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

Holonomy braidings, biquandles and quantum invariants of links with $SL_2(\mathbb C)$ flat connections