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

TL;DR

This paper investigates conditions under which certain surgeries on knots in homology 3-spheres produce Seifert fibered spaces, revealing relationships between the number of singular fibers and properties of the universal abelian cover.

Contribution

It establishes new criteria linking the number of singular fibers in Seifert fibered spaces to the properties of their universal abelian covers and specifies when surgeries are integral based on Alexander polynomial conditions.

Findings

If the first homology of the universal abelian cover is finite cyclic, then N≥4 if and only if this homology is infinite.

Under certain Alexander polynomial conditions, Seifert fibered spaces from surgery imply the surgery coefficient q is ±1.

The results connect the topology of the knot complement with the algebraic properties of the covering spaces.

Abstract

For a knot $K$ in a homology $3$-sphere $\Sigma$, let $M$ be the result of $2/q$-surgery on $K$, and let $X$ be the universal abelian covering of $M$. Our first theorem is that if the first homology of $X$ is finite cyclic and $M$ is a Seifert fibered space with $N\ge 3$ singular fibers, then $N\ge 4$ if and only if the first homology of the universal abelian covering of $X$ is infinite. Our second theorem is that under an appropriate assumption on the Alexander polynomial of $K$, if $M$ is a Seifert fibered space, then $q=\pm 1$ (i.e.\ integral surgery).