The Universal Order One Invariant of Framed Knots in the Total Spaces of S-bundles over Orientable Surfaces

It is well-known that self-linking is the only Z-valued Vassiliev invariant of framed knots in S. However for most 3-manifolds, in particular for the total spaces of S-bundles over an orientable surface F 6= S, the space of Z-valued order one invariants is infinite dimensional. We give an explicit formula for the order one invariant I of framed knots in orientable total spaces of S-bundles over an orientable not necessarily compact surface F 6= S. We show that if F 6= S, S × S, then I is the universal order one invariant, i.e. it distinguishes every two framed knots that can be distinguished by order one invariants with values in an Abelian group.