M Theory Lesson 266

Thinking once again about knotted and coloured string networks, it seems that trivalent vertices on three colours are a lot like a (Tutte) dual representation of a Pauli quandle for a trefoil, where each arc of the knot is a different colour.

These networks also allow crossings, such that the undercrossing preserves the colour. As a quandle rule, such colourings correspond to the action $b \circ a = a$. An example of a quandle that obeys this simple law is one generated by a single invertible operator $M$, such that $M^{n} = 1$. The quandle operation (as is usual for a group) is conjugation, and this acts trivially on $M^{k}$ because we have only powers of $M$ to play with. Without the trivalent vertices, this quandle generates $n$ separate link components, because the arcs at a crossing never mix.