Non-commutative holonomies in 2 + 1 LQG and Kauffman's brackets

We investigate the canonical quantization of 2+1 gravity with Λ > 0 in the canonical framework of LQG. A natural regularization of the constraints of 2+1 gravity can be defined in terms of the holonomies of where the SU(2) connection A and the triad field e are the conjugated variables of the theory. As a first step towards the quantization of these constraints we study the canonical quantization of the holonomy of the connection Aλ A + λe Ae acting on spin network links of the kinematical Hilbert space of LQG. We provide an explicit construction of the quantum holonomy operator, exhibiting a close relationship between the action of the quantum holonomy at a crossing and Kauffman's q-deformed crossing identity. The crucial difference is that the result is completely described in terms of standard SU(2) spin network states.