Khovanov homology and the cinquefoil

Abstract

We prove that Khovanov homology with coefficients in $\mathbb{Z}/2\mathbb{Z}$ detects the $(2,5)$ torus knot. Our proof makes use of a wide range of deep tools in Floer homology, Khovanov homology, and Khovanov homotopy. We combine these tools with classical results on the dynamics of surface homeomorphisms to reduce the detection question to a problem about mutually braided unknots, which we then solve with computer assistance.

21 pages, 1 figure; v2: added new Theorem 1.2, greatly simplified the computations in section 4; v3: accepted version

References in corpus (3)

• Knot Floer homology detects fibred knots
• Right-veering diffeomorphisms of compact surfaces with boundary II
• Small genus knots in lens spaces have small bridge number

Cited by in corpus (5)

• An enhanced Euler characteristic of sutured instanton homology
• Small Dehn surgery and SU(2)
• Fixed-point-free pseudo-Anosov homeomorphisms, knot Floer homology and the cinquefoil
• Closures of $3$-braids and detection
• A deformation of Asaeda-Przytycki-Sikora homology