Twisting, Stabilization and Bordered Floer homology

Abstract

Consider an unknot $c$ in $S^3$ and a knot $K$ in ${S^3-N(c)}$. Twisting the knot $K$ along $c$, or equivalently applying $\frac{1}{m}$-surgery on $c$, produces a family of knots $\{K_m\}_{m \in \mathbb{Z}}$. We use bordered Floer homology and the theory of immersed curve invariants to show that for $|m|\gg0$, total dimension of $\widehat{\mathrm{HFK}}(K_m)$, $\tau(K_{m})$ and thickness of $K_{m}$ are linear functions of $m$. Furthermore, we prove that the extremal coefficients of the Alexander polynomial and extremal knot Floer homologies of $K_m$ stabilize as $m$ goes to infinity. This generalizes results of Chen, Lambert-Cole, Roberts, Van Cott and the author on coherent twist families.