Khovanov homology and rational unknotting

Abstract

Building on 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.

69 pages, many figures. Comments welcome! v2: Minor changes, accepted for publication at Quantum Topology

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Cited by in corpus (1)

• Ma-Qiu index, presentation distance, and local moves in knot theory