The Knight Move Conjecture is false

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 $\mathbb {Q}^2$ 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 non-trivial differential of bidegree $(1,8)$.