Non-abelian Reidemeister torsion for twist knots

TL;DR

This paper derives an explicit formula for the non-abelian Reidemeister torsion of twist knots, linking it to the cusp shape for hyperbolic cases, advancing understanding of knot invariants.

Contribution

It provides the first explicit formula for SL_2(C)-non-abelian Reidemeister torsion for twist knots, connecting it to geometric structures.

Findings

Explicit formula for non-abelian Reidemeister torsion of twist knots

Expression of torsion as a rational function of cusp shape for hyperbolic twist knots

Enhanced understanding of the relationship between torsion and knot geometry

Abstract

This paper gives an explicit formula for the SL_2(C)-non-abelian Reidemeister torsion as defined in [Dub06] in the case of twist knots. For hyperbolic twist knots, we also prove that the non-abelian Reidemeister torsion at the holonomy representation can be expressed as a rational function evaluated at the cusp shape of the knot.