ON HOMOTOPIES WITH TRIPLE POINTS OF CLASSICAL KNOTS

Abstract

We consider a knot homotopy as a cylinder in 4-space. An ordinary triple point p of the cylinder is called coherent if all three branches intersect at p pairwise with the same intersection index. A triple unknotting of a classical knot K is a homotopy which connects K with the trivial knot and which has as singularities only coherent triple points. We give a new formula for the first Vassiliev invariant v2(K) by using triple unknottings. As a corollary we obtain a very simple proof of the fact that passing a coherent triple point always changes the knot type. As another corollary we show that there are triple unknottings which are not homotopic as triple unknottings even if we allow more complicated singularities to appear in the homotopy of the triple homotopy.