  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).
