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Discrete & Computational Geometry

, Volume 59, Issue 3, pp 663–679 | Cite as

On the Links of Vertices in Simplicial d-Complexes Embeddable in the Euclidean 2d-Space

  • Salman Parsa
Article
  • 79 Downloads

Abstract

We consider d-dimensional simplicial complexes which can be PL embedded in the 2d-dimensional Euclidean space. In short, we show that in any such complex, for any three vertices, the intersection of the link-complexes of the vertices is linklessly embeddable in the \((2d-1)\)-dimensional Euclidean space. In addition, we use similar considerations on links of vertices to derive a new asymptotic upper bound on the total number of d-simplices in an (continuously) embeddable complex in 2d-space with n vertices, improving known upper bounds, for all \(d \ge 2\). Moreover, we show that the same asymptotic bound also applies to the size of d-complexes linklessly embeddable in the \((2d+1)\)-dimensional space.

Keywords

Embeddability Simplicial complex f-vector Linkless embedding 

Mathematics Subject Classification

57Q35 52C45 68U05 

Notes

Acknowledgements

The author is indebted to Herbert Edelsbrunner for bringing Lemma 4.3 to his notice. The author also thanks Tamal Dey and Uli Wagner for helpful discussions about the problem of this paper.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesSharif University of TechnologyTehranIran

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