  In a topological description of elementary matter proposed by
Bilson-Thompson, the leptons and quarks of a single generation, together with
the electroweak gauge bosons, are represented as elements of the framed braid
group of three ribbons. By identifying these braids with emergent topological
excitations of ribbon networks, it has been possible to encode this braid model
into the framework of quantum geometry provided by loop quantum gravity. In the
case of trivalent networks, it has not been possible to generate particle
interactions, because the braids correspond to noiseless subsystems, meaning
they commute with the evolution algebra generated by the local Pachner moves.
In the case of tetravalent networks, interactions are only possible when the
model's original simplicity, in which interactions take place via the
composition of braids, is sacrificed. We demonstrate that it possible to
preserve both the original classification of fermions, as well as their
interaction via the braid product, if we embed the braid in a trivalent scheme,
and supplement the local Pachner moves, with a non-local and graph changing
1-handle attachment. Moreover, we use Kauffman-Lins recoupling theory to obtain
invariants of braided networks that distinguish topological configurations
associated to particles in the Bilson-Thompson model.
