M Theory Lesson 58

A 5-edged 2-level tree can have at most 4 input leaves. These trees define two dimensional polytopes as shown. The pentagon is familiar from 1-operad tilings, or as the Mac Lane pentagon, and the hexagon from the braiding rule for braided monoidal categories. Only polygons of six sides or less are capable of tiling the plane regularly. Here are some purely pentagonal tilings.

Higher dimensional analogues of the hexagon polytope are known as permutohedra, because they describe permutations on $(d + 1)$ letters. Loday showed that by cutting hypercubes with hyperplanes one can naturally obtain both associahedra and then permutohedra in any dimension. Postnikov et al have done a lot of work on the combinatorics of generalised permutohedra.

As a shameless thief of pretty pictures, today I offer the reader the $6_3$ knot on the Klein surface! A six punctured Klein surface has a moduli space of real dimension 24, which is a number we like a lot.