Intrinsic knotting and linking of complete graphs

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 $| a_{2} (Q) | \geq m$ , where $a_{2} (Q)$ denotes the second coefficient of the Conway polynomial of $Q$ .

Keywords

embedded graphs, intrinsic knotting, intrinsic linking

Mathematical Subject Classification 2000

Primary: 57M25

Secondary: 05C10

References

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Publication

Received: 13 March 2002

Accepted: 28 March 2002

Published: 21 May 2002

Authors

Erica Flapan
Department of Mathematics Pomona College Claremont, CA 91711 USA

Volume 2, issue 1 (2002)

Erica Flapan