The Knight Move Conjecture is false

TL;DR

This paper disproves the Knight Move Conjecture by providing a counterexample where the Lee spectral sequence exhibits a nontrivial differential, challenging previous assumptions about Khovanov homology decompositions.

Contribution

It introduces a specific knot that serves as a counterexample, demonstrating the conjecture does not hold universally.

Findings

Counterexample knot with nontrivial Lee differential

Disproof of the universal validity of the Knight Move Conjecture

Implications for the structure of Khovanov homology

Abstract

The Knight Move Conjecture claims that the Khovanov homology of any knot decomposes as direct sums of some "knight move" pairs and a single "pawn move" pair. This is true for instance whenever the Lee spectral sequence from Khovanov homology to Q^2 converges on the second page, as it does for all alternating knots and knots with unknotting number at most 2. We present a counterexample to the Knight Move Conjecture. For this knot, the Lee spectral sequence admits a nontrivial differential of bidegree (1,8).