Alexander polynomials of doubly primitive knots

We provide explicit formulas for the Alexander polynomial of Pretzel knots and establish several immediate corollaries, including the characterization of Pretzel knots with a trivial Alexander polynomial.

The twisted Alexander polynomial of a knot is defined associated to a linear representation of the knot group. If there exists a surjective homomorphism of a knot group onto a finite group, then we obtain a representation of the knot group by the composition of the surjective homomorphism and the regular representation of the finite group. In this paper, we provide several formulas of the twisted Alexander polynomial of a knot associated to such representations in terms of th...

We introduce a version of the Alexander polynomial for singular knots and tangles and show how it can be strengthened considerably by introducing a perturbation. For singular long knots, we also prove that our Alexander polynomial agrees with previous definitions.

This article provides an overview of relative strengths of polynomial invariants of knots and links, such as the Alexander, Jones, Homflypt, Kaufman two-variable polynomial, and Khovanov polynomial.

We give a survey of some recent papers by the authors and Masaaki Wada relating the twisted Alexander polynomial with a partial order on the set of prime knots. We also give examples and pose open problems.

We give a volume formula of hyperbolic knot complements using twisted Alexander invariants.

The Alexander polynomial of a knot has been generalized in three different ways to give twisted invariants. The resulting invariants are usually referred to as twisted Alexander polynomials, higher-order Alexander polynomials and $L^2$-Alexander invariants of knots. We quickly recall the definitions and we summarize and compare some of their properties. We also report on work by the authors on $L^2$-Alexander torsions and we conclude the paper with several conjectures on $L^2...

In this paper we use twisted Alexander polynomials to prove that the exterior of a particular graph knot is not fibered. Then we build three 2-component graph links out of this knot, and use similar techniques to discuss their fiberedness.

Twisted torus knots are a generalization of torus knots, obtained by introducing additional full twists to adjacent strands of the torus knots. In this article, we present an explicit formula for the Alexander polynomial of twisted torus knots. Our approach utilizes a presentation of the knot group of twisted torus knots combined with Fox's free differential calculus. As applications, we provide a lower bound for the genus of certain families of twisted torus knots and identi...

A formula for the Alexander polynomial of a 2-bridge knot or link given by Hartley and also by Minkus has a beautiful interpretation as a walk on the integers. We extend this to the 2-variable Alexander polynomial of a 2-bridge link, obtaining a formula that corresponds to a walk on the 2-dimensional integer lattice.