M Theory Lesson 175

By placing each knot crossing in a box, we see 4 output lines for each box, defining two ribbon strands. Thus there are always twice as many extra faces (as squares) on an associated polytope in $\mathbb{R}^{3}$. The associahedron satisfies this condition, as does the deformed octahedron of cubic triality (which has four globule faces). The Euler characteristic defines a sequence of such polytopes via $E = V + F - 2$.

The ribbon diagram for the trefoil knot is the familiar once punctured torus (elliptic curve). Maps relating elliptic curves to the Riemann sphere go back a long way. In particular, the Weierstrass function $P: E(w_{1} , w_{2}) \rightarrow \mathbb{P}^{1}$ is defined via theta functions (for $\tau = \frac{w_{2}}{w_{1}}$) by

$P (z, \tau) = \pi^{2} \theta^{2} (0, \tau) \theta_{10}^{2} (0, \tau) \frac{\theta_{01}^{2} (0, \tau)}{\theta_{11}^{2} (0, \tau)} - \frac{\pi^{2}}{3} (\theta^{4} (0, \tau) + \theta_{10}^{4} (0, \tau))$

Recall that it is the functional relation on $\theta (0, \tau)$ which gives the functional relation for the Riemann zeta function, and these theta functions also define the triality of the j invariant.