Quadrisecant approximation of hexagonal trefoil knots

Abstract

A polygonal knot is a simple closed curve in the Euclidean space R3 obtained by joining finitely many points with straight line segments. The polygon index of a knot k ,denoted by p(k), is the minimal number of edges among all polygonal knots equivalent to k. It is know that if k is a nontrivial knot, then p(k) ¸ 6. Furthermore, the trefoil knot is the only knotted hexagonal knot. An n-secant line for a knot k is an oriented line whose intersection with k has at least n components. An n-secant is an ordered n-tuple of points in k which lie in order on an n-secant line. A 4-secant is called a quadrisecant. It is known that every non-trivial tame knot in $R^3$ has a quadrisecant. Let k be a knot which has finitely many quadrisecants. Then they cut k into finitely many subarcs. Straightening each of the subarcs with the end points fixed, we obtain a polygonal knot $\hat{k}$ which may have self-intersections. We call $\hat{k}$ the quadrisecant approximation of k. The main results show that every hexagonal trefoil knot has exactly three quadrisecants and the quadrisecant approximation of a hexagonal trefoil is a trefoil knot.