where by formula_197 we mean the loop formula_195 traversed in the opposite direction and formula_199 means the loop obtained by going around the loop formula_95 and then along formula_195. See figure below. Given that the matrices are unitary one has that formula_202. Also given the cyclic property of the matrix traces (i.e. formula_203) one has that formula_204. These identities can be combined with each other into further identities of increasing complexity adding more loops. These identities are the so-called Mandelstam identities. Spin networks certain are linear combinations of intersecting Wilson loops designed to address the over-completeness introduced by the Mandelstam identities (for trivalent intersections they eliminate the over-completeness entirely) and actually constitute a basis for all gauge invariant functions.