M Theory Lesson 180

The Bar-Natan paper continues with a definition of chorded braids and an algebra over (bracketed) chord diagrams which satisfies, in particular, the 4T relation. On $3$ strands, the algebra is given by combinations of the bracket symbols as shown. We can represent the 4T relation by paths on a triangle. Note that the $\left[12\right]$ terms act on either the left or right by composition, giving a direction to the edges $\left[01\right]$ and $\left[02\right]$, illustrated by the red and green arrows. Thus the 4T relation says that the span and cospan diagrams are equal, in some sense.