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Known and new results on intrinsically knotted and linked graphs
by
Ramin Naimi
Occidental College, Los Angeles, CA 90041
Coauthors: Erica Flapan, Blake Mellor
A graph is called intrinsically knotted (linked) if every embedding of it in S3 contains a nontrivial knot (link). In the early 1980's, Sachs, and independently, Conway and Gordon, showed that K6, the complete graph on six vertices, is intrinsically linked. Conway and Gordon also showed that K7 is intrinsically knotted. In 1995 Robertson, Seymour, and Thomas classified all intrinsically linked graphs by proving a conjecture of Sachs: a graph is intrinsically linked if and only if it contains as a minor either K6 or a graph obtained from K6 by triangle-Y and Y-triangle moves. We will start with the Conway and Gordon theorems and then go through a (partial) survey of old and recent results including this theorem: Let m be any positive integer; then for all sufficiently large n every embedding of Kn contains an m-component link such that any two components of the link have linking number at least m and the second coefficient a2 of the Conway polynomial of every component is at least m.
Paper reference: math/0610501
Date received: October 7, 2008
Copyright © 2008 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # caxq-02.