Khovanov homology is an unknot-detector

2011/02/10 by P. B. Kronheimer, Tomasz Mrowka · 4 citations
Mathematics · Medicine · #Geometric and Algebraic Topology #Botulinum Toxin and Related Neurological Disorders #Unknot #Khovanov homology #Spectral sequence #Knot (papermaking) #Mathematics #Floer homology #Knot complement #Knot invariant #Pure mathematics #Homology (biology) #Cohomology #Combinatorics #Knot theory #Algebra over a field #Genetics #Biology

paper · pdf · doi:10.1007/s10240-010-0030-y

openalex publication_date 2011/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

Abstract

We prove that a knot is the unknot if and only if its reduced Khovanov cohomology has rank 1. The proof has two steps. We show first that there is a spectral sequence beginning with the reduced Khovanov cohomology and abutting to a knot homology defined using singular instantons. We then show that the latter homology is isomorphic to the instanton Floer homology of the sutured knot complement: an invariant that is already known to detect the unknot.

Citations

• Floer homology and knot complements
• The orientation of Yang-Mills moduli spaces and 4-manifold topology
• Knot homology groups from instantons
• Holomorphic disks and knot invariants
• On the Heegaard Floer homology of branched double-covers
• An instanton-invariant for 3-manifolds
• Floer homology and surface decompositions
• A categorification of the Jones polynomial
• Floer’s work on instanton homology, knots and surgery

Cited by

• Traceless SU(2) representations of 2-stranded tangles
• The symplectic arc algebra is formal
• Khovanov homology detects the trefoils
• Unknotting problem [wikipedia]

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• Khovanov homology and rational unknotting
• Filtrations on instanton homology