Intrinsic linking with linking numbers of specified divisibility

Abstract

Let n, q and r be positive integers, and let K n N be the n-skeleton of an (N − 1)-simplex. We show that for N sufficiently large every embedding of K n N in R 2n+1 contains a link L 1 ∪ · · · ∪ L r consisting of r disjoint n-spheres, such that the linking number ℓk(L i , L j ) is a nonzero multiple of q for all i = j. This result is new in the classical case n = 1 (graphs embedded in R 3 ) as well as the higher dimensional cases n ≥ 2; and since it implies the existence of a link L 1 ∪ · · · ∪ L r such that |ℓk(L i , L j )| ≥ q for all i = j, 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 L 1 ∪L 2 such that ℓk(L 1 , L 2 ) is a nonzero multiple of q. Our new bound has growth O(nq 2 ), in contrast to the previous bound of growth O( √ n4 n q n+2 ).