相关论文: Weights of Markov traces for Alexander polynomials…
We use Reidemeister torsion to study a twisted Alexander polynomial, as defined by Turaev, for links in the projective space. Using sign-refined torsion we derive a skein relation for a normalized form of this polynomial.
In recent years, twisted Alexander polynomial has been playing an important role in low-dimensional topology. For Montesinos links, we develop an efficient method to compute the twisted Alexander polynomial associated to any linear…
We give an explicit formula of the Alexander polynomial of the link obtained by adding an arbitrary number of full twists to positively oriented parallel n-strands in terms of the Alexander polynomials of the links obtained by adding…
The Alexander polynomial (1928) is the first polynomial invariant of links devised to help distinguish links up to isotopy. Fox's conjecture (1962) -- stating that the absolute values of the coefficients of the Alexander polynomial for any…
The mock Alexander polynomial is an extension of the classical Alexander polynomial, defined and studied for (virtual) knots and knotoids by the second and third authors. In this paper we consider the mock Alexander polynomial for…
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…
As a generalization of a fundamental result about the Alexander polynomial of links, we give a description of a Torres condition for the twisted Alexander polynomial of links associated to a unimodular representation.
We derive a formula for the weight system of the multivariable Alexander polynomial using determinants, show that it obeys known relations, and satisfies some of the same relations as the single variable polynomial.
In this paper, we construct mock Alexander polynomials for starred links and linkoids in surfaces. These polynomials are defined as specific sums over states of link or linkoid diagrams that satisfy $f=n$, where $f$ denotes the number of…
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 $SL(2, \mathbb C)$-character variety. We also discuss similar things for the higher…
We consider the Alexander polynomial of a plane algebraic curve twisted by a linear representation. We show that it divides the product of the polynomials of the singularity links, for unitary representations. Moreover, their quotient is…
Inspired by the combinatorial constructions in earlier work of the authors that generalized the classical Alexander polynomial to a large class of spatial graphs with a balanced weight on edges, we show that the value of the Alexander…
We present a reduced Burau-like representation for the mixed braid group on one strand representing links in lens spaces and show how to calculate the Alexander polynomial of a link directly from the mixed braid.
In this paper we present a sequence of link invariants, defined from twisted Alexander polynomials, and discuss their effectiveness in distinguish knots. In particular, we recast and extend by geometric means a recent result of Silver and…
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…
Fox's conjecture from 1962, that the absolute values of the coefficients of the Alexander polynomial of an alternating link are trapezoidal, has remained stubbornly open to this date. Recently Fox's conjecture was settled for all special…
We calculate the twisted Alexander polynomials of all tunnel number one Montesinos knots associated to their $SL_2(\mathbb{C})$-representations and obtain their leading coefficients and degrees. As a corollary, we get some interesting…
For arborescent links, we present an efficient method of computing their Alexander polynomials. Applying this method, we express the Alexander polynomials of Montesinos links in terms of certain functions associated to rational tangles…
The Torres formula, which relates the Alexander polynomial of a link to the Alexander polyomial of its sublinks, admits a generalization to the twisted setting due to Morifuji. This paper uses twisted Reidemeister torsion to obtain a second…
Recently, Bigelow defined a diagrammatic method for calculating the Alexander polynomial of a knot or link by resolving crossings in a planar algebra. I will present my multivariate version of Bigelow's calculation. The advantage to my…