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

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).