  We 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.
