Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology

Abstract

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 combinatorics presented here, we conjecture that a certain family of knots is characterized by their Floer homology. Coupled with 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.

27 pages, 8 figures; Expositional improvements, corrected normalization of A grading in proof of Lemma 4.10

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• Lens space surgeries and L-space homology spheres
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Cited by in corpus (11)

• Lens space surgeries and L-space homology spheres
• On Floer homology and the Berge conjecture on knots admitting lens space surgeries
• An enhanced Euler characteristic of sutured instanton homology
• The lens space realization problem
• Grid Diagrams and Legendrian Lens Space Links
• GridPyM: a Python module to handle grid diagrams
• Legendrian Contact Homology in Seifert Fibered Spaces
• Tunnel One, Fibered Links
• On grid homology for lens space links: combinatorial invariance and integral coefficients
• Virtual quandle for links in lens spaces
• Links in the spherical 3-manifold obtained from the quaternion group and their lifts