Generalization of Arnold’s J+-invariant for pairs of immersions

This paper introduces the [Formula: see text]-invariant for oriented pairs of generic immersions. This invariant behaves like Arnold’s [Formula: see text]-invariant for generic immersions as it is invariant when going through inverse tangencies and triple points, but changes when traversing direct tangencies. It has several useful properties, for example, its independence of the [Formula: see text]-invariants of the single immersions forming the pair. Also, it is invariant under simultaneous orientation change. Therefore, one can define two [Formula: see text]-invariants for each pair depending on its orientation, those two invariants are not independent from each other. Furthermore, the invariant is extended to the [Formula: see text]-invariant for links of [Formula: see text] oriented immersions.