Twisted embeddings of tori have small extrinsic systole

Abstract

We prove a type of systolic inequality for embeddings of $T^2$ in ${\mathbb{R}}^3$. In particular, a highly twisted $T^2$ embedded in ${\mathbb{R}}^3$ must contain a non-contractible loop of small ${\mathbb{R}}^3$-diameter.

20 pages, v2: improved exposition