Thin position and planar surfaces for graphs in the 3-sphere

Li, Tao

doi:10.1090/S0002-9939-09-09878-5

Thin position and planar surfaces for graphs in the 3-sphere
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Proc. Amer. Math. Soc. 138 (2010), 333-340 Request permission

Abstract:

We show that given a trivalent graph in $S^3$, either the graph complement contains an essential almost meridional planar surface or, after edge slides, thin position for the graph is also bridge position. This can be viewed as an extension of a theorem of Thompson to graphs. It follows that any graph complement always contains a useful planar surface.

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