Lens space surgeries along certain 2-component links related with Park's rational blow down, and Reidemeister-Turaev torsion

TL;DR

This paper classifies lens space surgeries along specific 2-component links related to rational blow downs, revealing new and unexpected surgery coefficients using Reidemeister-Turaev torsion techniques.

Contribution

It determines which Dehn surgery coefficients produce lens spaces for two families of links, including unexpected cases, and extends results via Alexander polynomial considerations.

Findings

$A_{m,n}$ yields lens spaces only at $r=mn$ and unexpectedly at $r=7$ for (2,3)

$B_{p,q}$ yields lens spaces for infinitely many $r$

Reidemeister-Turaev torsion is effective in classifying lens space surgeries

Abstract

We study lens space surgeries along two different families of 2-component links, denoted by $A_{m,n}$ and $B_{p,q}$, related with the rational homology 4-ball used in J.\ Park's (generalized) rational blow down. We determine which coefficient $r$ of the knotted component of the link yields a lens space by Dehn surgery. The link $A_{m,n}$ yields a lens space only by the known surgery with $r=mn$ and unexpectedly with $r=7$ for $(m,n)=(2,3)$. On the other hand, $B_{p,q}$ yields a lens space by infinitely many $r$. Our main tool for the proof is the Reidemeister-Turaev torsions, i.e.\ Reidemeister torsions with combinatorial Euler structures. Our results can be extended to the links whose Alexander polynomials are same with those of $A_{m,n}$ and $B_{p,q}$.