Khovanov homology and rational unknotting
Building on work by Alishahi-Dowlin, we extract a new knot invariant $λ\ge 0$ from universal Khovanov homology. While $λ$ is a lower bound for the unknotting number, in fact more is true: $λ$ 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 $λ(K) = n$. Along the way, following Thompson, we compute the Bar-Natan complexes of rational tangles.
math.GT↗