The space of immersions $\R^1 \to \R^n$ with an arbitrary fixed type of transversal self-intersections is studied. All {\em first-order cohomology groups} of such a space with an arbitrary $n \ge 3$ are calculated, in particular two its lowest non-trivial cohomology groups for $n>4$; these groups are expressed in terms of the {\em intersection graph of the chord diagram} related with this self-intersection type. The homological obstructions to the existence of integer Gau\ss-like combinatorial formulas for such cohomology classes are calculated, in particular the first obstruction to the existence of an integer Polyak-Viro formula for invariants of knots in $\R^3$ is found.