Twisting, Stabilization and Bordered Floer homology

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 $\mathrm{\mid }m\mathrm{\mid }\gg 0$, total dimension of $\widehat{\mathrm{H}\mathrm{F}\mathrm{K}}(K_m)$, $\tau (K_m)$ 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.