Symmetry of Reidemeister torsion on $SU_2$-representation spaces of knots

TL;DR

This paper investigates symmetries of Reidemeister torsion on the space of irreducible $SU_2$-representations of knot groups, revealing a symmetry of the torsion function related to algebraic and automorphism actions.

Contribution

It demonstrates the symmetry of Reidemeister torsion on a 1-dimensional subspace under specific involution and automorphism actions, linking algebraic structure to geometric invariants.

Findings

Reidemeister torsion function exhibits symmetry about the metrization.

The symmetry is established on a 1-dimensional smooth, oriented, and metrized subspace.

Actions considered include an involution from $SU_2$ structure and outer automorphisms.

Abstract

We study two sorts of actions on the space of conjugacy classes of irreducible $SU_2$-representations of a knot group. One of them is an involution which comes from the algebraic structure of $SU_2$ and the other is the action by the outer automorphism group of the knot group. In particular, we consider them on an 1-dimensional smooth part of the space, which is canonically oriented and metrized via a Reidemeister torsion volume form. As an application we show that the Reidemeister torsion function on the 1-dimensional subspace has symmetry about the metrization.