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 are determined by their complements. We also prove the surgery characterisation of the unknot for null-homologous knots in –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 in the Brieskorn sphere are also determined by their complements.
"Heegaard Floer homology and knots determined by their complements." Algebr. Geom. Topol. 18 (1) 69 - 109, 2018. https://doi.org/10.2140/agt.2018.18.69