Generalizations of the Kolmogorov–Barzdin embedding estimates

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.