Intrinsic linking with linking numbers of specified divisibility

Abstract

Let n, q and r be positive integers, and let K_Nⁿ be the n-skeleton of an (N − 1)-simplex. We show that for N sufficiently large every embedding of K_Nⁿ in ℝ^2n + 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)4ⁿq^{n + 2}).