On a complexity of a spatial graph(Knots and soft-matter physics: Topology of polymers and related topics in physics, mathematics and biology)

Abstract

In a research of proteins, molecules, or polymers, it is important to understand geometrically and topologicallyspatial graphs possibly with degree one vertices including knotted arcs. In this article, we introduce a concept of a complexity and related topological invariants for a spatial graph without degree one vertices, called the γ-warping and warping degrees as well as the γ-unknotting and unknotting numbers generalizing the usual unknotting number of a knot. These invariants define geometric invariants for a spatial graph with degree one vertices, meaningful even for a knotted arc.