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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Abstract
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 |
|---|---|
| Status: | Published |
| Refereed: | Yes |
| 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 |
| Publisher: | Springer |
| ISSN: | 1022-1824 |
| 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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