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

Tuffley, Christopher

doi:10.2140/agt.2013.13.1579

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

Christopher Tuffley

Algebraic & Geometric Topology 13 (2013) 1579–1612

DOI: 10.2140/agt.2013.13.1579

Abstract

Define the complete $n$ –complex on $N$ vertices, $K_{N}^{n}$ , to be the $n$ –skeleton of an $(N - 1)$ –simplex. We show that embeddings of sufficiently large complete $n$ –complexes in $ℝ^{2 n + 1}$ necessarily exhibit complicated linking behaviour, thereby extending known results on embeddings of large complete graphs in $ℝ^{3}$ (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 $q$ .

Keywords

intrinsic linking, $n$–complexes, Ramsey theory

Mathematical Subject Classification 2010

Primary: 57Q45

Secondary: 57M15, 57Q35

References

Publication

Received: 8 February 2012

Revised: 17 January 2013

Accepted: 24 January 2013

Published: 16 May 2013

Authors

Christopher Tuffley
Institute of Fundamental Sciences Massey University Private Bag 11222 Palmerston North 4442 New Zealand
http://www.massey.ac.nz/~ctuffley/

Volume 13, issue 3 (2013)