Holomorphic disks, link invariants and the multi-variable Alexander polynomial

2005/12/31 by Peter Ozsváth, Peter Ozsvath, Zoltán Szabó +1 · 201 citations
Mathematics · #Advanced Combinatorial Mathematics #Alexander polynomial #Algebra over a field #Combinatorics #Composite material #Floer homology #Geometric and Algebraic Topology #Holomorphic function #Homology (biology) #Homotopy and Cohomology in Algebraic Topology #Identity theorem #Invariant (physics) #Khovanov homology #Knot (papermaking) #Knot theory #Link (geometry) #Mathematical physics #Mathematics #Pure mathematics #math.GT #math.SG #msc:57R

paper · pdf · doi:10.2140/agt.2008.8.615

published in Algebraic & Geometric Topology 8(2), 615-692 (Mathematical Sciences Publishers) · Many minor revisions

arxiv created 2007/11/07 · openalex publication_date 2008/05/24 · arxiv updated 2014/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08

Abstract

The knot Floer homology is an invariant of knots in S 3 whose Euler characteristic is the Alexander polynomial of the knot. In this paper we generalize this to links in S 3 giving an invariant whose Euler characteristic is the multi-variable Alexander polynomial. We study basic properties of this invariant, and give some calculations.

Citations

• Floer homology and knot complements
• The support of the Khovanov's invariants for alternating knots
• Morse theory for Lagrangian intersections
• Holomorphic disks and knot invariants
• Heegaard Floer homology and alternating knots
• Pseudo holomorphic curves in symplectic manifolds
• Holomorphic disks and topological invariants for closed three-manifolds
• Introduction to Symplectic Field Theory
• A categorification of the Jones polynomial
• Khovanov homology and the slice genus
• Whitehead torsion

Cited by

• Knot Floer homology of positive braids
• Heegaard–Floer homology and string links
• The Thurston polytope for four-stranded pretzel links
• Functoriality of the EH class and the LOSSinvariant under Lagrangian concordances