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2018 Heegaard Floer homology and knots determined by their complements
Fyodor Gainullin
Algebr. Geom. Topol. 18(1): 69-109 (2018). DOI: 10.2140/agt.2018.18.69

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

Citation

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Fyodor Gainullin. "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

Information

Received: 13 October 2015; Revised: 7 May 2017; Accepted: 24 July 2017; Published: 2018
First available in Project Euclid: 1 February 2018

zbMATH: 06828000
MathSciNet: MR3748239
Digital Object Identifier: 10.2140/agt.2018.18.69

Subjects:
Primary: 57M25
Secondary: 57M27

Rights: Copyright © 2018 Mathematical Sciences Publishers

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Vol.18 • No. 1 • 2018
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