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On Floer homology and the Berge conjecture on knots admitting lens space surgeries


Author: Matthew Hedden
Journal: Trans. Amer. Math. Soc. 363 (2011), 949-968
MSC (2000): Primary 57M25, 57M27
DOI: https://doi.org/10.1090/S0002-9947-2010-05117-7
Published electronically: September 22, 2010
MathSciNet review: 2728591
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Abstract: We complete the first step in a two-part program proposed by Baker, Grigsby, and the author to prove that Berge's construction of knots in the three-sphere which admit lens space surgeries is complete. The first step, which we prove here, is to show that a knot in a lens space with a three-sphere surgery has simple (in the sense of rank) knot Floer homology. The second (conjectured) step involves showing that, for a fixed lens space, the only knots with simple Floer homology belong to a simple finite family. Using results of Baker, we provide evidence for the conjectural part of the program by showing that it holds for a certain family of knots. Coupled with work of Ni, these knots provide the first infinite family of non-trivial knots which are characterized by their knot Floer homology. As another application, we provide a Floer homology proof of a theorem of Berge.


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Additional Information

Matthew Hedden
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: mhedden@math.msu.edu

DOI: https://doi.org/10.1090/S0002-9947-2010-05117-7
Keywords: Knots, lens spaces, Dehn surgery, Berge knot, Floer homology
Received by editor(s): January 5, 2009
Received by editor(s) in revised form: May 23, 2009
Published electronically: September 22, 2010
Additional Notes: The author was supported in part by NSF Grant #DMS-0706979.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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