Heegaard Floer homology and alternating knots

2002/09/30 by Peter Ozsváth, Peter Ozsvath, Zoltán Szabó +1 · 239 citations
Mathematics · Medicine · #Alexander polynomial #Algebra over a field #Amino acid #Botulinum Toxin and Related Neurological Disorders #Combinatorics #Floer homology #Geometric and Algebraic Topology #Homology (biology) #Homotopy and Cohomology in Algebraic Topology #Jones polynomial #Khovanov homology #Knot (papermaking) #Knot complement #Knot invariant #Knot polynomial #Knot theory #Mathematics #Pure mathematics #Quantum invariant #Seifert surface #Symplectic geometry #Trefoil knot #Tricolorability #math.GT #math.SG #msc:53D40 #msc:57M25 #msc:57M27 #msc:57R58

paper · pdf · doi:10.2140/gt.2003.7.225

published in Geometry & Topology 7(1), 225-254 (Mathematical Sciences Publishers) · Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper6.abs.html

openalex publication_date 2003/03/24 · arxiv created 2003/05/23 · arxiv updated 2014/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

In an earlier paper, we introduced a knot invariant for a null-homologous knot K in an oriented three-manifold Y , which is closely related to the Heegaard Floer homology of Y . In this paper we investigate some properties of these knot homology groups for knots in the three-sphere. We give a combinatorial description for the generators of the chain complex and their gradings. With the help of this description, we determine the knot homology for alternating knots, showing that in this special case, it depends only on the signature and the Alexander polynomial of the knot (generalizing a result of Rasmussen for two-bridge knots). Applications include new restrictions on the Alexander polynomial of alternating knots.

Citations

• An application of gauge theory to four-dimensional topology
• The support of the Khovanov's invariants for alternating knots
• Heegaard Floer homologies and contact structures
• Holomorphic disks and knot invariants
• Holomorphic disks and topological invariants for closed three-manifolds
• A categorification of the Jones polynomial
• An Introduction to Knot Theory
• An inequality for the h-invariant in instanton Floer theory

Cited by

• Trapezodial property of the generalized Alexander polynomial
• Holomorphic disks and genus bounds
• Knot Floer homology of Whitehead doubles
• Khovanov–Seidel quiver algebras and Ozsváth–Szabó's bordered theory
• On knot Floer homology in double branched covers
• Knot Floer homology of positive braids
• sl3–foam homology calculations
• Holomorphic disks, link invariants and the multi-variable Alexander polynomial
• Heegaard–Floer homology and string links
• Knot Floer homology and Seifert surfaces