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A Short Exposition of Salman Parsa’s Theorems on Intrinsic Linking and Non-realizability

  • Arkadiy SkopenkovEmail author
Article

Abstract

We present a short exposition of the following result by Salman Parsa: LetLbe a graph such that the join\(L*\{1,2,3\}\) (i.e., the union of three cones overLalong their common bases) piecewise linearly (PL) embeds into\(\mathbb {R}^4\). ThenLadmits a PL embedding into\(\mathbb {R}^3\)such that any two disjoint cycles have zero linking number. We also clarify its relation to earlier publications.

Keywords

Linking number Join Embedding 

Mathematics Subject Classification

57M25 57Q45 

Notes

References

  1. 1.
    Parsa, S.: On links of vertices in simplicial \(d\)-complexes embeddable in the euclidean \(2d\)-space. Discrete Comput. Geom. 59(3), 663–679 (2018). (arXiv:1512.05164v4 up to numbering of sections, theorems, etc.; we refer to numbering in the arxiv version)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Parsa, S.: On links of vertices in simplicial \(d\)-complexes embeddable in the euclidean \(2d\)-space (2015). arXiv:1512.05164v6
  3. 3.
    Skopenkov, A.: Realizability of hypergraphs and Ramsey link theory (2014). arXiv:1402.0658
  4. 4.
    Skopenkov, A.: A short exposition of S. Parsa’s theorems on intrinsic linking and non-realizability (2018). arXiv:1808.08363v2
  5. 5.
    Skopenkov, M.: Embedding products of graphs into Euclidean spaces. Fundam. Math. 179(3), 191–198 (2003). arXiv:0808.1199 MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Moscow Institute of Physics and TechnologyDolgoprudnyiRussia
  2. 2.Independent University of MoscowMoscowRussia

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