Volume 18, issue 1 (2018)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 25, 1 issue

Volume 24, 9 issues

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN 1472-2739 (online)
ISSN 1472-2747 (print)
Author Index
To Appear
 
Other MSP Journals
Heegaard Floer homology and knots determined by their complements

Fyodor Gainullin

Algebraic & Geometric Topology 18 (2018) 69–109
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 P3 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.

Keywords
Heegaard Floer homology
Mathematical Subject Classification 2010
Primary: 57M25
Secondary: 57M27
References
Publication
Received: 13 October 2015
Revised: 7 May 2017
Accepted: 24 July 2017
Published: 10 January 2018
Authors
Fyodor Gainullin
Department of Mathematics
Imperial College London
London
United Kingdom