -action.The boundary of the knot exterior X is an incompressible torus T , andthe restriction of γ φ to π T determines γ φ -twisted absolute and relativehomology groups H ∗ ( T ; V γ φ ) and H ∗ ( X, T ; V ). The latter groups are definedby the quotient chain complex C ∗ ( X, T ; V ) = C ∗ ( X ; V ) /C ∗ ( T ; V ) (see, forexample, [19]). A long exact sequence relating the absolute and relative27omology groups is found from the natural short exact sequence of chaincomplexes 0 → C ∗ ( T ; V ) → C ∗ ( X ; V ) → C ∗ ( X, T ; V ) → . Lefschetz duality for twisted homology is presented in [19] (see also[18]). We apply a version in our situation. Tensoring the chain complex6.1 with the field Z / (2), we obtain twisted homology groups that we de-note by H ∗ ( X ; V (2) ) and H ∗ ( X, T ; V (2) ). Let ¯ V be the module V with right π -action given v · g = v ( γ φ ( g ) − ) τ , where τ denotes transpose. Substituting ¯ V for V in the chain complex 6.1,and tensoring with Z / (2), yields a new complex with homology groups thatwe denote by H ∗ ( X ; ¯ V (2) ) and H ∗ ( X, T ; ¯ V (2) ).The inner product { , } : V × ¯ V → Z / (2) given by usual dot product isnon-degenerate, and satisfies { rv, w } = r { v, w } = { v, rw }{ v · g, w } = { v, wg − } , for all r ∈ Z / (2) and g ∈ π . (Readers of [19] should note the missing inversesign in Equation (2.5).) As in Section 5.1 of [19], we have H q ( X ; V (2) ) ∼ = H − q ( X, T ; ¯ V (2) ) for all q .Taking inverse transpose of matrices X, Y modulo 2 merely shifts the off-diagonal blocks to the opposite side. The ranks of the resulting homologygroups are unaffected. We have
Proposition 6.8. rk H i ( X ; V (2) ) = rk H − i ( X, T ; V (2) ) , for each i . The following lemma will enable us to determine several boundary mapsin appropriate chain complexes. Recall that the group π of k has a presen-tation of the form h x, y | r i , where r has the form W x = yW . We denotethe class of an oriented longitude by l . Lemma 6.9. (1) γ φ ( l ) is a block matrix of m × m integer matrices L = (cid:18) − I E − I (cid:19) , where I, are the identity and zero matrices, respectively, and E hasonly even entries. The matrix of Fox partial derivative matrix ( ∂r/∂x ) γ φ has the form (cid:18) ∗ ∗ (cid:19) , where all blocks are m × m . Similarly, ( ∂r/∂y ) γ φ has (up to sign) theform (cid:18) I ∗ (cid:19) . Proof.
Statement (1) follows immediately from Theorem 2 of [28].We prove statement (2). Fox’s fundamental formula (2.3) of [7] impliesthat ( r − γ φ = ( ∂r/∂x ) γ φ ( X − I ) + ( ∂r/∂y ) γ φ ( Y − I ) . The left hand side is trivial, since r = 1 in π . Using the forms for X, Y , wehave (cid:18) (cid:19) = ( ∂r/∂x ) γ φ (cid:18) (cid:19) + ( ∂r/∂y ) γ φ (cid:18) C (cid:19) . It follows immediately that ( ∂r/∂x ) γ φ and ( ∂r/∂y ) γ φ have the forms (cid:18) ∗ ∗ (cid:19) and (cid:18) ∗ ∗ (cid:19) , respectively.In order to see that the upper left-hand block of ( ∂r/∂y ) γ φ is the identitymatrix, we use form W x = yW of the relation r . Fox calculus yields( ∂r/∂y ) γ φ = ( ∂W/∂y ) γ φ − I − Y ( ∂W/∂y ) γ φ . The right-hand side can be written as (cid:18) − C (cid:19) ( ∂W/∂y ) γ φ − I, and from this the desired form follows immediately. Remark 6.10.
Similar arguments show that, in fact, the upper right-handblock of ( ∂r/∂x ) γ φ is unimodular. We will not require this fact here.We compute H i ( T ; V ), for i = 0 ,
1. Beginning with the presentation π T = h x, l | xl = lx i , we construct a canonical 2-complex with a single 0-cell*, 1-cells x , l corresponding to generators and a single 2-cell s correspondingto the relation. Consider the associated γ φ -twisted chain complex0 → C ( T ; V ) ∼ = V ∂ −→ C ( T ; V ) ∼ = V ⊕ V ∂ −→ C ( T ; V ) ∼ = V → , v ⊗ s ; 1-chain generators v ⊗ x and v ⊗ l ; and 0-chaingenerators v ⊗ ∗ . Here v ranges freely over a basis e , . . . , e m for V = Z m .We have ∂ ( v ⊗ x ) = v ⊗ ∂x = v ⊗ ( x ∗ −∗ ) = v ( X − I ) ⊗ ∗ . Writing v = ( v , v ) ∈ Z m ⊕ Z m , we can express the 0-chain boundaries obtained thisway as (0 , v ) ⊗ ∗ , where v is arbitrary. Similarly, ∂ ( w ⊗ l ) = v ( L − I ) ⊗ ∗ =( − w , w E − w ) ⊗ ∗ , where w = ( w , w ) ∈ V ⊕ V . It follows that H ( T ; V ) ∼ = ( Z / m , with generators represented by the 0-chains e i ⊗ ∗ ,where 1 ≤ i ≤ m .The computation of H ( X ; V ) is similar. From the previous descriptionof ∂ ( v ⊗ x ) and ∂ ( w ⊗ l ), it is easy to see that 1-cycles have the form(2 w , v ) ⊗ x + (0 , w ) ⊗ l. (6.2)Applying Fox calculus to the relation xl = lx , we find that ∂ ( v ⊗ s ) = v ⊗ ∂ s = v ⊗ (1 − l ) x + v ⊗ ( x − l = v ( I − L ) ⊗ x + v ( X − I ) ⊗ l. ByLemma 6.9, the latter expression reduces to(2 v , − v E + 2 v ) ⊗ x + (0 , v ) ⊗ l. (6.3)Comparing the expressions (6.2) and (6.3), we see that H ( T ; V ) ∼ = ( Z / m ,with generators represented by 1-chains e i ⊗ x , where m < i ≤ m .Consider part of the long exact sequence of γ φ -twisted homology groupsassociated to the pair ( X, T ): H ( T ; V ) → H ( X ; V ) → H ( X, T ; V ) → H ( T ; V ) → H ( X ; V )Recall that H ( X ; V ) vanishes.We compute H ( X, T ; V ). Consider the following presentation of π X : h x, y, l | q, r, s i , where r and s are given above, and q is the relation l = u expressing thelongitude l as a word u in x ± and y ± . An explicit word u is not difficult towrite (see [28]), but we will not require this. The γ φ -twisted chain complexassociated to this presentation contains the γ φ -twisted chain complex for T as a subcomplex. For the purpose of computing homology below dimension2, we may substitute this complex for C ∗ ( X ; V ).The relative complex C ∗ ( X, T ; V ) has the form0 → C ( X, T ; V ) ∼ = V ⊕ V ∂ −→ C ( X, T ; V ) ∼ = V → , with relative 2-chain generators represented by v ⊗ r and v ⊗ s ; relative1-chain generators represented by v ⊗ y . As before v ranges freely over abasis for Z m . 30e have ∂ ( v ⊗ r ) = v ⊗ ( ∂r/∂y ) y = v ( ∂r/∂y ) γ φ ⊗ y . By Lemma 6.9, ∂ ( v ⊗ r ) has the form ( v , ⊗ y , where v ∈ Z m is arbitrary. Hence therank of H ( X, T ; V γ φ ) cannot exceed m .Similarly, ∂ ( v ⊗ s ) = v ⊗ ( ∂s/∂y ) y = v ( ∂s/∂y ) γ φ ⊗ y . Recall that therelation s has the form l = u . ∂ ( v ⊗ s ) = v ( ∂u/∂y ) γ φ ⊗ y . Hence by Fox’sfundamental formula,( u − γ φ = ( ∂u/∂x ) γ φ ( X − I ) + ( ∂u/∂y ) γ φ ( Y − I ) . The left-hand side is equal to ( l − γ φ = L − I , since u = l in π ; by Lemma6.9, this matrix has the form (cid:18) − I E − I (cid:19) . Writing ( ∂u/∂x ) γ φ = (cid:18) U x U x U x U x (cid:19) , ( ∂u/∂y ) γ φ = (cid:18) U y U y U y U y (cid:19) , and using the forms for X, Y , the right-hand side has the form (cid:18) U x U x (cid:19) + (cid:18) U y C U y C (cid:19) . Since C is unimodular (see Remark 5.2), we see that that ( ∂u/∂y ) γ φ has theform (cid:18) ∗ − C − ∗ (cid:19) . Hence ∂ ( v ⊗ s ) has the form ( ∗ , w ) ⊗ y , where w ranges over (2 Z ) m as v varies over Z m . It follows that H ( X, T ; V ) ∼ = ( Z / m , with generatorsrepresented by e i ⊗ y, where m < i ≤ m. We see also that the map H ( X, T ; V ) → H ( T ; V ) in the long exactsequence of the pair ( X, T ) is an isomorphism. From this it follows that themap H ( T ; V ) → H ( X ; V ) is surjective. Since H ( T ; V ) is 2-torsion, so is H ( X ; V ).We can now prove Theorem 6.1. Proof.
By Lefschetz duality,rk H ( X ; V (2) ) = rk H ( X, T ; V (2) ) = m. H ( X ; V (2) ) ∼ = H ( X ; V ) ⊗ Z / (2). Since H ( X ; V ) is trivial, rk H ( X ; V (2) ) = 0. Also,rk H ( X ; V (2) ) = rk H ( X, T ; V (2) ) = 0.The alternating sum rk H ( X ; V (2) ) − rk H ( X ; V (2) ) + rk H ( X ; V (2) ) − rk H ( X ; V (2) ) coincides with the Euler characteristic of X , which is 0.Hence rk H ( X ; V (2) ) = m . By the universal coefficient theorem for ho-mology, H ( X ; V (2 ) ∼ = H ( X ; V ) ⊗ Z /
2. Since H ( X ; V ) is 2-torsion, it isisomorphic to ( Z / (2)) m . Therefore | ∆ k,γ φ (1) | = | H ( X ; V ) | = 2 m .In the beginning of Section 5 we saw that for a 2-bridge knot k withexterior X , the group H ( X ′ ; V ) vanishes for every total representation γ φ associated to a nonabelian parabolic representation of the group of k . Theresult depends on the fact that the roots w of any Riley polynomial arealgebraic units.For 2-bridge links, the algebraic integer w need not be a unit (see Exam-ple 5.7). Nevertheless, we have the following, which is useful for computationin view of Proposition 3.6. Proposition 6.11.
Assume that ℓ is a -bridge link with exterior X . Let γ φ be a total representation associated to a nonabelian parabolic SL C rep-resentation of the group of ℓ . Then H ( X ′ ; V ) ∼ = Z / ( φ (0)) ⊕ Z / ( φ (0)) . Inparticular, the group is finite.Proof. We use notation similar to that above. Beginning with the pre-sentation h x, y | W x = xW i for the link group, we construct a canonical2-complex with a single 0-cell *, 1-cells x, y corresponding to generators,and a single 2-cell corresponding to the relation. We consider the associated γ φ -twisted chain complex C ∗ ( X ′ ; V ) = ( Z [ Z ] ⊗ V ) ⊗ Z [ π ] C ∗ ( ˜ X ) , with 1-chain generators 1 ⊗ v ⊗ x and 1 ⊗ v ⊗ y ; and 0-chain generators1 ⊗ v ⊗ ∗ . As before, v ranges freely over a basis e , . . . , e m for V = Z m ,where m is the degree of φ . We will not need to consider 2-chains.The γ φ -twisted homology group H ( X ′ ; V ) is the quotient of Z [ Z ] ⊗ V ,the 0-chains, modulo the image of the boundary operator ∂ . Since ∂ (1 ⊗ v ⊗ x ) = t ⊗ vX ⊗∗− ⊗ v ⊗∗ and ∂ (1 ⊗ v ⊗ y ) = t ⊗ vY ⊗∗− ⊗ v ⊗∗ , it followsimmediately that t n ⊗ v ⊗ ∗ = 1 ⊗ vX − n ⊗ ∗ and t n ⊗ v ⊗ ∗ = 1 ⊗ vY − n ⊗ ∗ for all n ∈ Z . We can define an abelian group homomorphism f : Z [ Z ] ⊗ V / (im ∂ ) → V /V ( XY − Y X )32y sending p ( t , t ) ⊗ v ⊗ ∗ 7→ vp ( ¯ X, ¯ Y ), for any polynomial p ∈ Z [ Z ], andextending linearly. The assignment v ⊗ v ⊗ ∗ , for v ∈ V , induces aninverse homomorphism that is well defined, since1 ⊗ vXY ⊗ ∗ = t − t − ⊗ v ⊗ ∗ = t − t − ⊗ v ⊗ ∗ = 1 ⊗ vY X ⊗ ∗ for all v ∈ V . Hence f is an isomorphism.The matrix XY − Y X has the form (cid:18) C − C (cid:19) , where C is the m × m -companion matrix of φ and 0 is the zero matrix of thatsize. The determinant of XY − Y X is (det C ) , which is the square of theconstant coefficient of φ . It follows that H ( X ′ ; V ) ∼ = Z / ( φ (0)) ⊕ Z / ( φ (0)). The group π of any fibered knot has a presentation of the form h x, a , . . . , a g | xa i x − = µ ( a i ) (1 ≤ i ≤ g ) i , where a , . . . , a g are free generators of the fundamental group of the fiber S , and µ : π S → π S is the automorphism induced by the monodromy.Let γ : π → GL N R be a representation, where R is a Noetherian UFD.The twisted homology computations in Section 3 can be performed using theabove presentation. We find that the based γ -twisted Alexander module A γ is a free R -module of rank 2 gN , with generators 1 ⊗ v ⊗ a i , where v rangesover a basis for R N . Multiplication by t induces an automorphism describedby an N × N -block matrix M with 2 g × g blocks X − ∂µ ( a i ) ∂a j γ ! . As before, X is the image of the meridian x , and the notation γ indicatesthat the terms of the Fox partial derivatives are to be evaluated by γ . Thecharacteristic polynomial is the γ -twisted Alexander polynomial of k .In [28], Riley describes nonabelian parabolic representations of any torusknot k . The (2 n +1 ,
2) torus knots are 2-bridge knots with α = 2 n +1 , β = 1.33heir Riley polynomials Φ n +1 , ( w ) can be found recursively: Φ , ( w ) =1 , Φ , ( w ) = w + 1, andΦ n +5 , ( w ) = − Φ n +1 , ( w ) + ( w + 2)Φ n +3 , ( w ) , n ≥ . For arbitrary torus knots, the situation is more complicated. Nevertheless,Riley shows that any parabolic representation for a torus knot group isconjugate to a representation with image in SL A , where A is the ring ofalgebraic integers. Consequently, any parabolic representation γ determinesa total representation γ φ and hence a total twisted Alexander polynomial∆ k,γ φ ( t ), as above. Theorem 7.1.
For any parabolic representation γ of the group of a torusknot k , the total twisted Alexander polynomial ∆ k,γ φ ( t ) is a product of cy-clotomic polynomials.Proof. The monodromy µ of a ( p, q )-torus knot is periodic of order pq . More-over, µ pq is inner automorphism by the longitude l ∈ π ′ . Hence, for anygenerator a i of the fundamental group of the fiber S , as above, we have µ pqr ( a i ) = l r µ ( a i ) l − r . Let M be the matrix describing the action of t on A γ φ , as above. A standardcalculation using Fox calculus shows that the ij th block of ∂µ pqr ( a i ) ∂a j γ φ ! is equal to( I + L + · · · + L r − ) ∂l∂a j γ φ + L r ∂µ ( a i ) ∂a j γ φ − L r A i ( L − + · · · + L − r ) ∂l∂a j γ φ , where A i denotes µ ( a i ) and L = γ φ ( l ). The sum can be rewritten as( I − L r A i L − r )( I + L + · · · + L r − ) ∂l∂a j γ φ + L r ∂µ ( a i ) ∂a j γ φ . Recall from Lemma 6.9 that L has the form (cid:18) − I E − I (cid:19) . Hence the entries of L r have polynomial growth as r goes to infinity. Con-sequently, the entries of M r have polynomial growth. It follows that thespectral radius of M is 1. Since the characteristic polynomial, which is∆ k,γ φ ( t ), has integer coefficients, it is a product of cyclotomic polynomialsby Kronecker’s theorem. 34 emark 7.2. Hirasawa and Murasugi [14] have shown that the total twistedAlexander polynomial of a (2 q + 1 , k has the form∆ k,γ Φ ( t ) = ( t + 1)( t q +2 + 1) q − , a product of cyclotomic polynomials, as predicted by Theorem 7.1.Since the γ -twisted Alexander polynomial ∆ k,γ ( t ), which is well de-fined up to a unit in C [ t ± ], divides the total twisted Alexander polynomial∆ k,γ φ ( t ), the following is immediate. Corollary 7.3.
For any parabolic representation γ of a torus knot k , the γ -twisted Alexander polynomial is a product of cyclotomic polynomials. With minor modifications, the above arguments apply for any permuta-tion representation γ : π → GL N Z . Corollary 7.4.
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Department of Mathematics and Statistics, ILB325, University of South Alabama, Mobile AL 36688 USA