M Theory Lesson 241

For polygon graphs with $n$ sides one quickly finds that the Tutte polynomial $T_{n}$ is given by

$T_{n} = x^{n-1} + T_{n-1} = x^{n-1} + x^{n-2} + \cdots + x + y$

For $x = -t$ and $y = -1/t$ this becomes

$T_{n} = (-1)^{n-1}t^{n-1} + (-1)^{n-2}t^{n-2} + \cdots - t - \frac{1}{t}$

With the choice $y = 1/x$ the expansion looks similar in form to the infinite Fourier expansion of a function such as the j invariant $J(q) = j(q) - 744$, although $J(q)$ has positive integer coefficients. Naturally, in M theory we would like to associate the polygon graphs with MUB cycles, generalising the Pauli MUB triangle.

Note that $T_{4}$ at a cubed root of unity $t = \omega$ spits out the cubed root $\omega^{2}$. Similarly, $T_{5}$ at a fourth root of unity ($t = i$) is equal to $i$. The Jones invariant for the trefoil equals $-3$ at $t = -1$. The Tutte component, $T_{3} = x^{2} + x + 1/x$, contributes the $3$, because here $x = -t = 1$. Similarly, $T_{n} = n$ whenever $x = y = 1$. That is, setting $t = -1$ is one way to express the usual ordinals $n$ as dimensions of MUB spaces.