Knot distortion via curve combinatorics

Abstract

Distortion is a geometric knot invariant introduced by Gromov in 1983. Since then, the question of estimating distortion for general knots has remained widely open. In this paper, we develop a combinatorial language to study this question. We apply our method to prove new lower bounds for knot distortion. Specifically, we prove that the distortion of $T_{p,p+1}\#K$ is at least $p$ up to a constant, independent of $K$. In the appendix, we also prove that any embedding of a minimal genus Seifert surface for $T_{p,p+1}\#K$ in ${\mathbb{R}}^3$ has small extrinsic systole, in the sense that it contains a non-contractible loop with small ${\mathbb{R}}^3$-diameter relative to the length of the knot. These results are consequences of combinatorial properties of the monodromy map associated to torus knots.

arXiv comment45 pages, v2: fixed referencing issues caused by arXiv's latex compiler, v3: significantly revised exposition