Twisted embeddings of tori have small extrinsic systole

TL;DR

The paper establishes a systolic inequality for torus embeddings in three-dimensional space, showing that highly twisted tori necessarily contain small non-contractible loops.

Contribution

It introduces a new systolic inequality specific to embeddings of tori in three-dimensional space, highlighting the geometric constraints of twisting.

Findings

Highly twisted tori contain small non-contractible loops.

Systolic inequality relates twisting to loop size.

Embedding complexity affects geometric properties.

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.