Khovanov homology detects the trefoils

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, proven 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.

54 pages, 23 figures; v2: accepted version

References in corpus (4)

• Right-veering diffeomorphisms of compact surfaces with boundary II
• Lectures on open book decompositions and contact structures
• Contact structures, sutured Floer homology and TQFT
• A note on the knot Floer homology of fibered knots

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• An enhanced Euler characteristic of sutured instanton homology
• Khovanov homology and the cinquefoil
• Gluing maps and cobordism maps for sutured monopole Floer homology
• Instanton Floer homology, sutures, and Euler characteristics
• Small Dehn surgery and SU(2)
• Knot Floer homology, link Floer homology and link detection
• Categorical lifting of the Jones polynomial: a survey
• Khovanov homology detects split links
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• Links of Second Smallest Knot Floer Homology
• Earrings, sutures and pointed links
• Knot surgery formulae for instanton Floer homology II: applications
• SU(2) representations and a large surgery formula
• Floer homology and non-fibered knot detection
• Two detection results of Khovanov homology on links
• Floer homology and right-veering monodromy
• Knot surgery formulae for instanton Floer homology I: the main theorem
• An instanton take on some knot detection results
• Classification of tight contact structures on a solid torus