Heegaard Floer homology and knots determined by their complements

Fyodor Gainullin

doi:10.2140/agt.2018.18.69

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arXiv · 1504.06180

Heegaard Floer homology and knots determined by their complements

Fyodor Gainullin

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Abstract

In this paper 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 $\mathbb{R}P^3$'s 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.

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BibTeX RIS

Fyodor Gainullin. 2017-08-05. Heegaard Floer homology and knots determined by their complements. https://doi.org/10.2140/agt.2018.18.69

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