(A) study on trisecants of spatial graphs

Abstract

A spatial graph is an embedding of a finite graph into the 3-dimensional Euclidean space $＼mathbb{R}^3$. It is trivial if it is ambient isotopic to an embedding into $＼mathbb{R}^2 ＼subset ＼mathbb{R}^3$. So knots and links in $＼mathbb{R}^3$ can be considered as spatial graphs. A secant line is a straight line which intersects the spatial graph in at least two distinct places. Trisecant, quadrisecant and quintisecant lines are straight lines which intersect the spatial graph in at least three, four, and five distinct places, respectively. A little thought will reveal that non-trivial knots must have uncountably many trisecants. Also, it is easy to see that there exist non-trivial spatial graphs which have no quadrisecants. The relationship between spatial graphs and trisecants is not so immediately clear. The Main Theorem shows that every non-trivial polygonal graph in general position has uncountably many trisecants. As a corollary, we know that a polygonal graph in general position is trivial if and only if it is ambient isotopic to a spatial graph with no trisecants. That is, we must make use of trisecants to seek the geometric meaning of a trivial spatial graph.