symmetries of spatial graphs and rational twists along spheres and tori空间图形的对称性和理性曲折球体和花床.pdfVIP

Symmetry 2012, 4, 26-38; doi:10.3390/sym4010026 OPEN ACCESS symmetry ISSN 2073-8994 /journal/symmetry Article Symmetries of Spatial Graphs and Rational Twists along Spheres and Tori Toru Ikeda Department of Mathematics, Faculty of Science, Kochi University, 2-5-1 Akebono-cho, Kochi-Shi, Kochi 780-8520, Japan; E-Mail: ikedat@kochi-u.ac.jp Received: 14 November 2011; in revised form: 12 January 2012 / Accepted: 13 January 2012 / Published: 20 January 2012 Abstract: A symmetry group of a spatial graph Γ in S is a finite group consisting of orientation-preserving self-diffeomorphisms of S which leave Γ setwise invariant. In this paper, we show that in many cases symmetry groups of Γ which agree on a regular neighborhood of Γ are equivalent up to conjugate by rational twists along incompressible spheres and tori in the exterior of Γ . Keywords: 3-manifold; geometric topology; symmetry; finite group action; spatial graph; rational twist 1. Introduction There are several approaches to the theory of graphs embedded in the 3-sphere, which are often motivated by molecular chemistry, since the chemical properties of a molecule depend on the symmetries of its molecular bond graph (see, for example, [1]). The symmetries of an abstract graph Γ are described by automorphisms. If Γ is embedded in S , some of these automorphisms are induced from self-diffeomorphisms of S . For example, [2–6] studied the extendabilities of the automorphisms