Khovanov homology and rational unknotting

Building on the work by Alishahi–Dowlin, we extract a new knot invariant \lambda \geq 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\geq 0 , there exists a knot K with \lambda(K) = n . Along the way, following Thompson, we compute the Bar-Natan complexes of rational tangles.