  R. Kashaev and N. Reshetikhin introduced the notion of holonomy braiding
extending V. Turaev's homotopy braiding to describe the behavior of cyclic
representations of the unrestricted quantum group $U_qsl_2$ at root of unity.
In this paper, using quandles and biquandles we develop a general theory for
Reshetikhin-Turaev ribbon type functor for tangles with quandle
representations. This theory applies to the unrestricted quantum group
$U_qsl_2$ and produces an invariant of links with a gauge class of quandle
representations.
