Heegaard Floer homology and knots determined by their complements

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 $ℝP^3$ 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 $\mathrm{\Sigma }\left(2,3,7\right)$ are also determined by their complements.