Khovanov homology and rational unknotting

Abstract

Building on the work by Alishahi–Dowlin, we extract a new knot invariant $\lambda \ge 0$ from universal Khovanov homology. While $\lambda$ is a lower bound for the unknotting number, in fact more is true: $\lambda$ is a lower bound for the proper rational unknotting number (the minimal number of rational tangle replacements preserving connectivity necessary to relate a knot to the unknot). Moreover, we show that, for all $n\ge 0$, there exists a knot $K$ with $\lambda (K)=n$. Along the way, following Thompson, we compute the Bar-Natan complexes of rational tangles.