We can draw the Jacobi rule of knotty gauge theory a slightly different way.

Recall the connection to knots in renormalisation theory. The loop formed by the traced braid diagram forms the loop of a chord diagram. The example shown on the right is a trefoil knot, with zeta value $\zeta (3)$. A double copy of the right hand side of the Jacobi rule appears in the 4T relation. Now the trivalent vertex on the left of the Jacobi rule is interpreted as a resolution (into trivalent vertices) of the diagram that is half way between the two diagrams on the right (namely the $4$-valent diagram of the node cross over). The pair of Jacobi rules gave us the color kinematic duality of $N = 8$ supergravity.

7 years ago

Very nice book on Vassiliev invariants.

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