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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

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 S3 in terms of homological data derived from CFK(K). This allows us to prove some results about Dehn surgery on knots in S3. 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.

Heegaard Floer homology, Dehn surgery
Mathematical Subject Classification 2010
Primary: 57M27, 57M25
Received: 7 July 2015
Revised: 26 November 2016
Accepted: 13 December 2016
Published: 3 August 2017
Fyodor Gainullin
Department of Mathematics
Imperial College London
South Kensington Campus
United Kingdom