The torsion order in knot Floer homology is multiplicative for L-space knots under cabling

Abstract: The knot Floer order $\operatorname{Ord}(K)$ is a knot invariant derived from the knot Floer homology. It bounds many invariants that are difficult to calculate. For all L-space cables of L-space knots, we prove that $\operatorname{Ord}(K) + 1$ is multiplicative in $p$ when $g(K) > 1$. The same holds for $g(K) = 1$, provided $q > 2p$. We further compute the knot Floer torsion order in the range $p < q < 2p$, thereby determining $\operatorname{Ord}(K_{p,q})$ from $\operatorname{Ord}(K)$ for all L-space cables of L-space knots. We discuss possible applications to the BB-conjecture.