Twisted Alexander polynomials of genus one two-bridge knots

Abstract

Morifuji [Mo2] computed the twisted Alexander polynomial of twist knots for nonabelian representations. In this paper we compute the twisted Alexander polynomial and the Reidemeister torsion of genus one two-bridge knots, a class of knots which includes twist knots. As an application, we give a formula for the Reidemeister torsion of the 3-manifold obtained by a Dehn surgery on a genus one two-bridge knot.2010 Mathematics Classification: Primary 57N10. Secondary 57M25.

On the volume and Chern–Simons invariant for 2-bridge knot orbifolds

“…In [17], explicit volume formulae for the link 7 2 3 (α, α) cone-manifolds are computed. In [49,50], explicit volume formulae for double twist knot cone-manifolds and double twist link cone-manifolds are introduced without explicit computations.…”

Twisted Iwasawa invariants of knots

“…Tran's result [56, Theorem 1] that we cited for twist knots in Proposition 8.4 is originally stated for general two‐bridge knots with genus 1. Although his assertion is stated for representations over $${\mathbb {C}}$$, the argument makes sense over $${\mathbb {Z}}$$, and hence it is applicable to the case over any field.…”

Adjoint twisted Alexander polynomials of genus one two-bridge knots

“…Suppose ρ : π 1 (K) → SL 2 (C) is a nonabelian representation. Up to conjugation, we may assume that The formulas in the following proposition are taken from [19].…”