#### Volume 21, issue 5 (2021)

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Surgeries, sharp $4$–manifolds and the Alexander polynomial

### Duncan McCoy

Algebraic & Geometric Topology 21 (2021) 2649–2676
##### Abstract

Work of Ni and Zhang has shown that, for the torus knot ${T}_{r,s}$ with $r>s>1$, every surgery slope $p∕q\ge \frac{30}{67}\left({r}^{2}-1\right)\left({s}^{2}-1\right)$ is a characterizing slope. We show that this can be lowered to a bound which is linear in $rs$, namely $p∕q\ge \frac{43}{4}\left(rs-r-s\right)$. The main technical ingredient in this improvement is to show that if $Y$ is an $L$–space bounding a sharp $4$–manifold which is obtained by $p∕q$–surgery on a knot $K$ in ${S}^{3}$ and $p∕q$ exceeds $4g\left(K\right)+4$, then the Alexander polynomial of $K$ is uniquely determined by $Y$ and $p∕q$. We also show that if $p∕q$–surgery on $K$ bounds a sharp $4$–manifold, then ${S}_{{p}^{\prime }∕{q}^{\prime }}^{3}\left(K\right)$ bounds a sharp $4$–manifold for all ${p}^{\prime }∕{q}^{\prime }\ge p∕q$.

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##### Keywords
Dehn surgery, sharp 4–manifolds, characterizing slopes, changemaker lattices
##### Mathematical Subject Classification
Primary: 57K10, 57K18
##### Publication
Received: 2 August 2020
Revised: 2 October 2020
Accepted: 20 October 2020
Published: 31 October 2021
##### Authors
 Duncan McCoy Département de mathématiques Université du Québec à Montréal Montreal, QC Canada