### 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 language | English (US) |
---|---|

Pages (from-to) | 2549-2603 |

Number of pages | 55 |

Journal | Duke Mathematical Journal |

Volume | 161 |

Issue number | 13 |

DOIs | |

State | Published - 2012 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Duke Mathematical Journal*,

*161*(13), 2549-2603. https://doi.org/10.1215/00127094-1812840

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

Research output: Contribution to journal › Article

*Duke Mathematical Journal*, vol. 161, no. 13, pp. 2549-2603. https://doi.org/10.1215/00127094-1812840

}

TY - JOUR

T1 - Generalizations of the Kolmogorov-Barzdin embedding estimates

AU - Gromov, Mikhael

AU - Guth, Larry

PY - 2012

Y1 - 2012

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84871107634&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84871107634&partnerID=8YFLogxK

U2 - 10.1215/00127094-1812840

DO - 10.1215/00127094-1812840

M3 - Article

VL - 161

SP - 2549

EP - 2603

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 13

ER -