M Theory Lesson 199

Let $\omega$ be the primitive cubed root of unity. Using the ordinary matrix product one finds that the prospective $B_3$ braid generator satisfies $\sigma_{2}^{2} = 1$, but as Lieven points out one can consider fancier matrix products, such as and it follows that instead $\sigma_{2}^{3}$ might be a permutation matrix. Anyhow, one easily verifies that the braid relation $\sigma_{1} \sigma_{2} \sigma_{1} = \sigma_{2} \sigma_{1} \sigma_{2}$ holds. Moreover, in this case of cubed roots of unity, using ordinary matrix product one gets the relation which reduces the braid group to the modular group. Recall that this process views the group $B_{3}$ as the fundamental group of the complement of the trefoil knot in three dimensional space. Note that the generator $\sigma_{1}$ behaves similarly, reduced by the properties of $\omega$, but never quite to the identity. For powers of $\sigma_{1}$ we have the relations where the big dot means the permutation operation, which has no knowledge of the crossing. What a nice way of looking at the modular group! Category theorists have a fancy way of thinking of semidirect products as a piece of two dimensional group structure, but these simple matrices are enough to see what is going on.