On homotopies with triple points of classical knots

TL;DR

This paper introduces a new approach to understanding classical knots using coherent triple points in homotopies, providing a formula for the Vassiliev invariant and insights into knot transformations.

Contribution

It presents a novel formula for the Vassiliev invariant $v_2(K)$ based on triple unknottings and explores properties of triple unknottings in knot homotopies.

Findings

New formula for $v_2(K)$ using triple unknottings

Passing a coherent triple point changes knot type

Existence of non-homotopic triple unknottings with complex singularities

Abstract

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.