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 $J\left(q\right)=j\left(q\right)-744$, although $J\left(q\right)$ 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.