Intrinsic linking and knotting are arbitrarily complex

Abstract

We show that, given any $n$ and $\alpha$, any embedding of any sufficiently large complete graph in $\mathbb{R}^3$ contains an oriented link with components $Q_1, \ldots, Q_n$ such that for every $i\not =j$, $|{\rm lk}(Q_i,Q_j)|\geq\alpha$ and $|a_2(Q_i)|