Integrality of Kauffman brackets of trivalent graphs

Abstract

We show that Kauffman brackets of colored framed graphs (also known as quantum spin networks) can be renormalized to a Laurent polynomial with integer coefficients by multiplying it by a coefficient which is a product of quantum factorials depending only on the abstract combinatorial structure of the graph. Then we compare the shadow-state sums and the state-sums based on $R$-matrices and Clebsch-Gordan symbols, reprove their equivalence and comment on the integrality of the weight of the states. We also provide short proofs of most of the standard identities satisfied by quantum $6j$-symbols of $U_q(sl_2)$.