Counterexamples to the quadrisecant approximation conjecture

A quadrisecant line of a knot K is a straight line which intersects K in four points, and a quadrisecant is a 4-tuple of points of K which lie in order along the quadrisecant line. If K has a finite number of quadrisecants, take W to be the set of points of K which are in a quadrisecant. Replace each subarc of K between two adjacent points of W along K with the straight line segment between them. This gives the quadrisecant approximation of K. It was conjectured that the quadrisecant approximation is always a knot with the same knot type as the original knot. We show that every knot type contains two knots, the quadrisecant approximation of one knot has self-intersections while the quadrisecant approximation of the other knot is a knot with a different knot type.