2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/144711We propose a new notation for the states in some models of quantum gravity, namely 4-valent spin networks embedded in a topological three manifold. With the help of this notation, equivalence moves, namely translations and rotations, can be defined, which relate the projections of diffeomorphic embeddings of a spin network. Certain types of topological structures, viz 3-strand braids as local excitations of embedded spin networks, are defined and classified by means of the equivalence moves. This paper formulates a mathematical approach to the further research of particle-like excitations in quantum gravity.24 pages, 16 figures, 5 tablesHigh Energy Physics - TheoryGeneral Relativity and Quantum CosmologyMathematical Physicstext