Whereas the composition of a particle and antiparticle braid results in a braid diagram that is equivalent to the identity braid (on three strands), the composition of a particle and a mirror particle switches two out of the four crossings, so that the full braid is no longer equivalent to the identity.

The three braids shown are constructed from mirror pairs, and are all equivalent to the three joined rings. The underlying permutation for all these braids in $B_3$ is the identity permutation $(123)$.

8 years ago

Note also that the $3 \times 3$ fun representation of this (left hand) braid is a product which evaluates to the diagonal $(1, \omega, \overline{\omega})$. This is the diagonal that we associated to the Z boson, under the quantum Fourier transform.

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