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Khovanov homology and the cinquefoil

John A. Baldwin, Ying Hu, Steven Sivek

TL;DR

The article proves that reduced Khovanov homology with coefficients in $\mathbb{Z}/2\mathbb{Z}$ detects the torus knot $T(2,5)$ by combining Dowlin’s spectral sequence to knot Floer homology, pseudo-Anosov monodromy analysis, and the Lipshitz–Sarkar stable Khovanov homotopy type. It derives that any candidate knot must be genus-2, fibered, and strongly quasipositive with $\widehat{HFK}$ matching $T(2,5)$, then shows that the monodromy cannot admit fixed points, forcing a braid–axis lifting description in a branched double cover. The argument weaves together Floer theory, annular Khovanov homology, and spectral sequences to reduce to a finite computer-assisted check on mutually braided unknots, ultimately ruling out all nontrivial possibilities and identifying the knot as $T(2,5)$. The work also yields broader corollaries about monodromies of genus-2 knots, Dehn surgeries yielding lens or prism manifolds, and connections to recent developments in Khovanov homotopy theory and annular invariants. This combination of deep Floer, Khovanov, and computational techniques marks a new approach to knot-detection problems beyond genus-1 cases.

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.

Khovanov homology and the cinquefoil

TL;DR

The article proves that reduced Khovanov homology with coefficients in detects the torus knot by combining Dowlin’s spectral sequence to knot Floer homology, pseudo-Anosov monodromy analysis, and the Lipshitz–Sarkar stable Khovanov homotopy type. It derives that any candidate knot must be genus-2, fibered, and strongly quasipositive with matching , then shows that the monodromy cannot admit fixed points, forcing a braid–axis lifting description in a branched double cover. The argument weaves together Floer theory, annular Khovanov homology, and spectral sequences to reduce to a finite computer-assisted check on mutually braided unknots, ultimately ruling out all nontrivial possibilities and identifying the knot as . The work also yields broader corollaries about monodromies of genus-2 knots, Dehn surgeries yielding lens or prism manifolds, and connections to recent developments in Khovanov homotopy theory and annular invariants. This combination of deep Floer, Khovanov, and computational techniques marks a new approach to knot-detection problems beyond genus-1 cases.

Abstract

We prove that Khovanov homology with coefficients in detects the 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.

Paper Structure

This paper contains 9 sections, 12 theorems, 110 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof outline
  3. Other applications
  4. Organization
  5. Acknowledgements
  6. Pseudo-Anosov maps
  7. Floer homology and fixed points
  8. Exchangeable braids and computations
  9. The proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Suppose $K\subset S^3$ is a knot whose reduced Khovanov homology over $\mathbb{Z}/2\mathbb{Z}$ is 5-dimensional and is supported in a single positive $\delta$-grading. Then $K = T(2,5)$.

Figures (1)

  • Figure 1: From left to right, one of the invariant foliations as seen in neighborhoods of: $n$-pronged interior singularities for $n= 3,4,5$; a singularity $x$ on $\partial \Sigma$ (the thickened segment represents a portion of $\partial \Sigma$); and a 1-pronged marking/puncture.

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 18 more