On the multiplicity of Knot Floer order under cabling

Abstract

The knot Floer order $\operatorname{Ord}(K)$ is a knot invariant derived from knot Floer homology that provides bounds on many other invariants, such as the bridge index $\operatorname{br}(K)$ for which $\operatorname{Ord}(K) + 1 \leq \operatorname{br}(K)$. For all $(p,q)$-cables of L-space knots, we show that $\operatorname{Ord}(K) + 1$ 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 $\operatorname{Ord}(K_{p,q})$ in terms of $\operatorname{Ord}(K)$ for all cables of L-space knots. We establish upper bounds under cabling for $\operatorname{Ord}(K)$ 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 $\operatorname{Ord}(K)+1$ is multiplicative.