扭曲Hopf链环的表示簇

Stratification of $\mathrm{SU}(r)$-character varieties of twisted Hopf links

We describe the geometry of the character variety of representations of the fundamental group of the complement of a Hopf link with $n$ twists, namely $Γ_{n}=\langle x,y \,| \, [x^n,y]=1 \rangle$ into the group $\mathrm{SU}(r)$. For arbitrary rank, we provide geometric descriptions of the loci of irreducible and totally reducible representations. In the case $r = 2$, we provide a complete geometric description of the character variety, proving that this $\mathrm{SU}(2)$-character variety is a deformation retract of the larger $\mathrm{SL}(2,\mathbb{C})$-character variety, as conjectured by Flo…

twisted Hopf链接基本群补集的AGL_r(C)表示变种的分层

Stratification of $\mathrm{AGL}_r(\mathbb{C})$-representation varieties of twisted Hopf links

本文通过GL_r(C)表示变种的分层来分层twisted Hopf链接基本群补集的AGL_r(C)表示变种，并对秩1和2的情况进行了显式描述和计算。

Monodromy of rank 2 twisted Hitchin systems and real character varieties

We introduce a new approach for computing the monodromy of the Hitchin map and use this to completely determine the monodromy for the moduli spaces of $L$-twisted $G$-Higgs bundles, for the groups $G = GL(2,\mathbb{C})$, $SL(2,\mathbb{C})$ and $PSL(2,\mathbb{C})$. We also determine the twisted Chern class of the regular locus, which obstructs the existence of a section of the moduli space of $L$-twisted Higgs bundles of rank $2$ and degree $deg(L)+1$. By counting orbits of the monodromy action with $\mathbb{Z}_2$-coefficients, we obtain in a unified manner the number of components of the chara…

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 and character varieties of 2-bridge knot groups

We study the twisted Alexander polynomial from the viewpoint of the SL(2,C)-character variety of nonabelian representations of a knot group. It is known that if a knot is fibered, then the twisted Alexander polynomials associated with nonabelian SL(2,C)-representations are all monic. In this paper, we show that the converse holds for 2-bridge knots. Furthermore we show that for a 2-bridge knot there exists a curve component in the SL(2,C)-character variety such that if the knot is not fibered then there are only finitely many characters in the component for which the associated twisted Alexand…

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

We give an extension of Fox's formula of the Alexander polynomial for double branched covers over the three-sphere. Our formula provides the Reidemeister torsion of a double branched cover along a knot for a non-trivial one dimensional representation 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. As an application, we show that these products distinguish isotopy classes of two-bridge knots up to mirror images.