Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology

By Kenneth L. Baker, J. E. Grigsby, M. Hedden. 2007

83 citations recorded by the source corpus.

Similar to knots in S 3 , any knot in a lens space has a grid diagram from which one can combinatorially compute all of its knot Floer homology invariants. We give an explicit description of the generators, differentials, and rational Maslov and Alexander gradings in terms of combinatorial data on the grid diagram. Motivated by existing results for the Floer homology of knots in S 3 and the similarity of the resulting combinatorics presented here, we conjecture that a certain family of knots is characterized by their Floer homology. Coupled with the work of the third author, an affirmative answer to this would prove the Berge conjecture, which catalogs the knots in S 3 admitting lens space surgeries.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ff36544e588a93da02fe8220819d18885787c859

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

L-space surgeries, genus bounds, and the cabling conjecture (2010, methodology): “In addition, Baker-Grigsby-Hedden [2] and Rasmussen [33] have proposed programs to prove the completeness of Berge’s construction using Floer homology.”
On the geography and botany of knot Floer homology (2014, background): “The Berge conjecture on which knots admit lens space surgeries has been translated into a finiteness conjecture for the botany problem of simple knots in lens spaces [3,29,78], and recent work of Li and Ni [48] similarly reformulate finite filling questions in terms of botany conjectures.”
On Knot Floer Homology and Cabling: 2 (2008, background): “One area of low-dimensional topology where the Ozsváth-Szabó invariants have had a significant impact is in the study of Dehn surgery [1, 5, 7, 20, 28, 23, 22, 34, 32, 39].”
Cabling, contact structures and mapping class monoids (2010, background): “The known non-torus knots in lens spaces admitting a Dehn surgery yielding S3 all have grid number one (see [7], [3]).”
The lens space realization problem (2010, methodology): “In addition, one step involved in both approaches to the Berge conjecture outlined in [2, 32] is to argue the non-existence of a non-trivial knot K for which K2g(K)−1 is a lens space.”
Lens space surgeries and L-space homology spheres (2007, methodology): “the similar program put forward by Baker, Grigsby, and Hedden in [3]).”