M Theory Lesson 181

As The Everything Seminar pointed out, the 4T relation may be thought of in terms of trivalent knotted diagrams. The chorded circle below is obtained by shifting the internal node down onto the circle, where is it resolved into two trivalent vertices. See this paper by Bar-Natan. Observe that a chorded braid, as drawn in the last lesson, becomes a chorded circle upon composition with a braid such as $(312)$ in $B_3$. A chord diagram can be turned into a knot, allowing self intersection. One rule is to send the endpoints of a chord to a self intersection. The Vassiliev invariants discussed by Bar-Natan use the idea that smooth paths of deformations of embedded knots in three dimensional space should naturally pass through such self intersecting knots.