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

Hedden, Matthew

doi:10.1090/S0002-9947-2010-05117-7

On Floer homology and the Berge conjecture on knots admitting lens space surgeries
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Trans. Amer. Math. Soc. 363 (2011), 949-968 Request permission

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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