On Transforming a Spatial Graph into a Plane Graph(Statistical Physics and Topology of Polymers with Ramifications to Structure and Function of DNA and Proteins)

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This article is a revised detailed version of the research announcement [Bussei Kenkyu 92 (2009), 16] introducing a complexity of a spatial graph, which is useful to transform a spatial graph (without degree one vertices) into a plane graph. We also introduce related topological invariants for every spatial graph, called the warping degree, and γ-warping degree and (γ,Γ)-warping degree. We also generalize the usual unknotting number of a knot to every spatial graph and introduce related topological invariants for every spatial graph, called the γ-unknotting number, Γ-unknotting number and (γ,Γ)-unknotting number. These invariants are used to define "semi-topological" invariants for a spatial graph with degree one vertices, meaningful even for a knotted arc.
2011-12-16