  As it is well-known, all Vassiliev invariants of degree one of a knot
$K\subset R^3$ are trivial. There are nontrivial Vassiliev invariants of degree
one, when the ambient space is not $R^3$. Recently, T. Fiedler introduced such
invariants of a knot in an $R^1$-fibration over a surface $F$. They take values
in the free $Z$-module generated by all the free homotopy classes of loops in
$F$. Here, we generalize them to the most refined Vassiliev invariant of degree
one. The ranges of values of all these invariants are explicitly described.
  We also construct a similar invariant of a two-component link in an
$\R^1$-fibration. It generalizes the linking number.
