Reidemeister torsion and lens surgeries on knots in homology 3-spheres I

Abstract

Let $\Sigma(K; p/q)$ be the result of $p/q$-surgery along a knot $K$ in a homology 3-sphere $\Sigma$. We investigate the Reidemeister torsion of $\Sigma(K; p/q)$. Firstly, when the Alexander polynomial of $K$ is the same as that of a torus knot, we give a necessary and sufficient condition for the Reidemeister torsion of $\Sigma(K; p/q)$ to be that of a lens space. Secondly, when the Alexander polynomial of $K$ is of degree $2$, we show that if the Reidemeister torsion of $\Sigma(K; p/q)$ is the same as that of a lens space, then $\varDelta_K(t)=t^2-t+1$.