Low Dimensional Topology

May 21, 2008

Update: Spacial graphs

Filed under: Uncategorized — Jesse Johnson @ 6:12 pm

A few months ago I asked whether it was possible to embed a theta graph in the 3-sphere so that all the edge loops were isotopic to the trefoil knot. Well it turns out that the answer was already known (and moreover, my intutition was way off.) A much stronger result was proved by Kouki Taniyama and Akira Yasuhara [1]. Given a graph, they ask you to associate to each edge loop an isotopy class of knots. They call a graph adaptable if for any choice of knots, the graph can be embedded in the 3-sphere so that each edge loop is sent to a knot in the associated isotopy class. They not only find a reasonably large class of adaptable knots (all of which contain theta graphs as minors) but they find all graphs that are minor-minimal non-adaptable. (Such a graph is not adaptable, but each of its minors is.) Thus a theta graph can be embedded to contain three trefoils, or any choice of three knots. I haven’t read the paper carefully enough to figure out how to construct such a graph, but I’m curious how complicated a three-trefoil embedding would look.

[1] Realization of knots and links in a spatial graph. (English summary) Topology Appl. 112 (2001), no. 1, 87–109.

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