Underlying Stokes and de Rham structures for Arnold-type invariants
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| Format: | Preprint |
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2026
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| author | Ito, Noboru Mizuno, Hiroki |
| author_facet | Ito, Noboru Mizuno, Hiroki |
| contents | We introduce a framework on dual complexes for studying Arnold-type invariants of immersed curves and immersed surfaces via local finite-difference structures associated with Alexander numberings. For generic immersed plane curves and generic immersed surfaces, we define locally normalized maps $d^k ϕ$ on dual skeleta and show that suitable evaluations recover the Arnold-type invariants $St_{(1)}$ and $St_{(2)}$. In particular, we establish normalized discrete Stokes-type compatibilities between adjacent dual skeleta and derive corresponding Shumakovitch-type identities for curves and surfaces. The normalization coefficients are determined by finite-difference factorial structures together with multiplicities of local configurations. We further interpret the iterated-integral-type structures appearing in Shumakovitch-type identities through finite-difference structures and highest-degree local Stokes compatibilities on dual complexes. We also reinterpret the slice formula for $St_{(2)}$ and $St_{(1)}$ as a compatibility relation between slicing and local operations on the dual complex. These results provide a unified framework in which global Arnold-type invariants arise as distribution-type evaluations of local data on dual complexes. The framework further clarifies the distinction between untwisted local closures and globally twisted structures such as the original Arnold invariant $St$, and suggests the existence of higher-degree local operations associated with the same dual-complex structure. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_12834 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Underlying Stokes and de Rham structures for Arnold-type invariants Ito, Noboru Mizuno, Hiroki Geometric Topology We introduce a framework on dual complexes for studying Arnold-type invariants of immersed curves and immersed surfaces via local finite-difference structures associated with Alexander numberings. For generic immersed plane curves and generic immersed surfaces, we define locally normalized maps $d^k ϕ$ on dual skeleta and show that suitable evaluations recover the Arnold-type invariants $St_{(1)}$ and $St_{(2)}$. In particular, we establish normalized discrete Stokes-type compatibilities between adjacent dual skeleta and derive corresponding Shumakovitch-type identities for curves and surfaces. The normalization coefficients are determined by finite-difference factorial structures together with multiplicities of local configurations. We further interpret the iterated-integral-type structures appearing in Shumakovitch-type identities through finite-difference structures and highest-degree local Stokes compatibilities on dual complexes. We also reinterpret the slice formula for $St_{(2)}$ and $St_{(1)}$ as a compatibility relation between slicing and local operations on the dual complex. These results provide a unified framework in which global Arnold-type invariants arise as distribution-type evaluations of local data on dual complexes. The framework further clarifies the distinction between untwisted local closures and globally twisted structures such as the original Arnold invariant $St$, and suggests the existence of higher-degree local operations associated with the same dual-complex structure. |
| title | Underlying Stokes and de Rham structures for Arnold-type invariants |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2605.12834 |