  It is known that for every knotted curve in space, there is a line
intersecting it in four places, a quadrisecant. Comparing the order of the four
points along the line and knot we can distinguish three types of quadrisecants;
the alternating ones have the most relevance for the geometry of a knot. In
this paper we prove that every (nontrivial tame) knot has an alternating
quadrisecant. This result had applications to the total curvature, second hull
and ropelength of knots.
