Volume 2, issue 1 (2002)

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Intrinsic knotting and linking of complete graphs

Erica Flapan

Algebraic & Geometric Topology 2 (2002) 371–380
 arXiv: math.GT/0205231
Abstract

We show that for every $m\in ℕ$, there exists an $n\in ℕ$ 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 $r\in ℕ$ such that every embedding of ${K}_{r}$ in ${ℝ}^{3}$ contains a knot $Q$ with $|{a}_{2}\left(Q\right)|\ge m$, where ${a}_{2}\left(Q\right)$ denotes the second coefficient of the Conway polynomial of $Q$.

Keywords
embedded graphs, intrinsic knotting, intrinsic linking
Primary: 57M25
Secondary: 05C10