### M Theory Lesson 208

Now consider the $6 \times 6$ Kasteleyn matrix given by This matrix is the unique such matrix with eigenvalues $(1) + (2) = (231)$ and $(1) - (2) = (312)$, the elements of $S_{3}$. It satisfies the relation $K^{2} = K + [0,(312) - (231)]$, 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.

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