Volume 18, issue 1 (2018)

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Heegaard Floer homology and knots determined by their complements

Fyodor Gainullin

Algebraic & Geometric Topology 18 (2018) 69–109

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 P3 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.

Heegaard Floer homology
Mathematical Subject Classification 2010
Primary: 57M25
Secondary: 57M27
Received: 13 October 2015
Revised: 7 May 2017
Accepted: 24 July 2017
Published: 10 January 2018
Fyodor Gainullin
Department of Mathematics
Imperial College London
United Kingdom