Heegaard Floer homology and knots determined by their complements

Gainullin, Fyodor

doi:10.2140/agt.2018.18.69

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Heegaard Floer homology and knots determined by their complements

Fyodor Gainullin

Algebraic & Geometric Topology 18 (2018) 69–109

DOI: 10.2140/agt.2018.18.69

Abstract

We investigate the question of when different surgeries on a knot can produce identical manifolds. We show that given a knot in a homology sphere, unless the knot is quite special, there is a bound on the number of slopes that can produce a fixed manifold that depends only on this fixed manifold and the homology sphere the knot is in. By finding a different bound on the number of slopes, we show that non-null-homologous knots in certain homology $ℝ P^{3}$ are determined by their complements. We also prove the surgery characterisation of the unknot for null-homologous knots in $L$ –spaces. This leads to showing that all knots in some lens spaces are determined by their complements. Finally, we establish that knots of genus greater than $1$ in the Brieskorn sphere $Σ (2, 3, 7)$ are also determined by their complements.

Keywords

Heegaard Floer homology

Mathematical Subject Classification 2010

Primary: 57M25

Secondary: 57M27

References

Publication

Received: 13 October 2015

Revised: 7 May 2017

Accepted: 24 July 2017

Published: 10 January 2018

Authors

Fyodor Gainullin
Department of Mathematics Imperial College London London United Kingdom

Volume 18, issue 1 (2018)