On the geography and botany of knot Floer homology

Hedden, M. and Watson, L. (2018) On the geography and botany of knot Floer homology. Selecta Mathematica - New Series, 24(2), pp. 997-1037. (doi: 10.1007/s00029-017-0351-5)

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This paper explores two questions: (1) Which bigraded groups arise as the knot Floer homology of a knot in the three-sphere? (2) Given a knot, how many distinct knots share its Floer homology? Regarding the first, we show there exist bigraded groups satisfying all previously known constraints of knot Floer homology which do not arise as the invariant of a knot. This leads to a new constraint for knots admitting lens space surgeries, as well as a proof that the rank of knot Floer homology detects the trefoil knot. For the second, we show that any non-trivial band sum of two unknots gives rise to an infinite family of distinct knots with isomorphic knot Floer homology. We also prove that the fibered knot with identity monodromy is strongly detected by its knot Floer homology, implying that Floer homology solves the word problem for mapping class groups of surfaces with non-empty boundary. Finally, we survey some conjectures and questions and, based on the results described above, formulate some new ones.

Item Type:Articles
Glasgow Author(s) Enlighten ID:Watson, Professor Liam
Authors: Hedden, M., and Watson, L.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Selecta Mathematica - New Series
ISSN (Online):1420-9020
Published Online:30 August 2017
Copyright Holders:Copyright © 2017 The Authors
First Published:First published in Selecta Mathematica - New Series 24(2): 997-1037
Publisher Policy:Reproduced under a Creative Commons License

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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
651711HFFUNDGRP: New connections in low-dimensional topology: Relating Heegaard Floer homology and the fundamental groupLiam WatsonEuropean Commission (EC)631364M&S - MATHEMATICS