It is well-known that self-linking is the only –valued Vassiliev invariant of framed knots in . However for most –manifolds, in particular for the total spaces of –bundles over an orientable surface , the space of –valued order one invariants is infinite dimensional. We give an explicit formula for the order one invariant of framed knots in orientable total spaces of –bundles over an orientable not necessarily compact surface . We show that if then 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.
"The universal order one invariant of framed knots in most $S^1$–bundles over orientable surfaces." Algebr. Geom. Topol. 3 (1) 89 - 101, 2003. https://doi.org/10.2140/agt.2003.3.89