A circular embedding of a graph in Euclidean 3-space

摘要

A spatial embedding of a graph G is an embedding of G into the 3-dimensional Euclidean space R~3. J.H. Conway and C.McA. Gordon proved that every spatial embedding of the complete graph on 7 vertices contains a nontrivial knot. A linear spatial embedding of a graph is an embedding which maps each edge to a single straight line segment. In this paper, we construct a linear spatial embedding of the complete graph on 2n - 1 (or 2n) vertices which contains the torus knot T(2n - 5,2) (n ≥ 4). A circular spatial embedding of a graph is an embedding which maps each edge to a round arc. We define the circular number of a knot as the minimal number of round arcs in R~3 among such embeddings of the knot. We show that a knot has circular number 3 if and only if the knot is a trefoil knot, and the figure-eight knot has circular number 4.