M Theory Lesson 154

The three pentagons on the upper half plane (ie. the front of the pair of pants) are marked by boundary pieces from the fundamental domain tesselation of the modular group. The compact pentagon has three edges on the real axis as shown, where the point $\omega$ is the vertex where the three pentagons meet. The real points -1 and 2 define the ends of the larger circles on the ribbon graph. Thus there exist modular transformations mapping points from one pentagon into the others. We would like to consider a triality for the pentagons, such as that given by $(ST)^{3} = 1$, which fixes $\omega$.

Recall also that the three strand braid group $B_3$, which covers the modular group, is the fundamental group for the complement of a trefoil knot in $\mathbb{R}^{3}$. Embedding the Riemann sphere in $\mathbb{R}^{3}$, we can draw the trefoil on the sphere and choose to place the three knot crossings near the points $0$, $1$ and $\infty$, which are the squares of the associahedron. The cell complex dual to this associahedron in $\mathbb{R}^{3}$ turned up in the description of the moduli for the 6-valent ribbon vertex of Mulase et al.

Aside: The critical line of the Riemann zeta function, which is self dual under the functional relation, is the line that divides the hexagon into two pentagons on the front of the pair of pants.