Intrinsic linking and knotting are arbitrarily complex

Abstract

We show that, given any $n$ and $\alpha$, every embedding of any sufficiently large complete graph in ${\mathbb{R}}^3$ contains an oriented link with components $Q_1$, ..., $Q_n$ such that for every $i\ne j$, $|\lk(Q_i,Q_j)|\geqα$ and $|a_2(Q_i)|\ge \alpha$, where $a_2(Q_i)$ denotes the second coefficient of the Conway polynomial of $Q_i$.

18 pages, 5 figures. Proposition 2 has been strengthened, and Corollary 1 and Proposition 3 have been added to answer a question of Taniyama's

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• Every graph has an embedding in $S^3$ containing no non-hyperbolic knot