# LOOM

Unitary matrix for 3 anyons of charge T is F and its inverse F* (F** = F)

$$F = \left[ \begin{array}{ccccc} \phi^{-1} & \sqrt{\phi^{-1}} \\ \sqrt{\phi^{-1}} & -\phi^{-1} \end{array} \right]$$
$$F^{-1} = \left[ \begin{array}{ccccc} \frac{\phi}{\phi + 1} & \frac{1}{(\frac{1}{\phi})^{3/2}(\phi + 1)} \\ \frac{1}{ (\frac{1}{\phi})^{3/2}(\phi + 1)} & -\frac{\phi}{\phi+1} \end{array} \right]$$

R matrix to apply for bottom anyons braiding and its inverse R*

$$R = \left[ \begin{array}{ccccc} e^{-4i\pi/5} & 0 \\ 0 & -e^{-2i\pi/5} \end{array} \right]$$
$$R^{-1} = \left[ \begin{array}{ccccc} e^{4i\pi/5} & 0 \\ 0 & -e^{2i\pi/5} \end{array} \right]$$

B matrix to apply for top anyons braiding and its inverse B*

$$B = F^{-1}RF = \left[ \begin{array}{ccccc} \frac{\phi}{\phi + 1} & \frac{1}{(\frac{1}{\phi})^{3/2}(\phi + 1)} \\ \frac{1}{ (\frac{1}{\phi})^{3/2}(\phi + 1)} & -\frac{\phi}{\phi+1} \end{array} \right] \left[ \begin{array}{ccccc} e^{-4i\pi/5} & 0 \\ 0 & -e^{-2i\pi/5} \end{array} \right] \left[ \begin{array}{ccccc} \phi^{-1} & \sqrt{\phi^{-1}} \\ \sqrt{\phi^{-1}} & -\phi^{-1} \end{array} \right]$$
$$B = \left[ \begin{array}{ccccc} \frac{e^{-4i\pi/5}}{\phi + 1} & 0 \\ 0 & -\frac{e^{-2i\pi/5}}{\phi + 1} \end{array} \right]$$
$$B^{-1} = \left[ \begin{array}{ccccc} e^ {4i\pi/5}(\phi + 1) & 0 \\ 0 & -e^ {2i\pi/5}(\phi + 1) \end{array} \right]$$
We can see that matrix operations R and B keep anyons unmeasured.