Reidemeister torsion for linear representations and Seifert surgery on knots

TL;DR

This paper investigates a Reidemeister torsion invariant for 3-manifolds derived from linear representations through finite groups, providing a Dehn surgery formula and applications to Seifert fibered spaces, offering new insights beyond abelian torsion.

Contribution

It introduces a new Reidemeister torsion invariant for linear representations, establishes a Dehn surgery formula, and applies it to identify Seifert fibered surgeries on knots.

Findings

Derived a Dehn surgery formula for the invariant.

Computed the invariant for Seifert manifolds over S^2.

Provided a necessary condition for Seifert fibered surgeries that surpasses abelian torsion limitations.

Abstract

We study an invariant of a 3-manifold which consists of Reidemeister torsion for linear representations which pass through a finite group. We show a Dehn surgery formula on this invariant and compute that of a Seifert manifold over $S^2$. As a consequence we obtain a necessary condition for a result of Dehn surgery along a knot to be Seifert fibered, which can be applied even in a case where abelian Reidemeister torsion gives no information.