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

TL;DR

This paper investigates the structure of knot concordance classes using Khovanov homology, showing that the (4,5) torus knot's homology appears as a summand in all knots within its class, shedding light on the slice-ribbon conjecture.

Contribution

It demonstrates that the reduced rational Khovanov homology of the (4,5) torus knot is a summand in the homology of any knot concordant to it, revealing a new invariant property.

Findings

Khovanov homology of (4,5) torus knot is a summand in all concordant knots.

Supports the idea of a minimal element in the concordance class.

Provides evidence related to the slice-ribbon conjecture.

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.