Trisecants of Polygonal Knots

Let K be a polygonal knot. A triple abc is a trisecant of K if a, b and c are points in K, no two of which lie on a common edge of K, that are collinear, in this order, in R^3. Fix x in K and let t_x denote the number of trisecants having x as a common point. We show that t_x \u3e= (2cr(K)+1)/(3), where cr(K) is the minimal crossing number of K, when x is the ending point of the trisecants. If we let x appear not only as an end point but also as a middle point in the trisecants, we have conjectured that t_x \u3e= cr(K). In this talk, we will present our progress towards proving this conjecture