An explicit slice formula for surface invariants via curve invariants

TL;DR

This paper presents an explicit formula linking surface invariants of generic immersions in 3D to curve invariants from planar slices, enabling combinatorial computation and relating curve and surface invariants.

Contribution

It introduces a new explicit slice formula for surface invariants expressed through curve invariants, connecting local changes during singular events to global surface properties.

Findings

Change in surface invariant at quadruple points is 2j - 4, with j the number of outward coorientations.

Provides a computable, combinatorial description of surface invariants via slice data.

Clarifies the relation between curve invariants and finite-order surface invariants.

Abstract

We give an explicit slice formula for a surface invariant of generic immersions in $\mathbb{R}^3$, expressed in terms of curve invariants arising from planar slices. Using a motion-picture viewpoint, we introduce differential measures that record local changes of the curve invariant $St_{(1)}$ and the surface invariant $St_{(2)}$ across singular slice transitions. Our main result shows that, for a quadruple-point event, if $j$ denotes the number of outward coorientations before the event, then the change of the surface invariant satisfies $dSt_{(2)} = 2j - 4$. This yields a computable and combinatorial description of the surface invariant via slice data. In particular, the formula makes explicit the relation between curve-level invariants and finite-order invariants of surface immersions in the sense of Nowik.