The stable crossing number of a twist family of knots and the satellite crossing number conjecture

Twisting a given knot $K$ about an unknotted circle $c$ a full $n\in \mathbb{N}$ times, we obtain a “twist family” of knots $\{K_n\}$. Work of Kouno-Motegi-Shibuya implies that for a non-trivial twist family the crossing numbers $\{c(K_n)\}$ of the knots in a twist family grows unboundedly. However potentially this growth is rather slow and may never become monotonic. Nevertheless, based upon the apparent diagrams of a twist family of knots, one expects the growth should eventually be linear. Indeed we conjecture that if $\eta$ is the geometric wrapping number of $K$ about $c$, then the crossing number of $K_n$ grows like $n\eta (\eta -1)$ as $n\to \mathrm{\infty }$. To formulate this, we introduce the “stable crossing number” of a twist family of knots and establish the conjecture for (i) coherent twist families where the geometric wrapping and algebraic winding of $K$ about $c$ agree and (ii) twist families with wrapping number $2$ subject to an additional condition. Using the lower bound on a knot’s crossing number in terms of its genus via Yamada’s braiding algorithm, we bound the stable crossing number from below using the growth of the genera of knots in a twist family. (This also prompts a discussion of the “stable braid index”.) As an application, we prove that highly twisted satellite knots in a twist family where the companion is twisted as well satisfy the Satellite Crossing Number Conjecture.