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 simplic...

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Bibliographic Details
Published in:Duke mathematical journal 2012-10, Vol.161 (13), p.2549-2603
Main Authors: Gromov, Misha, Guth, Larry
Format: Article
Language:English
Subjects:
53C23
53C99
but in this section
differential geometric analysis on metric spaces
Global geometric and topological methods (á la Gromov)
None of the above
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Summary: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.
ISSN:0012-7094
1547-7398
DOI:10.1215/00127094-1812840