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

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We present a lower bound on the stable 4-genus of a knot based on Casson-Gordon τ -signatures and compute the lower bound for an infinite family of knots, the twist knots.