On a conjecture of Dunfield, Friedl and Jackson

Twisted Alexander Polynomials of $(-2,3,2n+1)$-Pretzel Knots

We calculate the twisted Alexander polynomials of $(-2,3,2n+1)$-pretzel knots associated to their holonomy representations. As a corollary, we obtain new supporting evidences of Dunfield, Friedl and Jackson's conjecture, that is, the twisted Alexander polynomials of hyperbolic knots associated to their holonomy representations determine the genus and fiberedness of the knots.

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.

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 of hyperbolic links

In this paper we apply the twisted Alexander polynomial to study the fibering and genus detecting problems for oriented links. In particular we generalize a conjecture of Dunfield, Friedl and Jackson on the torsion polynomial of hyperbolic knots to hyperbolic links, and confirm it for an infinite family of hyperbolic 2-bridge links. Moreover we consider a similar problem for parabolic representations of 2-bridge link groups.

纽结的单值化特征的数量有限性

Quantitative finiteness of monic characters of knots

本文针对纽结的 $SL(2, \mathbb{C})$ 特征簇，证明了具有单值化扭曲 Alexander 多项式的特征数量有限，并给出了以纽结群表示和交叉数为参数的显式上界。