An explicit slice formula for surface invariants via curve invariants
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866913005667614720 |
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| author | Ito, Noboru Mizuno, Hiroki |
| author_facet | Ito, Noboru Mizuno, Hiroki |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_03942 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An explicit slice formula for surface invariants via curve invariants Ito, Noboru Mizuno, Hiroki Geometric Topology 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. |
| title | An explicit slice formula for surface invariants via curve invariants |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2604.03942 |