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

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 $B_{p,q}$.