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

Work of Ni and Zhang has shown that, for the torus knot Tr,s with r > s > 1, every surgery slope pq 30 67(r2 1)(s2 1) is a characterizing slope. We show that this can be lowered to a bound which is linear in rs, namely pq 43 4 (rs 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 pq–surgery on a knot K in S3 and pq exceeds 4g(K) + 4, then the Alexander polynomial of K is uniquely determined by Y and pq. We also show that if pq–surgery on K bounds a sharp 4–manifold, then Spq3(K) bounds a sharp 4–manifold for all pq pq.

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