Twisted Alexander Polynomials Research Articles

A note on the topological sliceness of some 2-bridge knots

AbstractWe use twisted Alexander polynomials to show that certain algebraically slice 2-bridge knots are not topologically slice, even though all prime power Casson–Gordon signatures vanish. We also provide some computations indicating the efficacy of Casson–Gordon signatures in obstructing the smooth sliceness of 2-bridge knots.

Fintushel–Stern knot surgery in torus bundles

Suppose that X is a torus bundle over a closed surface with homologically essential fibers. Let X K be the manifold obtained by Fintushel–Stern knot surgery on a fiber using a knot K ⊂ S 3 . We prove that X K has a symplectic structure if and only if K is a fibered knot. The proof uses Seiberg–Witten theory and a result of Friedl–Vidussi on twisted Alexander polynomials.

Adjoint twisted Alexander polynomials of genus one two-bridge knots

We give explicit formulas for the adjoint twisted Alexander polynomial and nonabelian Reidemeister torsion of genus one two-bridge knots.

Introduction to twisted Alexander polynomials and related topics

This article is based on the lectures in the Winter Braids V (Pau, Feb. 2015). Main puposel of this is to explain how to compute twisted Alexander polynomials for non-experts.

Twisted Alexander polynomials, character varieties and Reidemeister torsions of double branched covers

We give an extension of Fox's formula of the Alexander polynomial for a double branched cover over the three-sphere. Our formula provides the Reidemeister torsion of the double branched cover along a knot for a non-trivial 1-dimensional representation. In our formula, the Reidemeister torsion is given by the product of two factors derived from the knot group. One of the factors is determined by the twisted Alexander polynomial and the other is determined by a rational function on the character variety of the knot group. As an application, we show that these products distinguish the isotopy classes of two-bridge knots up to their mirror images.

On the Seifert fibered space link group

We introduce generalized arrow diagrams and generalized Reidemeister moves for diagrams of links in Seifert fibered spaces. We give a presentation of the fundamental group of the link complement. As a corollary we are able to compute the first homology group of the complement and the twisted Alexander polynomials of the link.

Representations of knot groups into SLn(ℂ) and twisted Alexander polynomials

Let $\Gamma$ be the fundamental group of the exterior of a knot in the three-sphere. We study deformations of representations of $\Gamma$ into $\mathrm{SL}_n(\mathbf{C})$ which are the sum of two irreducible representations. For such representations we give a necessary condition, in terms of the twisted Alexander polynomial, for the existence of irreducible deformations. We also give a more restrictive sufficient condition for the existence of irreducible deformations. We also prove a duality theorem for twisted Alexander polynomials and we describe the local structure of the representation and character varieties.

Twisted Alexander Polynomials and Ideal Points Giving Seifert Surfaces

The coefficients of twisted Alexander polynomials of a knot induce regular functions of the \(SL_{2}(\mathbb {C})\)-character variety. We prove that the function of the highest degree has a finite value at an ideal point which gives a minimal genus Seifert surface by Culler–Shalen theory. It implies a partial affirmative answer to a conjecture by Dunfield, Friedl, and Jackson.

Twisted Alexander polynomials with the adjoint action for some classes of knots

We calculate the twisted Alexander polynomial with the adjoint action for torus knots and twist knots. As consequences of these calculations, we obtain the formula for the nonabelian Reidemeister torsion of torus knots in [J. Dubois, Nonabelian twisted Reidemeister torsion for fibered knots, Canad. Math. Bull.49(1) (2006) 55–71] and a formula for the nonabelian Reidemeister torsion of twist knots that is better than the one in [J. Dubois, V. Huynh and Y. Yamaguchi, Nonabelian Reidemeister torsion for twist knots, J. Knot Theory Ramifications18(3) (2009) 303–341].

Twisted Alexander polynomials of 2-bridge knots for parabolic representations

In this paper we show that the twisted Alexander polynomial associated to a parabolic representation determines fiberedness and genus of a wide class of 2-bridge knots. As a corollary we give an affirmative answer to a conjecture of Dunfield, Friedl and Jackson for infinitely many hyperbolic knots.

A vanishing theorem for twisted Alexander polynomials with applications to symplectic 4-manifolds

In this paper we show that given any 3-manifold N and any non-fibered class in H1(N;Z) there exists a representation such that the corresponding twisted Alexander polynomial is zero. We obtain this result by extending earlier work of ours and by combining this with recent results of Agol and Wise on separability of 3-manifold groups. This result allows us to completely classify symplectic 4-manifolds with a free circle action, and to determine their symplectic cones.

The Thurston norm and twisted Alexander polynomials

Abstract Using recent results of Agol, Przytycki–Wise and Wise we show that twisted Alexander polynomials detect the Thurston norm of any irreducible 3-manifold which is not a closed graph manifold.

Metabelian SL(n, ) representations of knot groups IV: twisted Alexander polynomials

AbstractIn this paper we will study properties of twisted Alexander polynomials of knots corresponding to metabelian representations. In particular we answer a question of Wada about the twisted Alexander polynomial associated to the tensor product of two representations, and we settle several conjectures of Hirasawa and Murasugi.

CHARACTER VARIETIES OF ONCE-PUNCTURED TORUS BUNDLES WITH TUNNEL NUMBER ONE

We determine the PSL2(ℂ) and SL2(ℂ) character varieties of the once-punctured torus bundles with tunnel number one, i.e. the once-punctured torus bundles that arise from filling one boundary component of the Whitehead link exterior. In particular, we determine "natural" models for these algebraic sets, identify them up to birational equivalence with smooth models, and compute the genera of the canonical components. This enables us to compare dilatations of the monodromies of these bundles with these genera. We also determine the minimal polynomials for the trace fields of these manifolds. Additionally, we study the action of the symmetries of these manifolds upon their character varieties, identify the characters of their lens space fillings, and compute the twisted Alexander polynomials for their representations to SL2(ℂ).

A NOTE ON DEGREES OF TWISTED ALEXANDER POLYNOMIALS II

In this short note we discuss degrees of twisted Alexander polynomials and demonstrate an explicit example of a closed 3-manifold which is related to the degree formula due to Friedl, Kim and Kitayama.

TWISTED ALEXANDER POLYNOMIALS ON CURVES IN CHARACTER VARIETIES OF KNOT GROUPS

For a fibered knot in the 3-sphere the twisted Alexander polynomial associated to an SL(2, ℂ)-character is known to be monic. It is conjectured that for a nonfibered knot there is a curve component of the SL(2, ℂ)-character variety containing only finitely many characters whose twisted Alexander polynomials are monic, i.e. finiteness of such characters detects fiberedness of knots. In this paper, we discuss the existence of a certain curve component which relates to the conjecture when knots have nonmonic Alexander polynomials. We also discuss the similar problem of detecting the knot genus.

TWISTED ALEXANDER POLYNOMIALS OF 2-BRIDGE KNOTS

We investigate the twisted Alexander polynomial of a 2-bridge knot associated to a Fox coloring. For several families of 2-bridge knots, including but not limited to, torus knots and genus-one knots, we derive formulae for these twisted Alexander polynomials. We use these formulae to confirm a conjecture of Hirasawa and Murasugi for these knots.

Twenty years of twisted Alexander polynomials

Twenty years of twisted Alexander polynomials

On knots and links in lens spaces

In this paper we study some aspects of knots and links in lens spaces. Namely, if we consider lens spaces as quotient of the unit ball B3 with suitable identification of boundary points, then we can project the links on the equatorial disk of B3, obtaining a regular diagram for them. In this context, we obtain a complete finite set of Reidemeister type moves establishing equivalence, up to ambient isotopy, a Wirtinger type presentation for the fundamental group of the complement of the link and a diagrammatic method giving the first homology group. We also compute Alexander polynomial and twisted Alexander polynomials of this class of links, showing their correlation with Reidemeister torsion.

Twisted Alexander Polynomials of Hyperbolic Knots

We study a twisted Alexander polynomial naturally associated to a hyperbolic knot in an integer homology 3-sphere via a lift of the holonomy representation to . It is an unambiguous symmetric Laurent polynomial whose coefficients lie in the trace field of the knot. It contains information about genus, fibering, and chirality, and moreover, is powerful enough to sometimes detect mutation. We calculated this invariant numerically for all 313 209 hyperbolic knots in S 3 with at most 15 crossings, and found that in all cases it gave a sharp bound on the genus of the knot and determined both fibering and chirality. We also study how such twisted Alexander polynomials vary as one moves around in an irreducible component X 0 of the -character variety of the knot group. We show how to understand all of these polynomials at once in terms of a polynomial whose coefficients lie in the function field of X 0. We use this to help explain some of the patterns observed for knots in S 3, and explore a potential relationship between this universal polynomial and the Culler–Shalen theory of surfaces associated to ideal points.