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Khovanov concordance minima and the (4,5) torus knot

Andrew Lobb

TL;DR

This paper investigates global minima in knot concordance under ribbon concordance and its algebraic refinement via a knot homology $H_*$. Focusing on reduced Khovanov homology over ${\mathbb Q}$, it analyzes two spectral sequences, Kronheimer–Mrowka and Lee, which relate ${\rm Kh}$ to instanton homology and to a one-dimensional limit, respectively. By modeling a self-concordance of the $(4,5)$ torus knot $T$ as a basepointed movie and tracking the induced maps through five steps, the authors show that ${\rm Kh}$-maps are isomorphisms on all bigradings except a few, which are later ruled out by Lee spectral sequence constraints. Consequently, ${\rm Kh}(T)$ embeds as a summand in ${\rm Kh}(K)$ for every knot $K$ concordant to $T$, establishing that $T$ is a global minimum for $\leq_{\mathrm{Kh}}$ in its concordance class and providing a concrete, generalizable method for proving algebraic minima via spectral sequence analysis.

Abstract

Ribbon concordance gives a partial order on knot types, and applying a knot homology functor to a ribbon concordance gives an inclusion of the homologies. The question of the existence of global ribbon minima in each concordance class is a generalization of the slice-ribbon conjecture, which asserts that the unknot is the global minimum in its class. We show that the (reduced rational) Khovanov homology of the (4,5) torus knot is a summand in the Khovanov homology of any knot in its concordance class.

Khovanov concordance minima and the (4,5) torus knot

TL;DR

This paper investigates global minima in knot concordance under ribbon concordance and its algebraic refinement via a knot homology . Focusing on reduced Khovanov homology over , it analyzes two spectral sequences, Kronheimer–Mrowka and Lee, which relate to instanton homology and to a one-dimensional limit, respectively. By modeling a self-concordance of the torus knot as a basepointed movie and tracking the induced maps through five steps, the authors show that -maps are isomorphisms on all bigradings except a few, which are later ruled out by Lee spectral sequence constraints. Consequently, embeds as a summand in for every knot concordant to , establishing that is a global minimum for in its concordance class and providing a concrete, generalizable method for proving algebraic minima via spectral sequence analysis.

Abstract

Ribbon concordance gives a partial order on knot types, and applying a knot homology functor to a ribbon concordance gives an inclusion of the homologies. The question of the existence of global ribbon minima in each concordance class is a generalization of the slice-ribbon conjecture, which asserts that the unknot is the global minimum in its class. We show that the (reduced rational) Khovanov homology of the (4,5) torus knot is a summand in the Khovanov homology of any knot in its concordance class.
Paper Structure (4 sections, 1 theorem, 10 equations)

This paper contains 4 sections, 1 theorem, 10 equations.

Table of Contents

  1. Introduction
  2. Three conjectures and a theorem.
  3. Two spectral sequences and the (4,5) torus knot.
  4. Khovanov homology and the (4,5) torus knot.

Key Result

Theorem 2.5

The $(4,5)$ torus knot $T$ is a global minimum for $\leq_{{\rm Kh}}$ in its concordance class.

Theorems & Definitions (7)

  • Definition 1.1
  • Conjecture 2.1: Slice-ribbon conjecture.
  • Conjecture 2.2
  • Conjecture 2.3
  • Remark 2.4
  • Theorem 2.5
  • proof : Proof of Theorem \ref{['thm:t45']}.