  Recent work suggests that topological features of certain quantum gravity
theories can be interpreted as particles, matching the known fermions and
bosons of the first generation in the Standard Model. This is achieved by
identifying topological structures with elements of the framed Artin braid
group on three strands, and demonstrating a correspondence between the
invariants used to characterise these braids (a braid is a set of
non-intersecting curves, that connect one set of $N$ points with another set of
$N$ points), and quantities like electric charge, colour charge, and so on. In
this paper we show how to manipulate a modified form of framed braids to yield
an invariant standard form for sets of isomorphic braids, characterised by a
vector of real numbers. This will serve as a basis for more complete
discussions of quantum numbers in future work.
