## Friday, February 4, 2011

### Theory Update 55

For the ribbon braid group $B_3$, we can twist ribbons and also perform braiding within the three strands of $B_3$. One cyclic set of generators is shown.
The generators $(12,23,31)$ are those given by the trefoil quandle. The ribbon strands are labeled $1$, $2$, $3$. We see that two qutrits may be used to label the generators, under the correspondence $1 = XX$ and $12 = \{ XY , YX \}$. In other words, the three ribbons are specified by the letters $X$, $Y$ and $Z$.

Observe that the trefoil quandle rule is now naturally associated to braids of the form $\sigma_{1} \sigma_{2}^{-1} = (12)(23)^{-1}$ $= 12 3^{-1} 2^{-1}$, which are used to specify the Bilson-Thompson particle spectrum. Since qutrits and triality are used to specify braid information, a $3 \times 3$ Koide mass matrix now lives in an exceptional bioctonion Jordan algebra, as do the neutrino and CKM mixing matrices.