On homotopies with triple points of classical knots

Details On homotopies with triple points of classical knots On homotopies with triple points of classical knots

2010-05-02

arxiv.org/abs/1005.0136v2

Mathematics

Geometric Topology

10 pages, 13 figures, bugs in figures corrected

Scientific paper

10.1142/S0218216511009911

We consider a knot homotopy as a cylinder in 4-space. An ordinary triple point $p$ of the cylinder is called {\em coherent} if all three branches intersect at $p$ pairwise with the same index. A {\em 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 $v_2(K)$ 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 homotopy.

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