Multiplicity-free Quantum 6j-Symbols for {U_q(\mathfrak{sl}_N)}

We conjecture a closed form expression for the simplest class of multiplicity-free quantum 6j-symbols for ${U_q(\mathfrak{sl}_N)}$ . The expression is a natural generalization of the quantum 6j-symbols for ${U_q(\mathfrak{sl}_2)}$ obtained by Kirillov and Reshetikhin. Our conjectured form enables computation of colored HOMFLY polynomials for various knots and links carrying arbitrary symmetric representations.     

Trivalent Graphs,Volume Conjectures and Character Varieties

The generalized volume conjecture and the AJ conjecture (a.k.a. the quantum volume conjecture) are extended to \({U_q(\mathfrak{sl}_2)}\) colored quantum invariants of the theta and tetrahedron graph. The \({\mathrm{SL}(2,\mathbb{C})}\) character variety of the fundamental group of the complement of a trivalent graph with E edges in S ³ is a Lagrangian subvariety of the Hitchin moduli space over the Riemann surface of genus g = E/3 + 1. For the theta and tetrahedron graph, we conjecture that the configuration of the character variety is locally determined by large color asymptotics of the quantum invariants of the trivalent graph in terms of complex Fenchel–Nielsen coordinates. Moreover, the q-holonomic difference equation of the quantum invariants provides the quantization of the character variety.