#### Volume 17, issue 4 (2017)

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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
##### 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 ${CFK}^{\infty }\left(K\right)$. 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