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 $[12]$ terms act on either the left or right by composition, giving a direction to the edges $[01]$ and $[02]$, illustrated by the red and green arrows. Thus the 4T relation says that the span and cospan diagrams are equal, in some sense.
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 $[12]$ terms act on either the left or right by composition, giving a direction to the edges $[01]$ and $[02]$, illustrated by the red and green arrows. Thus the 4T relation says that the span and cospan diagrams are equal, in some sense.
posted by Kea | 12:46 PM
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