Intrinsic linking with linking numbers of specified divisibility

Christopher Tuffley


Let n, q and r be positive integers, and let KNn be the n-skeleton of an (N − 1)-simplex. We show that for N sufficiently large every embedding of KNn in ℝ2n + 1 contains a link consisting of r disjoint n-spheres, such that every pairwise linking number is a nonzero multiple of q. This result is new in the classical case n = 1 (graphs embedded in ℝ3) as well as the higher dimensional cases n ≥ 2; and since it implies the existence of an r-component link with all pairwise linking numbers at least q in absolute value, it also extends a result of Flapan et al. from n = 1 to higher dimensions. Additionally, for r = 2 we obtain an improved upper bound on the number of vertices required to force a two-component link with linking number a nonzero multiple of q. Our new bound has growth O(nq2), in contrast to the previous bound of growth O(√(n)4nqn + 2).


Intrinsic linking, complete n-complex, Ramsey theory

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ISSN: 1855-3974

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