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.
Categories: math.GT