Arcadian Functor

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Marni D. Sheppeard

Wednesday, October 15, 2008

M Theory Lesson 234

Now let us try to picture how the trefoil and other knots correspond to the cyclic MUB relations, which are quandle relations in the case of the trefoil.

First, mark each arc segment of the knot with a suitable symbol. Starting with the Pauli trefoil, the overcrossing marked $X$ reads as the composition $Y \circ Z$. For the 3d MUBs, note that the first knot in the braid group $B_{4}$ is the 6 crossing $6_{1}$ knot. One can mark the 6 arcs of this knot with elements of the relations, but this only covers two relations at a time. In this case, we have used $Z^{2} = \omega X$ and $Z^{3} = 1$. A similar diagram describes the variables $Y$ and $T$. The 3d MUB relations do not fit as quandle rules on the figure 8 knot, which belongs to $B_{3}$. Note that $\omega$ times the identity matrix is the same as $(YT)^{-1}$, which is another way of saying that $Z^{-1} X Z^{-1} = YT$.


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