Generalizations of the Kolmogorov-Barzdin embedding estimates

TL;DR

This paper explores geometric complexity measures for embeddings of simplicial complexes into Euclidean space, generalizes existing inequalities relating thickness and simplices, and discusses knot distortion with new proofs leveraging hyperbolic manifolds.

Contribution

It generalizes Kolmogorov-Barzdin's embedding estimates to higher-dimensional complexes and provides an alternative proof for large knot distortion using hyperbolic geometry.

Findings

Inequalities relating thickness and simplices in embeddings.

Generalization of Kolmogorov-Barzdin estimates beyond graphs.

Alternative proof of large knot distortion using hyperbolic manifolds.

Abstract

We consider several ways to measure the `geometric complexity' of an embedding from a simplicial complex into Euclidean space. One of these is a version of `thickness', based on a paper of Kolmogorov and Barzdin. We prove inequalities relating the thickness and the number of simplices in the simplicial complex, generalizing an estimate that Kolmogorov and Barzdin proved for graphs. We also consider the distortion of knots. We give an alternate proof of a theorem of Pardon that there are isotopy classes of knots requiring arbitrarily large distortion. This proof is based on the expander-like properties of arithmetic hyperbolic manifolds.