# Intrinsic linking with linking numbers of specified divisibility

## TL;DR

This paper proves that large enough embeddings of certain high-dimensional simplices in Euclidean space always contain links of multiple spheres with specified divisibility properties in their linking numbers, extending classical results to higher dimensions.

## Contribution

It introduces new topological linking results with divisibility constraints for embeddings of skeletons of simplices in Euclidean space, generalizing known low-dimensional cases.

## Key findings

- Existence of links with linking numbers divisible by q in high-dimensional embeddings.
- Improved upper bounds on the number of vertices needed to force such links.
- Extension of classical linking results from graphs in 3D to higher-dimensional complexes.

## 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 $\mathbb{R}^{2n+1}$ contains a link $L_1\cup\cdots\cup L_r$ consisting of $r$ disjoint $n$-spheres, such that the linking number $link(L_i,L_j)$ is a nonzero multiple of $q$ for all $i\neq j$. This result is new in the classical case $n=1$ (graphs embedded in $\mathbb{R}^3$) as well as the higher dimensional cases $n\geq 2$; and since it implies the existence of a link $L_1\cup\cdots\cup L_r$ such that $|link(L_i,L_j)|\geq q$ for all $i\neq j$, 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 $L_1\cup L_2$ such that $link(L_1,L_2)$ is a nonzero multiple of $q$. Our new bound has growth $O(nq^2)$, in contrast to the previous bound of growth $O(\sqrt{n}4^nq^{n+2})$.