Some Ramsey-type results on intrinsic linking ofn–complexes

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 2n+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 non-zero 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.

Intrinsic linking with linking numbers of specified divisibility

“…The proposition plays the role of Flapan, Mellor and Naimi's Lemma 2, and the statement and proof are heavily modelled on theirs, making modifications as needed for it to work in all dimensions and achieve the divisibility condition. From an arithmetic standpoint, realising the divisibility condition largely boils down to repeatedly applying the following simple number-theoretic observation, used by both Fleming [5] and Tuffley [10]:…”

“…The result is a disc D n , and the triangulation D n = ∆ 1 ∪ · · · ∪ ∆ ℓ satisfies Definition 2.2 by construction. In Lemma 2.6 of [10] it is shown that a disc constructed in this way has ℓ + n vertices, and the number of (n − 1)-simplices in ∂D n is ℓ(n − 1) + 2. We note that for n ≥ 2 a path does not necessarily have this form: for instance, for n = 2 the triangulation of a regular n-gon by radii may be given the structure of a path.…”

“…Let K n N be the n-skeleton of an (N − 1)-simplex, which we call the complete n-complex on N vertices. Then K n 2n+4 is intrinsically linked, in the sense that every embedding in R 2n+1 contains a pair of disjoint n-spheres that have nonzero linking number [7,9]; and moreover, the results described above can all be extended to embeddings of sufficiently large complete n-complexes in R 2n+1 [10].…”

“…Since a nonzero multiple of q has magnitude at least q, it also extends the Flapan-Mellor-Naimi result to n ≥ 2. The techniques used to prove it draw heavily on those of Flapan, Mellor and Naimi (for the construction of many-component links with all pairwise linking numbers nonzero), as well as those of our previous paper [10] (for intrinsic linking with n ≥ 2, and constructing links with linking numbers divisible by q). We also obtain a vastly improved upper bound on the number of vertices required in the case r = 2.…”

“…The r = 2 case of Theorem 1.1 is proved as Theorem 1.4 of [10], with an upper bound of growth O( √ n4 n q n+2 ) on the number of vertices required. We re-prove this result with a greatly improved bound with growth O(nq 2 ): Theorem 1.3.…”