Surgeries, sharp 4–manifolds and the Alexander polynomial

McCoy, Duncan

doi:10.2140/agt.2021.21.2649

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

Duncan McCoy

Algebraic & Geometric Topology 21 (2021) 2649–2676

DOI: 10.2140/agt.2021.21.2649

Abstract

Work of Ni and Zhang has shown that, for the torus knot $T_{r, s}$ with $r > s > 1$ , every surgery slope $p ∕ q \geq \frac{30}{67} (r^{2} - 1) (s^{2} - 1)$ is a characterizing slope. We show that this can be lowered to a bound which is linear in $r s$ , namely $p ∕ q \geq \frac{43}{4} (r s - r - s)$ . 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 $4 g (K) + 4$ , 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 $S_{p^{'} ∕ q^{'}}^{3} (K)$ bounds a sharp $4$ –manifold for all $p^{'} ∕ q^{'} \geq p ∕ q$ .

Keywords

Dehn surgery, sharp 4–manifolds, characterizing slopes, changemaker lattices

Mathematical Subject Classification

Primary: 57K10, 57K18

References

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

Volume 21, issue 5 (2021)