Knotty Gauge Theory VI

We see that the gauge theory Jacobi rule may be written in two ways, according to braid crossing choice. Using the higher categorical associated braids, the link to Khovanov homology becomes clearer, since the basic tree diagram becomes a smoothing operation.

Now the combination of the two planar terms in the Jacobi rule is explicit in a crossing diagram, which splits the four legs onto two strands. That is, since the usual skein rule for knots accounts for the three ringed diagram segments, the double Jacobi rule is encoding the joining of knot diagrams via a sum.