Generalization of Arnold's $J^+$-invariant for pairs of immersions

TL;DR

This paper extends Arnold's $J^+$-invariant to pairs and links of immersions, introducing the $J^{2+}$- and $J^{n+}$-invariants, which are invariant under certain transformations and independent of single immersion invariants.

Contribution

It introduces the $J^{2+}$-invariant for pairs of immersions and extends it to $J^{n+}$-invariants for links, providing new tools for studying immersion invariants.

Findings

$J^{2+}$-invariant is invariant under inverse tangencies and triple points.

$J^{2+}$-invariant changes under direct tangencies.

Extension to $J^{n+}$-invariants for links of multiple immersions.

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.