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

By M. Hedden. 2007

87 citations recorded by the source corpus.

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 threesphere 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.

Subfields: Geometry and Topology, Mathematical Physics, Neurology

View Paper: https://www.semanticscholar.org/paper/96db3c48427382ab32d550c9aa1135921fe52e97

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Citations in the corpus

On Knot Floer Homology and Cabling: 2 (2008, result): “To this end, our results provide the following obstructions: Corollary 1.6.”
Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology (2007, background): “In Section 4 of [ He07 ], two knots T1, T2 ⊂ L(p, q) satisfying rk( \”
Lens space surgeries and L-space homology spheres (2007, methodology): “The author would like to thank Matt Hedden for sharing his work in [17], Motoo Tange for kindly providing a copy of his preprint [32], and Ken Baker, Nathan Dunfield, Eli Grigsby, Zoltán Szabó, and Dylan Thurston for helpful conversations.”
Floer simple manifolds and L-space intervals (2015, unknown): “Recall that a knot K in a rational homology sphere Y is Floer simple [20] if the knot Floer homology ĤFK(K) ≃ Z1 .”
On the geography and botany of knot Floer homology (2014, background): “[29,78]) Suppose a knot K in the lens space L(p, q) satisfies”