Generalized connected sum formula for the Arnold invariants of generic plane curves

TL;DR

This paper introduces a generalized connected sum operation for generic closed plane curves, extending Arnold's original sum, and thoroughly analyzes how key Arnold invariants change under this new operation.

Contribution

It provides a comprehensive description of the behavior of Arnold invariants under the generalized connected sum, expanding the understanding of invariants for plane curves.

Findings

Derived formulas for Arnold invariants under the generalized sum

Extended Arnold's original sum to a broader class of curves

Enhanced the theoretical framework for plane curve invariants

Abstract

We define the generalized connected sum for generic closed plane curves, generalizing the strange sum defined by Arnold, and completely describe how the Arnold invariants $J^{\pm}$ and $\mathit{St}$ behave under the generalized connected sums.