Generalizations of the Kolmogorov-barzdin Embedding Estimates

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. In this paper we study quantitative geometric estimates about embedding different spaces into Euclidean space. The main theme is the connection between topology and geometry: if an embedding is topologically complicated, what kind of geometric estimates does that imply? Our results generalize a theorem by Kolmogorov and Barzdin [KB] from the 1960’s about embedding graphs into R3. Let’s begin by recalling what they did. Given a topological embedding from a graph Γ into R3, we say that the embedding has thickness at least T if the distance between any non-adjacent edges is at least T , the distance between two vertices is at least T , and the distance from an edge to a vertex not in the edge is at least T . Roughly speaking, one should imagine the vertices as balls of radius T and the edges as (curved) tubes of thickness T . Kolmogorov and Barzdin mention as examples “logical networks and neuron networks” ([KB] page 194). A logical network probably refers to a computer circuit where the edges correspond to wires and the vertices correspond to gates. A neuron network refers to a brain, where the vertices correspond to neurons and the edges correspond to axons connecting the neurons. Kolmogorov and Barzdin essentially proved the following theorem. Theorem 1. If Γ is a graph of degree at most d with N vertices, then Γ may be embedded with thickness 1 into a 3-dimensional Euclidean ball of radius R ≤ C(d)N1/2. On the other hand, let Γ be a random bipartite graph of degree 6 with 2N vertices. With high probability (tending to 1), there is no embedding of thickness 1 from Γ into a ball of radius cN1/2. Moreover, if we embed Γ into R3 with thickness 1, then the volume of the 1-neighborhood of the image is at least cN3/2. This result and its proof contain several interesting geometric ideas. The most important idea is the discovery of expanders. Kolmogorov and Barzdin essentially observed that a random graph is an expander. Then they proved that expanders are hard to embed in Euclidean space.