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It is known that every nontrivial knot has at least two quadrisecants. Given a knot, we mark each intersection point of each of its quadrisecants. Replacing each subarc between two nearby marked points with a straight line segment joining them, we obtain a polygonal closed curve which we will call the quadrisecant approximation of the given knot. We show that for any hexagonal trefoil knot, there are only three quadrisecants, and the resulting quadrisecant approximation has the same knot type.

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.