The mapping cone formula in Heegaard Floer homology and Dehn surgery on knots in S3

Gainullin, Fyodor

doi:10.2140/agt.2017.17.1917

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The mapping cone formula in Heegaard Floer homology and Dehn surgery on knots in $S^3$

Fyodor Gainullin

Algebraic & Geometric Topology 17 (2017) 1917–1951

DOI: 10.2140/agt.2017.17.1917

Abstract

We write down an explicit formula for the $+$ version of the Heegaard Floer homology (as an absolutely graded vector space over an arbitrary field) of the results of Dehn surgery on a knot $K$ in $S^{3}$ in terms of homological data derived from ${CFK}^{\infty} (K)$ . This allows us to prove some results about Dehn surgery on knots in $S^{3}$ . In particular, we show that for a fixed manifold there are only finitely many alternating knots that can produce it by surgery. This is an improvement on a recent result by Lackenby and Purcell. We also derive a lower bound on the genus of knots depending on the manifold they give by surgery. Some new restrictions on Seifert fibred surgery are also presented.

Keywords

Heegaard Floer homology, Dehn surgery

Mathematical Subject Classification 2010

Primary: 57M27, 57M25

References

Publication

Received: 7 July 2015

Revised: 26 November 2016

Accepted: 13 December 2016

Published: 3 August 2017

Authors

Fyodor Gainullin
Department of Mathematics Imperial College London South Kensington Campus London SW7 2AZ United Kingdom

Volume 17, issue 4 (2017)