Recent work [S. O. Bilson-Thompson, e-print arXiv:hep-th/0503213; Bilson-Thompson et al., Class. Quantum Grav. 24, 3975 (2007)] 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 characterize these braids (a braid is a set of nonintersecting curves, that connect one set of N points with another set of N points) and quantities such as electric charge, color charge, and so on [S. O. Bilson-Thompson, e-print arXiv:hep-th/0503213; Bilson-Thompson et al., e-print aXiv:0804.0037]. 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, characterized by a vector of real numbers. This will serve as a basis for more complete discussions of quantum numbers in future work.