Seifert surgery on knots via Reidemeister torsion and Casson-Walker-Lescop invariant

TL;DR

This paper investigates conditions under which certain surgeries on knots yield Seifert fibered spaces, using Reidemeister torsion and Casson-Walker-Lescop invariants to determine the surgery coefficient.

Contribution

It establishes new criteria linking Reidemeister torsion and Casson-Walker-Lescop invariants to Seifert surgeries on specific knots.

Findings

Identifies conditions for $q=\pm 1$ in surgeries on knots with specific Alexander polynomial

Connects invariants of universal abelian coverings to Seifert fibered space outcomes

Provides a method to detect Seifert fibered spaces via algebraic invariants

Abstract

For a knot $K$ with $\Delta_K(t)\doteq t^2-3t+1$ in a homology $3$-sphere, let $M$ be the result of $2/q$-surgery on $K$. We show that appropriate assumptions on the Reidemeister torsion and the Casson-Walker-Lescop invariant of the universal abelian covering of $M$ imply $q=\pm 1$, if $M$ is a Seifert fibered space.