M Theory Lesson 232

Recall that the cyclic quandle rules for the three Pauli MUBs are associated to the trefoil knot, where each generator $\sigma_{i}$ labels an arc of the knot diagram. The braiding uses the quandle rule $(XY)X=ZX=Y$.

For the four MUBs of dimension three, there is a unique knot of four crossings, namely the figure 8 knot. The minimal braid representation of this knot is a $(231) = (312)^{2}$ braid on three strands, familiar to M theorists as a basic ribbon diagram with two distinct crossings for each factor of $(312)$. Four crossings in total means four arcs labelled by MUBs. A braid type relation appears in an analogue of the Pauli quandle rules via, for instance,

$XYX = (ZYTZ)YX = ZYT(XTYX)YX$

but we will not bother multiplying out Lie brackets, because they are not particularly interesting.