Biquandle cocycle condition for invariants of immersed surface-links in the four-space
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866916867067609088 |
|---|---|
| author | Jablonowski, Michal |
| author_facet | Jablonowski, Michal |
| contents | We consider a biquandle-cohomological framework for invariants of oriented immersed surface-links in the four-space. After reviewing projections and Roseman moves for immersed surfaces, we prove that the move types (a,b,c,e,f,g,h) form a minimal generating set, showing in particular that the singular move (h) is independent of the embedded-case set (a,b,c,e,f,g). We extend biquandle colorings to broken surface diagrams with singular points and establish that coloring sets are in bijection for diagrams related by these moves, yielding a coloring number invariant for immersed surface-links. We introduce singular biquandle 3-cocycles: biquandle 3-cocycles satisfying an additional antisymmetry when the singular relations hold. Using such cocycles, we define a triple-point state-sum with Boltzmann weights and prove its invariance under all generating moves, including (h), thereby obtaining a state-sum invariant for immersed surface-links. The theory is illustrated on the Fenn-Rolfsen link example, where a computation yields non-trivial value, demonstrating nontriviality of the invariant in the immersed setting. These results unify and extend biquandle cocycle invariants from embedded to immersed surface-links. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20360 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Biquandle cocycle condition for invariants of immersed surface-links in the four-space Jablonowski, Michal Geometric Topology We consider a biquandle-cohomological framework for invariants of oriented immersed surface-links in the four-space. After reviewing projections and Roseman moves for immersed surfaces, we prove that the move types (a,b,c,e,f,g,h) form a minimal generating set, showing in particular that the singular move (h) is independent of the embedded-case set (a,b,c,e,f,g). We extend biquandle colorings to broken surface diagrams with singular points and establish that coloring sets are in bijection for diagrams related by these moves, yielding a coloring number invariant for immersed surface-links. We introduce singular biquandle 3-cocycles: biquandle 3-cocycles satisfying an additional antisymmetry when the singular relations hold. Using such cocycles, we define a triple-point state-sum with Boltzmann weights and prove its invariance under all generating moves, including (h), thereby obtaining a state-sum invariant for immersed surface-links. The theory is illustrated on the Fenn-Rolfsen link example, where a computation yields non-trivial value, demonstrating nontriviality of the invariant in the immersed setting. These results unify and extend biquandle cocycle invariants from embedded to immersed surface-links. |
| title | Biquandle cocycle condition for invariants of immersed surface-links in the four-space |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2507.20360 |