Generalization of Arnold's $J^+$-invariant for pairs of immersions
Hanna Haeussler
TL;DR
The paper extends Arnold's J^+ invariant from single immersions to pairs and then to links of n immersions. It introduces the J^{2+} invariant for oriented pairs, defined by J^{2+}(K)=2+n−∑_C ω_c(K)^2+∑_p ind_p(K)^2+u(K)^2, where u(K) measures the inter-immersion winding; J^{2+} is invariant under inverse tangencies and triple points and changes by ±2 on direct tangencies, while remaining computable and independent of the individual J^+ values of the component immersions. It further develops a pair-wise framework by proving inequalities between (J^+(S_1),J^+(S_2),J^{2+}(K),J^{2+}(ar K)) and providing algorithms (e.g., Algorithm 0) to realize prescribed invariants, then introduces J^{2-} as a complementary invariant and connects J^{2+} and J^{2-} through orientation-reversal relations. Finally, the work generalizes to n-component links via J^{n+} with a natural sum formula, establishing that J^{n+} changes by 2 under direct tangencies and extends the pairwise theory to multiparticle configurations, thus enabling a structured analysis of families of linked orbits and their regular-homotopy classes.
Abstract
This paper introduces the $J^{2+}$-invariant for oriented pairs of generic immersions. This invariant behaves like Arnold's $J^+$-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 $J^+$-invariants of the single immersions forming the pair. Also it is invariant under simultaneous orientation change. Therefore, one can define two $J^{2+}$-invariants for each pair depending on its orientation, those two invariants are not independent from each other. Furthermore the invariant is extended to the $J^{n+}$-invariant for links of n oriented immersions.