Intrinsic knotting and linking of complete graphs

Abstract

We show that for every $m\in \mathbb{N}$, there exists an $n\in \mathbb{N}$ such that every embedding of the complete graph $K_n$ in ${ℝ}^3$ contains a link of two components whose linking number is at least $m$. Furthermore, there exists an $r\in \mathbb{N}$ such that every embedding of $K_r$ in ${ℝ}^3$ contains a knot $Q$ with $\left|a_2\left(Q\right)\right|\ge m$, where $a_2\left(Q\right)$ denotes the second coefficient of the Conway polynomial of $Q$.