Khovanov homology detects the trefoils

TL;DR

This paper proves that Khovanov homology uniquely identifies trefoil knots, utilizing advanced tools from Floer homology, contact geometry, and instanton theory to establish this detection property.

Contribution

It introduces a novel proof that Khovanov homology detects trefoils, integrating multiple sophisticated techniques from Floer and contact topology.

Findings

Khovanov homology detects trefoil knots

Established a bypass exact triangle in sutured instanton homology

Strengthened results on $SU(2)$ representations of knot groups

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.