Braids, 3-manifolds, elementary particles: number theory and symmetry in particle physics

In this paper, we will describe a topological model for elementary particles based on 3-manifolds. Here, we will use Thurston's geometrization theorem to get a simple picture: fermions as hyperbolic knot complements (a complement $C(K)=S^{3}\setminus(K\times D^{2})$ of a knot $K$ carrying a hyperbolic geometry) and bosons as torus bundles. In particular, hyperbolic 3-manifolds have a close connection to number theory (Bloch group, algebraic K-theory, quaternionic trace fields), which~will be used in the description of fermions. Here, we choose the description of 3-manifolds by branched covers. Every 3-manifold can be described by a 3-fold branched cover of $S^{3}$ branched along a knot. In case of knot complements, one will obtain a 3-fold branched cover of the 3-disk $D^{3}$ branched along a 3-braid or 3-braids describing fermions. The whole approach will uncover new symmetries as induced by quantum and discrete groups. Using the Drinfeld--Turaev quantization, we will also construct a quantization so that quantum states correspond to knots. Particle properties like the electric charge must be expressed by topology, and we will obtain the right spectrum of possible values. Finally, we will get a connection to recent models of Furey, Stoica and Gresnigt using octonionic and quaternionic algebras with relations to 3-braids (Bilson--Thompson model).