Khovanov homology and rational unknotting

Abstract

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.

Rational tangle replacements and knot Floer homology

“…[AD19] and [CGL + 20]). Recently, Iltgen, Lewark and Marino proved that their invariant λ [ILM21], the best known unknotting bound from Khovanov homology, is in fact a lower bound for the proper rational distance u q , defined as follows:…”

“…More, recently, McCoy and Zenter adapted the so called Montesinos trick, to study proper rational unknotting as well [MZ21]. The work of Iltgen, Lewark and Marino [ILM21] is the first connection between (proper) rational unknotting and the quantum invariants.…”