A short exposition of S. Parsa's theorems on intrinsic linking and non-realizability

08/25/2018

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by   A. Skopenkov, et al.

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We present a short exposition of the following results by S. Parsa. Let L be a graph such that the join L*{1,2,3} (i.e. the union of three cones over L along their common bases) piecewise linearly (PL) embeds into R^4. Then L admits a PL embedding into R^3 such that any two disjoint cycles have zero linking number. There is C such that every 2-dimensional simplicial complex having n vertices and embeddable into R^4 contains less than Cn^5/3 simplices of dimension 2. We also present corrected statement and proof of the analogue of the second result for intrinsic linking.