Wednesday, February 2, 2011

Theory Update 50

Some years ago, mathematicians Albequerque and Majid described the octonions in terms of three bits. This is natural, since the $8$ basis elements fit onto a parity cube. Similarly, there is one bit of information in the choice of axis from a complex basis $(1,i)$. The quaternion basis looks more like a tetrahedron though, because a parity square does not put $i$, $j$ and $k$ on the same footing. But a tetrahedron fits nicely into the parity cube.

So moving onto the bioctonions, we would start with a four dimensional parity cube. Two $4$-simplices provide the vertices of the real quaternions and imaginary quaternions, and one cubic face a copy of the (real) octonions.

Using (unbracketed) words in $X$ and $Y$ of length four, we can label all $16$ vertices of a four dimensional cube. Alternatively, using left and right brackets we can also label $16$ vertices with length three words, such as $(XX)Y$. But if the words are cyclic, there should be three kinds of brackets (one around the ends), which introduces a trit index on the set of bits. In this case, there are a total of $24 = 3 \times 8$ vertices to label.

For a (cyclic) bracketing of three strands, we need to look at Bar-Natan's classic paper on nonassociative tangle diagrams. Here there are bits defined by the direction (up or down) of the arrow marked on a strand. Trit labels would instead require a $3$-coloring of each strand, omitting the arrow label. One way to do this would be to use ribbons with only three possible twist types. However, this breaks the cyclicity of three equivalent colors, much as the three cubed roots of unity contain an identity. On the other hand, the identity rule $\omega \cdot 1 = 1$ looks a bit like a Pauli quandle rule, and the Pauli quandle is neatly cyclic.

It from bit and trit.

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