M Theory Lesson 144

The 9 faced associahedron in $\mathbb{R}^{3}$ keeps popping up in M Theory. Today we'll turn it into a pair of pants, with the three discs at the boundary corresponding to the three squares of the polytope, which were occasionally marked with crossings of a trefoil knot. The numbers labelling the real axis (which are a bit hard to read) are $-1$, $0$, $\frac{1}{2}$, $1$, $2$ and $\infty$. The first image is an extension of the Riemann sphere of lesson 62 to a hexagon on the real axis. The second image is a Grothendieck ribbon graph associated to the j invariant. Note that the ribbons pass through $-1$, $\frac{1}{2}$ and $2$ on the real axis.

By splitting the ribbon into six pieces on the pair of pants, marked with a trivalent vertex on the back and front, and attaching vertices to the nodes of the projection onto the plane, we find exactly 14 vertices, six pentagons and three squares, describing the associahedron. This might just be a bit of fun, until we look at what happens when we glue four of these pants together to form a genus 3 surface. By adding vertices on the squares from each side of the gluing, the pentagons are turned into heptagons, and we get a 24 heptagon tiling of the Klein quartic. Who said operads weren't useful?