Extrinsic systole of Seifert surfaces and distortion of knots

Abstract

In 1983, Gromov introduced the notion of distortion of a knot, and asked if there are knots with arbitrarily large distortion. In 2011, Pardon proved that the distortion of $T_{p,q}$ is at least $\min\{p,q\}$ up to a constant factor. We prove that the distortion of $T_{p, p+1}\# K$ is at least $p$ up to a constant, independent of $K$. 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 related to combinatorial properties of the monodromy map associated to torus knots.