M Theory Lesson 188

The 1997 Broadhurst and Kreimer paper shows how knot crossing numbers correspond to the weight of the MZV. For example, the positive braid in $B_{2}$ defined by the word $\sigma_{1}^{5}$ is decorated with three chords, and this corresponds to $\zeta (5)$ at weight $w = 5$. The trefoil knot $\sigma_{1}^{3}$ is the simplest $B_{2}$ knot, which gives a three loop chord diagram (well, Feynman diagram, actually) using only two chords. The pattern of crossing and non-crossing chords gets more interesting for braids with $s$ strands where $s > 2$, via the relations of the MZV algebra, where depth corresponds to $(s - 1)$ (this is why the example of $\zeta (5)$ only has one argument). Who would have thought it was so easy to do QED with knots and number theory? Once upon a time physicists admitted that group and gauge theory was a complicated, messy business, so why bother with it? M Theory is much more fun. Observe that the number of points on the circle of the chord diagram is $2n$ (or $w + 1$) where $n$ is the number of chords, so $\zeta (5)$ is really a decorated hexagon, our favourite polygon, often used to label the vertices of the three dimensional associahedron.