The Topological Symmetry Group of a Canonically Embedded Complete Graph in $S^3$
Chie NODA · Tokyo Journal of Mathematics · 1997
We show that the topological symmetry group of a canonically embedded complete graph of $n\geq 7$ vertices in the 3-sphere is isomorphic to adihedral group of order $2n$ . Introduction.Throughout this paper graphs are assumed to be finite and simple.The topological symmetry group of an embedded graph in the three-sphere $S^{3}$ was introduced by Jonathan Simon on his lecture at Tokyo in June 1991.On the other hand Takashi Otsuki defined a canonical embedding of a complete graph $K_{n}$ of $n$ vertices into $S^{3}[2]$ [8].The purpose of this paper is to show that the topological symmetry group of a canonical embedding of $K_{n}$ is isomorphic to a dihedral group $D_{n}$ of order $2n$ for $n\geq 7$ . a bijection preserving the adjacency of the vertices}.Let $f:G\rightarrow S^{3}$ be an embedding.Then the topological symmetry group off, denoted byWe remark that $\varphi$ is not necessarily orientation preserving.Thus our definition of $TSG(f)$ is somewhat different from that in [7] and [9].Let $P_{1},$ $P_{2},$ $\cdots,$ $P_{m}$ be smoothly embedded disks in $S^{3}$ such that $P_{i}\cap P_{j}=\partial P_{i}=\partial P_{j}$ for $1\leq i