Heegaard Floer homology and knots determined by their complements

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 RP3 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(2, 3, 7) are also determined by their complements.