Finite slope cyclic surgeries along toroidal Brunnian links and generalized Properties P and R

TL;DR

This paper classifies when finite slope surgeries on Milnor links produce lens spaces, showing that for certain links only specific surgeries yield lens spaces, and extends results to Brunnian and toroidal Brunnian links using Alexander polynomials and Reidemeister torsions.

Contribution

It provides a complete classification of finite slope surgeries on Milnor links for lens space outcomes and generalizes to Brunnian and toroidal Brunnian links, characterizing their properties.

Findings

Finite slope surgeries on M_3 yield lens spaces in three infinite sequences.

For λ ≥ 4, no finite slope surgery yields a lens space.

M_λ for λ ≥ 3 does not produce S^3 or S^1×S^2 via finite slope surgeries.

Abstract

Let $M_{\lambda}$ be the $\lambda$-component Milnor link. For $\lambda \ge 3$, we determine completely when a finite slope surgery along $M_{\lambda}$ yields a lens space including $S^3$ and $S^1\times S^2$, where {\it finite slope surgery} implies that a surgery coefficient of every component is not $\infty$. For $\lambda =3$ (i.e.\ the Borromean rings), there are three infinite sequences of finite slope surgeries yielding lens spaces. For $\lambda \ge 4$, any finite slope surgery does not yield a lens space. As a corollary, $M_{\lambda}$ for $\lambda \ge 3$ does not yield both $S^3$ and $S^1\times S^2$ by any finite slope surgery. We generalize the results for the cases of {\it Brunnian type links} and toroidal Brunnian type links (i.e.\ Brunnian type links including essential tori in the link complement). Our main tools are Alexander polynomials and Reidemeister torsions. Moreover we characterized toroidal Brunnian links and toroidal Brunnian type links in some senses.