Holonomy braidings, biquandles and quantum invariants of links with $S{L}_2(\mathbb{C})$ flat connections

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 $U_{q}s{l}_2$ 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 $U_{q}s{l}_2$ and produces an invariant of links with a gauge class of quandle representations.