Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology

2007/10/31 by Kenneth L. Baker, J. Elisenda Grigsby, Matthew Hedden · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.GT #math.SG #msc:57M27 #msc:57R58

paper · pdf · doi:10.1093/imrn/rnn024

published as International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn024, 39 pages · 27 pages, 8 figures; Expositional improvements, corrected normalization of A grading in proof of Lemma 4.10

openalex publication_date 2008/01/01 · arxiv created 2008/08/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Similar to knots in S3, 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 S3 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 S3 admitting lens space surgeries.

Citations

• Floer homology and knot complements
• Lens space surgeries and L-space homology spheres
• Holomorphic disks and knot invariants
• Holomorphic disks and topological invariants for closed three-manifolds
• Link Floer homology and the Thurston norm
• On Floer homology and the Berge conjecture on knots admitting lens space surgeries
• A combinatorial description of knot Floer homology
• A concordance invariant from the Floer homology of double branched covers

Cited by

• Bennequin Type Inequalities in Lens Spaces