Arnold Strangeness of surface immersions

TL;DR

This paper introduces a new integer-valued function that classifies and detects quadruple point jumps in surface immersions, extending Arnold's strangeness invariant to higher dimensions and revealing finer geometric features.

Contribution

It defines a novel invariant for surface immersions that classifies quadruple point jumps and extends Arnold's invariant to three-dimensional surface immersions.

Findings

The function detects and classifies quadruple point jumps.

It reveals five distinct geometric cases based on coorientation.

The invariant reflects finer geometric features of surface immersions.

Abstract

It is known that for any smooth sphere eversion, the number of quadruple point jumps is always odd. In this paper, we define an integer-valued function that detects and classifies jumps involving quadruple points and triple-line tangencies. Our function provides a higher-dimensional analogue of the Arnold strangeness invariant for plane curves. It classifies quadruple point jumps into the five geometrically distinct cases based on coorientation data and reflects finer geometric features for generic immersions of closed surfaces into the 3-space.