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In 1983 Conway and Gordon proved that every embedding of the complete graph $K_6$ in $\mathbb{R}^3$ contains a pair of disjoint cycles that form a non-separable link --- a fact that is expressed by saying $K_6$ is \emph{intrinsically linked}. Since then, a number of authors have shown that embeddings of larger complete graphs necessarily exhibit more complicated linking behaviour, such as links with many components and/or large pairwise linking numbers.

With some adaptions to the proofs, similar results can be established for embeddings of large $n$-complexes in $\mathbb{R}^{2n+1}$. We will look at some of the adaptions required, in the context of proving the existence of two component links with linking number a nonzero multiple of a given integer $q$.

Extra seminar