Twisting, Stabilization and Bordered Floer homology

TL;DR

This paper investigates how twisting a knot along an unknot affects its Floer homology invariants, showing linear growth and stabilization of certain properties for large twisting parameters using bordered Floer homology.

Contribution

It introduces new results on the linear behavior and stabilization of knot Floer homology invariants under twisting, extending previous work on coherent twist families.

Findings

Total dimension of K and Tau grow linearly with the twisting parameter m.

Extremal coefficients of the Alexander polynomial stabilize as m approaches infinity.

Extremal knot Floer homologies stabilize for large |m|.

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.