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

TL;DR

This paper investigates conditions under which certain surgeries on knots yield Seifert fibered spaces, using Reidemeister torsion and the Casson-Walker-Lescop invariant, focusing on a specific knot with a given Alexander polynomial.

Contribution

It establishes a criterion linking Reidemeister torsion to the surgery coefficient for knots with a specific Alexander polynomial, advancing understanding of Seifert surgeries.

Findings

If the universal abelian cover's Reidemeister torsion satisfies a certain condition, then the surgery coefficient q must be ±1.

The result applies to knots with Alexander polynomial t^2 - 3t + 1 in homology 3-spheres.

The work extends previous results on Seifert surgeries using torsion and 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 an appropriate assumption on the Reidemeister torsion of the universal abelian covering of $M$ implies $q=\pm 1$, if $M$ is a Seifert fibered space.