The universal order one invariant of framed knots in most $S^1$–bundles over orientable surfaces

Abstract

It is well-known that self-linking is the only $\mathbb{Z}$–valued Vassiliev invariant of framed knots in $S^3$. However for most $3$–manifolds, in particular for the total spaces of $S^1$–bundles over an orientable surface $F\ne S^2$, the space of $\mathbb{Z}$–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 $S^1$–bundles over an orientable not necessarily compact surface $F\ne S^2$. We show that if $F\ne S^2,S^1\times S^1,$ 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.