# How to read the indicies of the Fusion Matrix F for Anyons

I am reading "Introduction to Topological Quantum Computation" from J. K. Pachos (first script from his Website: http://quince.leeds.ac.uk/~phyjkp/Files/IntroTQC.pdf).

I can't seem to understand where the indicies go on the Fusion Matrix F (for Anyons) or what they mean in their special place. The indicies on Page 62 in the book (70 in the pdf) make no sense to me compared to the earlier example on page 58 in the book (66 in the pdf).

Looking at the first example I would conclude that in the second example I should write again $F_{abc}$ instead of $F_{acb}$. I understand both the diagramm and ket vector notation for the fusion process but the $F$ matrix index notation makes no sense to me. I can't find much in the internet about that stuff ether, except his stuff.

I can't connect how I can read $F$ correctly to conclude what the new/old diagramm or ket looks like once I let it transform it. I thought I understood the $F$ notation from the first example but those ideas break down on the second example.

• On these matters P. Bonderson's thesis thesis.library.caltech.edu/2447/2/thesis.pdf is a far more reliable source. See eq. (2.67) and (2.68). – Meng Cheng Jun 23 '15 at 15:59
• Ok, looks like Bonderson has even crazier graphs in his thesis but I will try to figure out. Thank you for that recommendation. – Thomas Elliot Jun 23 '15 at 16:23
• Bonderson actually defined the diagrams more carefully (i.e. orientations). You can just read the relevant sections in the beginning of chapeter 2. – Meng Cheng Jun 23 '15 at 16:25