On the multiplicity of Knot Floer order under cabling

Abstract

The knot Floer order $\mathrm{Ord}(K)$ is a knot invariant derived from knot Floer homology that provides bounds on many other invariants, such as the bridge index $\mathrm{br}(K)$ for which $\mathrm{Ord}(K)+1\le \mathrm{br}(K)$. For all $(p,q)$-cables of L-space knots, we show that $\mathrm{Ord}(K)+1$ is multiplicative in $p$ when $g(K)>1$, and the same holds for $g(K)=1$ provided $q>2p$. We also compute the knot Floer order in the range $q<2p$, thereby determining $\mathrm{Ord}(K_{p,q})$ in terms of $\mathrm{Ord}(K)$ for all cables of L-space knots. We establish upper bounds under cabling for $\mathrm{Ord}(K)$ and discuss potential applications to a conjecture by Krishna and Morton, proving that the braid index of an L-space cable appears as an exponent in its Alexander polynomial if it does for its companion, provided $\mathrm{Ord}(K)+1$ is multiplicative.