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.

arXiv comment27 pages, 8 figures; Expositional improvements, corrected normalization of A grading in proof of Lemma 4.10

References in corpus (16)

• Floer homology and knot complements
• Holomorphic disks and genus bounds
• Knot Floer homology detects fibred knots
• Knot Floer homology and rational surgeries
• Legendrian knots, transverse knots and combinatorial Floer homology
• A cylindrical reformulation of Heegaard Floer homology
• Lens space surgeries and L-space homology spheres
• Link Floer homology detects the Thurston norm
• Some knots with surgeries yielding lens spaces
• An algorithm for computing some Heegaard Floer homologies
• On Floer homology and the Berge conjecture on knots admitting lens space surgeries
• A concordance invariant from the Floer homology of double branched covers
• A combinatorial description of knot Floer homology
• The slice-ribbon conjecture for 3-stranded pretzel knots
• Covering spaces and Q-gradings on Heegaard Floer homology
• Link Floer homology and the Thurston norm

Cited by in corpus (29)

• Lens space surgeries and L-space homology spheres
• Khovanov homology, sutured Floer homology, and annular links
• Surgery obstructions from Khovanov homology
• On concordances in 3-manifolds
• Bennequin type inequalities in lens spaces
• On Floer homology and the Berge conjecture on knots admitting lens space surgeries
• Heegaard Floer homology of surgeries on two-bridge links
• Rational cobordisms and integral homology
• The lens space realization problem
• An enhanced Euler characteristic of sutured instanton homology
• Constrained knots in lens spaces
• The Poincare homology sphere, lens space surgeries, and some knots with tunnel number two
• Grid Diagrams and Legendrian Lens Space Links
• Non-simple genus minimizers in lens spaces
• Heegaard Floer correction terms and rational genus bounds
• GridPyM: a Python module to handle grid diagrams
• Upsilon invariants from cyclic branched covers
• Legendrian Contact Homology in Seifert Fibered Spaces
• The Poincare homology sphere and almost simple knots in lens spaces
• Tunnel number one knots satisfy the Berge Conjecture
• A note on Grid Homology in lens spaces: $\mathbb{Z}$ coefficients and computations
• A Combinatorial Description of the Knot Concordance Invariant Epsilon
• A bound for rational Thurston-Bennequin invariants
• Links in the spherical 3-manifold obtained from the quaternion group and their lifts
• Tunnel One, Fibered Links
• On grid homology for lens space links: combinatorial invariance and integral coefficients
• On manifolds with multiple lens space filings
• A Cabling Conjecture for Knots in Lens Spaces
• Virtual quandle for links in lens spaces