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Planar graphs, family trees and braids. (English) Zbl 0924.57002
Boileau, Michel (ed.) et al., Progress in knot theory and related topics. Paris: Hermann. Trav. Cours. 56, 29-47 (1997).
In [J. Differ. Geom. 34, No. 2, 539-560 (1991; Zbl 0751.05033)], the author and A. Thompson proved:
Theorem 1.2: A finite abstractly planar graph \(\Gamma\subset S^3\) is planar if and only if for any subgraph \(\Gamma'\subseteq \Gamma\), \(\pi_1(S^3| \Gamma')\) is free.
In this article, the author gives an alternative proof of this theorem together with applications, namely to \(2\)-bridge graphs.
For the entire collection see [Zbl 0913.00020].
57M15 Relations of low-dimensional topology with graph theory
05C10 Planar graphs; geometric and topological aspects of graph theory
57M25 Knots and links in the \(3\)-sphere (MSC2010)