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 ${ℝ}^3$. 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 $P:E(w_1,w_2)\to {\mathbb{P}}^1$ is defined via theta functions (for $\tau =\frac{w_2}{w_1}$) by

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

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.