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It is well-known that self-linking is the only Z valued Vassiliev invariantof framed knots in S^3. However for most 3-manifolds, in particular for thetotal spaces of S^1-bundles over an orientable surface F not S^2, the space ofZ-valued order one invariants is infinite dimensional. We give an explicitformula for the order one invariant I of framed knots in orientable totalspaces of S^1-bundles over an orientable not necessarily compact surface F notS^2. We show that if F is not S^2 or S^1 X S^1, then I is the universal orderone invariant, i.e. it distinguishes every two framed knots that can bedistinguished by order one invariants with values in an Abelian group.

The universal order one invariant of framed knots in mostS1–bundles over orientable surfaces

The universal order one invariant of framed knots in mostS¹–bundles over orientable surfaces