Knotted 4-regular graphs II: Consistent application of the Pachner moves

A common choice for the evolution of the knotted graphs in loop quantum gravity is to use the Pachner moves, adapted to graphs from their dual triangulations. Here, we show that the natural way to consistently use these moves is on framed graphs with edge twists, where the Pachner moves can only be performed when the twists, and the vertices the edges are incident on, meet certain criteria. For other twists, one can introduce an algebraic object, which allow any knotted graph with framed edges to be written in terms of a generalized braid group.