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An explicit slice formula for surface invariants via curve invariants

Noboru Ito, Hiroki Mizuno

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.

An explicit slice formula for surface invariants via curve invariants

Abstract

We give an explicit slice formula for a surface invariant of generic immersions in , 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 and the surface invariant across singular slice transitions. Our main result shows that, for a quadruple-point event, if denotes the number of outward coorientations before the event, then the change of the surface invariant satisfies . 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.

Paper Structure

This paper contains 5 sections, 3 theorems, 26 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Definitions and notations
  3. Differential Measures and Slice Formulas
  4. Table of Local Changes
  5. Example

Key Result

Lemma 3.1

Let $j$ be the number of outward coorientations of the vanishing triangle before the move. Then $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 1: $\textup{(E)}$ : Elliptic tangency of two sheets.
  • Figure 2: $\textup{(H)}$ : Hyperbolic tangency of two sheets.
  • Figure 3: $\textup{(T)}$ : Tangency of the line of intersection of two sheets to another sheet.
  • Figure 4: $\textup{(Q)}$ : A sheet crossing over a triple point formed by three other sheets. For the quadruple point, four sheets intersect at the same point.

Theorems & Definitions (20)

  • Definition 1: immersion
  • Definition 2: event
  • Definition 3: codimension-one points and their coorientation
  • Definition 4: index for triple points and $T(S)$
  • Definition 5: $St_{(2)}$
  • Definition 6: $St_{(1)}$
  • Definition 7: Morse surface immersion
  • Remark 1: Independence of orientation of the slice curve
  • Definition 8: index of slice plane
  • Definition 9: differential measure $dSt_{(1)}$ for $St_{(1)}$
  • ...and 10 more