Tangle replacements and knot Floer homology torsions

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 $K$.