Map coloring, q-deformed spin networks, and Turaev-Viro invariants for 3-manifolds

This paper explores a range of ideas that interconnect map coloring, knots, links and 3-manifolds, statistical mechanics, Temperley-Lieb algebra, and generalizations of angular momentum theory to the study of q-deformed spin networks. The authors reformulate the Four Color Theorem in terms of an algebraic problem about the vector cross product algebra in three-dimensional space. The authors also review the bracket polynomial and its relation to the Potts model, chromatic polynomial, Jones polynomial and the Temperley-Lieb algebra. The authors discuss how the general bracket can be expanded in terms of the Temperley-Lieb algebra. This paper also discusses the relationship of the bracket polynomial with the SL(2) quantum group. The authors recall the framework of Penrose spin network theory and shows how this viewpoint correlates the coloring reformulations of previous sections. This paper details our generalization of spin networks to q-deformed spin networks. Finally, this paper describes joint work of the author and Sostenes Lins in constructing 3-manifold invariants of the Turaev-Viro type by using the recoupling theory of q-spin networks. The epilogue discusses connections and applications