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 ${\mathbb{R}}^3$, 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 $S{t}_{(1)}$ and the surface invariant $S{t}_{(2)}$ 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 $dS{t}_{(2)}=2j-4$. 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.

4 figures, 1 table