Braids, normed division algebras, and Standard Model symmetries

In this paper a structural similarity between a recent braid- and division algebraic description of the unbroken internal symmetries of a single generation of Standard Model (SM) fermions is identified. This unexpected connection between two independently motivated models provides the first step towards unifying them into a unified theory based on braid groups and normed division algebras (NDA). Each of the four NDAs over the reals is shown to contain a representation of a circular braid group. For the complex numbers and the quaternions, the represented circular braid groups are $B_2$ and $B_3^c$, precisely those used to represent leptons and quarks as framed braids in the Helon model of Bilson-Thompson. It is then shown that the twist structure of these framed braids representing fermions coincides exactly with the states that span the minimal left ideals of the complex (chained) octonions, shown by Furey to describe one generation of leptons and quarks with unbroken $SU{\left(3\right)}_c$ and $U{\left(1\right)}_{em}$ symmetry. This identification of basis states of minimal ideals with certain framed braids is possible because the braiding in $B_2$ and $B_3^c$ in the Helon model are interchangeable. It is shown that the framed braids in the Helon model can be written as pure braid words in $B_3^c$ with trivial braiding in $B_2$, something which is not possible for framed braids in general.