Some Ramsey-type results on intrinsic linking of n–complexes

Tuffley, Christopher¹

¹Institute of Fundamental Sciences, Massey University, Private Bag 11222, Palmerston North 4442, New Zealand

Algebraic and Geometric Topology, Tome 13 (2013) no. 3, pp. 1579-1612

Résumé

Define the complete n–complex on N vertices, KNn, to be the n–skeleton of an (N − 1)–simplex. We show that embeddings of sufficiently large complete n–complexes in ℝ2n+1 necessarily exhibit complicated linking behaviour, thereby extending known results on embeddings of large complete graphs in ℝ3 (the case n = 1) to higher dimensions. In particular, we prove the existence of links of the following types: r–component links, with the linking pattern of a chain, necklace or keyring; 2–component links with linking number at least λ in absolute value; and 2–component links with linking number a nonzero multiple of a given integer q. For fixed n the number of vertices required for each of our results grows at most polynomially with respect to the parameter r, λ or q.

Détail

DOI : 10.2140/agt.2013.13.1579

Classification : 57Q45, 57M15, 57Q35
Keywords: intrinsic linking, $n$–complexes, Ramsey theory

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