  In this work we investigate the canonical quantization of 2+1 gravity with
cosmological constant $\Lambda>0$ in the canonical framework of loop quantum
gravity. The unconstrained phase space of gravity in 2+1 dimensions is
coordinatized by an SU(2) connection $A$ and the canonically conjugate triad
field $e$. A natural regularization of the constraints of 2+1 gravity can be
defined in terms of the holonomies of $A+=A + \sqrt\Lambda e$. As a first step
towards the quantization of these constraints we study the canonical
quantization of the holonomy of the connection $A_{\lambda}=A+\lambda e$ on the
kinematical Hilbert space of loop quantum gravity. The holonomy operator
associated to a given path acts non trivially on spin network links that are
transversal to the path (a crossing). We provide an explicit construction of
the quantum holonomy operator. In particular, we exhibit 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 (being an operator
acting on the kinematical Hilbert space of LQG) the result is completely
described in terms of standard SU(2) spin network states (in contrast to
q-deformed spin networks in Kauffman's identity). We discuss the possible
implications of our result.
