Theory Update 65

We can break up the $21$ outer paths of the tetractys for three qutrits using three trefoil quandles. Recall that the trefoil rules may be expressed in the cyclic form $\{ a \circ b = c, b \circ c = a, c \circ a = b \}$, and this is applied to the multiplication $(123)(456) = ((14)(25)(36))$ on three qudit words. The three quandles are coloured black, red and green.
These $21$ paths can be mapped to a $3 \times 3$ planar matrix by interpreting the trefoil cycles as $1$-circulant components.
This is just a two qutrit matrix with the first coordinate doubled. There are also three trefoil quandles of the form $\{ XYZ, YYY, ZYX \}$, using the central $6$ path neutrino vertex along with the corner charged lepton vertices. A diagonal element $YYY$ belongs to two trefoil quandles, whereas an off diagonal element $XXY$ belongs to only one.

Note also that the other three trefoil quandles form the $2$-circulant pieces of the above matrix. Each $2$-circulant uses one path of each colour (black, red and green). Thus there are at least $9$ trefoil quandles in the tetractys path set.