Q-Series and quantum spin networks

The tail of a quantum spin network in the two-sphere is a [Formula: see text]-series associated to the network. We study the existence of the head and tail functions of quantum spin networks colored by [Formula: see text]. We compute the [Formula: see text]-series for an infinite family of quantum spin networks and give the relation between the tail of these networks and the tail of the colored Jones polynomial. Finally, we show that the family of quantum spin networks under study satisfies a natural product structure.

Morphisms between categorified spin networks

We introduce a graphical calculus for computing morphism spaces between the categorified spin networks of Cooper and Krushkal. The calculus, phrased in terms of planar compositions of categorified Jones–Wenzl projectors and their duals, is then used to study the module structure of spin networks over the colored unknots.