M Theory Lesson 208

Now consider the $6\times 6$ Kasteleyn matrix given by This matrix is the unique such matrix with eigenvalues $\left(1\right)+\left(2\right)=\left(231\right)$ and $\left(1\right)-\left(2\right)=\left(312\right)$, the elements of $S_3$. It satisfies the relation $K^2=K+[0,(312)-(231\left)\right]$, using the same notation as the last post. This comes close to being idempotent, but the real idempotents are of course Carl's particle operators. A graph for this $K$ looks like the tiling by hexagons and triangles, which is a rectification of the hexagonal tiling of the plane. Observe that the six edges of the top left (1) form a hexagon within this graph, as do the other circulant components. The graph can be factored into two Hamiltonian circuits of length 12.