Universal Vassiliev invariants of links in coverings of 3-manifolds

We study Vassiliev invariants of links in a 3-manifold M by using chord diagrams labeled by elements of the fundamental group of M . We construct universal Vassiliev invariants of links in M , where M = P 2×[0, 1] is a cylinder over the real projective plane P 2, M = Σ× [0, 1] is a cylinder over a surface Σ with boundary, and M = S1 × S2. A finite covering p : N −→ M induces a map π1(p) ∗ between labeled chord diagrams that corresponds to taking the preimage p−1(L) ⊂ N of a link L ⊂ M . The maps p−1 and π1(p) ∗ intertwine the constructed universal Vassiliev invariants.