Khovanov homology detects the trefoils
John A. Baldwin, Steven Sivek
Abstract
We prove that Khovanov homology detects the trefoils. Our proof incorporates an array of ideas in Floer homology and contact geometry. It uses open books; the contact invariants we defined in the instanton Floer setting; a bypass exact triangle in sutured instanton homology, proved here; and Kronheimer and Mrowka’s spectral sequence relating Khovanov homology with singular instanton knot homology. As a byproduct, we also strengthen a result of Kronheimer and Mrowka on SU(2) representations of the knot group.
Topics & Concepts
Khovanov homologyFloer homologyInstantonSpectral sequenceKnot (papermaking)Homology (biology)MathematicsMorse homologyPure mathematicsCellular homologyCombinatoricsAlgebra over a fieldMathematical physicsCohomologySymplectic geometryBiologyGeneticsGeneEngineeringChemical engineeringGeometric and Algebraic TopologyBotulinum Toxin and Related Neurological DisordersHomotopy and Cohomology in Algebraic Topology