Tangle replacements and knot Floer homology torsions

TL;DR

This paper establishes that the torsion order in knot Floer homology provides a lower bound on the minimal number of tangle replacements needed to unknot a knot, generalizing previous bounds related to unknotting number and bridge index.

Contribution

It introduces a new bound on tangle replacement complexity based on knot Floer homology torsion order, extending prior bounds on unknotting number and bridge index.

Findings

Torsion order bounds minimal tangle replacements for unknotting.

Generalizes previous bounds on unknotting number.

Connects Floer homology invariants with knot simplification processes.

Abstract

We show that the torsion order $\mathrm{Ord}(K)$ of a knot $K$ in knot Floer homology gives a lower bound on the minimum number $n$ such that an oriented $(n+1)$-tangle replacement unknots $K$. This generalizes earlier results by Alishahi and the author and by Juhasz, Miller and Zemke, that $\mathrm{Ord}(K)$ is a lower bound for both the unknotting number $u(K)$ and for $br(K)-1$, where $br(K)$ denotes the bridge index of $K$.