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Detecting unknotted graphs in 3-space. (English) Zbl 0751.05033
The authors call a finite graph $$G$$ abstractly planar if it is embeddable in $$S^ 2$$ (or equivalently, in the plane), and planar if it lies on an embedded surface in $$S^ 3$$ which is homeomorphic to $$S^ 2$$. The main result is a necessary and sufficient condition for a graph in $$S^ 3$$ to be planar: $$G$$ is planar if and only if it is abstractly planar and for every subgraph $$G'\subseteq G$$ the fundamental group $$\pi(S^ 3-G')$$ is free.
This result can be viewed as an unknotting theorem in the style of Papakyriakopoulos: a simple closed curve (i.e., a graph with one vertex and one edge) in $$S^ 3$$ is unknotted if and only if its complement in $$S^ 3$$ has free fundamental group. The above theorem generalizes also a more recent result of C. McA. Gordon [Topology Appl. 27, 285-299 (1987; Zbl 0634.57007)], where $$G$$ has a single vertex and an arbitrary number of edges.

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 57M25 Knots and links in the $$3$$-sphere (MSC2010) 05C75 Structural characterization of families of graphs
##### Keywords:
knotted graph; embedding in the 3-space; planar
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