Twisting, mutation and knot Floer homology

Let $\mathcal{L}$ be a knot with a fixed positive crossing and ${\mathcal{L}}_n$ the link obtained by replacing this crossing with $n$ positive twists. We prove that the knot Floer homology $\widehat{\text{HFK}}({\mathcal{L}}_n)$ `stabilizes’ as $n$ goes to infinity. This categorifies a similar stabilization phenomenon of the Alexander polynomial. As an application, we construct an infinite family of prime, positive mutant knots with isomorphic bigraded knot Floer homology groups. Moreover, given any pair of positive mutants, we describe how to derive a corresponding infinite family positive mutants with isomorphic bigraded $\widehat{\text{HFK}}$ groups, Seifert genera, and concordance invariant $\tau$.