Generalizations of the Kolmogorov-Barzdin embedding estimates

Mikhael Gromov, Larry Guth

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)2549-2603
Number of pages55
JournalDuke Mathematical Journal
Volume161
Issue number13
DOIs
StatePublished - 2012

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Simplicial Complex
Knot
Expander
Hyperbolic Manifold
Isotopy
Alternate
Estimate
Euclidean space
Graph in graph theory
Theorem
Generalization
Class

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Generalizations of the Kolmogorov-Barzdin embedding estimates. / Gromov, Mikhael; Guth, Larry.

In: Duke Mathematical Journal, Vol. 161, No. 13, 2012, p. 2549-2603.

Research output: Contribution to journalArticle

Gromov, Mikhael ; Guth, Larry. / Generalizations of the Kolmogorov-Barzdin embedding estimates. In: Duke Mathematical Journal. 2012 ; Vol. 161, No. 13. pp. 2549-2603.
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