Knotted symmetric graphs

Livingston, Charles

doi:10.1090/S0002-9939-1995-1273507-3

Knotted symmetric graphs
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by Charles Livingston PDF

Proc. Amer. Math. Soc. 123 (1995), 963-967 Request permission

Abstract:

For a knotted graph in ${S^3}$ we define the vertex constant group, a quotient of the fundamental group of the complement. For planar graphs the group is cyclic. For graphs with periodic symmetry the group is related to the fundamental group of the branched cover of ${S^3}$ branched over knots contained in the quotient of the graph under the symmetry. These tools are used to demonstrate that a large family of knotted graphs are not planar.

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