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.

arXiv comment21 pages, 1 figure; v2: added new Theorem 1.2, greatly simplified the computations in section 4; v3: accepted version

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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