The realization problem for noninteger Seifert fibered surgeries

Issa, Ahmad; McCoy, Duncan

doi:10.2140/agt.2023.23.1501

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The realization problem for noninteger Seifert fibered surgeries

Ahmad Issa and Duncan McCoy

Algebraic & Geometric Topology 23 (2023) 1501–1550

DOI: 10.2140/agt.2023.23.1501

Abstract

Conjecturally, the only knots in $S^{3}$ with noninteger surgeries producing Seifert fibered spaces are torus knots and cables of torus knots. We make progress on the associated realization problem. Let $Y$ be a small Seifert fibered space arising by $p ∕ q$ –surgery on a knot in $S^{3}$ , where $p ∕ q$ is positive and a noninteger. Let $e$ denote the weight of the central vertex in the minimal star-shaped plumbing that $Y$ bounds. We show that if $e \leq - 2$ or $e \geq 3$ , then $Y$ can be obtained by $p ∕ q$ –surgery on a torus knot or a cable of a torus knot.

Keywords

Dehn surgery, Seifert fibered spaces, changemaker lattices, exceptional surgeries, Heegaard Floer homology

Mathematical Subject Classification 2010

Primary: 57M25, 57M27

References

Publication

Received: 22 October 2018

Revised: 14 June 2021

Accepted: 24 October 2021

Published: 14 June 2023

Authors

Ahmad Issa
Department of Mathematics University of British Columbia Vancouver, BC Canada

Duncan McCoy
Département de mathématiques Université du Québec à Montréal Montreal, QC Canada

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Volume 23, issue 4 (2023)