A combinatorial description of knot Floer homology

2009/03/01 by Ciprian Manolescu, Peter Ozsváth, Sucharit Sarkar · 8 citations
Mathematics · Medicine · #Geometric and Algebraic Topology #Botulinum Toxin and Related Neurological Disorders #Homotopy and Cohomology in Algebraic Topology #Floer homology #Mathematics #Knot (papermaking) #Knot complement #Trefoil knot #Combinatorics #Tricolorability #Heegaard splitting #Knot invariant #Torus #Pure mathematics #Knot theory #Geometry #Fibered knot

paper · pdf · doi:10.4007/annals.2009.169.633

openalex publication_date 2009/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/15

Abstract

Given a grid presentation of a knot (or link) K in the three-sphere, we describe a Heegaard diagram for the knot complement in which the Heegaard surface is a torus and all elementary domains are squares. Using this diagram, we obtain a purely combinatorial description of the knot Floer homology of K.

Citations

• Floer homology and knot complements
• Holomorphic disks and knot invariants
• Holomorphic disks and topological invariants for closed three-manifolds
• A categorification of the Jones polynomial
• Khovanov homology and the slice genus
• Knot Floer homology detects fibred knots

Cited by

• Topological algebra of symplectic geometry of symmetric powers
• Concordance maps in knot Floer homology
• An algorithm for computing some Heegaard Floer homologies
• A surgery triangle for lattice cohomology
• L-space surgeries on links
• A link polynomial via a vertex-edge-face state model
• Once-punctured tori and knots in lens spaces
• A Prehistory of n-Categorical Physics
• A flexible construction of equivariant Floer homology and applications
• Floer homology [wikipedia]
• Unknotting problem [wikipedia]

Related

• A combinatorial description of knot Floer homology
• Heegaard diagrams and Floer homology
• An introduction to knot Floer homology
• Grid diagrams and Heegaard Floer invariants
• Braids and combinatorial knot Floer homology