Arnold Strangeness of surface immersions

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.