On Character varieties of two-bridge knot groups

TL;DR

This paper explicitly models the PSL(2,C) and SL(2,C) character varieties for a family of two-bridge knots, computes their genus, and explores fibered properties, providing new insights into knot group representations.

Contribution

It introduces explicit models for character varieties of two-bridge knot groups and resolves a conjecture relating fibered knots to their commensurability classes.

Findings

Explicit models for character varieties of two-bridge knots

Genus computations for components of these varieties

Characterization of fibered knot complements

Abstract

We find explicit models for the PSL(2,C)- and SL(2,C)-character varieties of the fundamental groups of complements in S^3 of an infinite family of two-bridge knots that contains the twist knots. We compute the genus of the components of these character varieties, and deduce upper bounds on the degree of the associated trace fields. We also show that these knot complements are fibered if and only if they are commensurable to a fibered knot complement in a Z/2Z-homology sphere, resolving a conjecture of Hoste and Shanahan.