Twisted Alexander polynomials of torus links

In this paper, we give an explicit formula for the twisted Alexander polynomial of any torus link and show that it is a locally constant function on the [Formula: see text]-character variety. We also discuss similar things for the higher-dimensional twisted Alexander polynomial and the Reidemeister torsion.

Classification of ribbon 2-knots presented by virtual arcs with up to four crossings

We consider classification of the oriented ribbon 2-knots presented by virtual arcs with up to four crossings. We show the difference by the 2-fold branched covering space, the Alexander polynomial, the number of representations of the knot group to SL[Formula: see text], [Formula: see text] a finite field, and the twisted Alexander polynomial.

Adjoint twisted Alexander polynomial of twisted Whitehead links

The adjoint twisted Alexander polynomial has been computed for twist knots [A. Tran, Twisted Alexander polynomials with the adjoint action for some classes of knots, J. Knot Theory Ramifications 23(10) (2014) 1450051], genus one two-bridge knots [A. Tran, Adjoint twisted Alexander polynomials of genus one two-bridge knots, J. Knot Theory Ramifications 25(10) (2016) 1650065] and the Whitehead link [J. Dubois and Y. Yamaguchi, Twisted Alexander invariant and nonabelian Reidemeister torsion for hyperbolic three dimensional manifolds with cusps, Preprint (2009), arXiv:0906.1500 ]. In this paper, we compute the adjoint twisted Alexander polynomial and nonabelian Reidemeister torsion of twisted Whitehead links.

Twisted Alexander polynomials of 2-bridge knots associated to dihedral representations

Let [Formula: see text] be a non-abelian semi-direct product of a cyclic group [Formula: see text] and an elementary abelian [Formula: see text]-group [Formula: see text] of order [Formula: see text], [Formula: see text] being a prime and [Formula: see text]. Suppose that the knot group [Formula: see text] of a knot [Formula: see text] in the [Formula: see text]-sphere is represented on [Formula: see text]. Then we conjectured (and later proved) that the twisted Alexander polynomial [Formula: see text] associated to [Formula: see text] is of the form: [Formula: see text], where [Formula: see text] is the Alexander polynomial of [Formula: see text] and [Formula: see text] is an integer polynomial in [Formula: see text]. In this paper, we present a proof of the following. For a [Formula: see text]-bridge knot [Formula: see text] in [Formula: see text], if [Formula: see text] and [Formula: see text], then [Formula: see text] is written as [Formula: see text], where [Formula: see text] is the set of [Formula: see text]-bridge knots whose knot groups map on that of [Formula: see text] with [Formula: see text] odd.

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.

Connected sum of representations of knot groups

When two boundary-parabolic representations of knot groups are given, we introduce the connected sum of these representations and show several natural properties including the unique factorization property. Furthermore, the complex volume of the connected sum is the sum of each complex volumes modulo iπ2 and the twisted Alexander polynomial of the connected sum is the product of each polynomials with normalization.

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].

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

In 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.

ON THE TWISTED ALEXANDER POLYNOMIAL FOR REPRESENTATIONS INTO SL2(ℂ)

We study the twisted Alexander polynomial ΔK,ρ of a knot K associated to a non-abelian representation ρ of the knot group into SL2(ℂ). It is known for every knot K that if K is fibered, then for every non-abelian representation, ΔK,ρ is monic and has degree 4g(K) – 2 where g(K) is the genus of K. Kim and Morifuji recently proved the converse for 2-bridge knots. In fact they proved a stronger result: if a 2-bridge knot K is non-fibered, then all but finitely many non-abelian representations on some component have ΔK,ρ non-monic and degree 4g(K) – 2. In this paper, we consider two special families of non-fibered 2-bridge knots including twist knots. For these families, we calculate the number of non-abelian representations where ΔK,ρ is monic and calculate the number of non-abelian representations where the degree of ΔK,ρ is less than 4g(K) – 2.