#### 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
##### 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 $\Sigma \left(2,3,7\right)$ are also determined by their complements.

##### Keywords
Heegaard Floer homology
Primary: 57M25
Secondary: 57M27