Maps on Surfaces as a Structural Framework for Genus-One Virtual Knot Classification
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866912840203370496 |
|---|---|
| author | Omelchenko, Alexander |
| author_facet | Omelchenko, Alexander |
| contents | We develop a purely combinatorial framework for the systematic enumeration of knot and link diagrams supported on the thickened torus $T^2\times I$. Using the theory of maps on surfaces, cellular $4$--regular torus projections are encoded by permutation pairs $(α,σ)$, and unsensed projection classes are enumerated completely and without duplication via canonical representatives. For a fixed projection, crossing assignments are encoded by bit data, and an immediate Reidemeister~II reduction supported by a bigon face is characterized directly in terms of these bits. The genus-one generalized Kauffman-type bracket is then evaluated as a state sum entirely within the permutation model, without drawing diagrams in a fundamental polygon.
The implementation is validated against published genus-one classifications for $N\le 5$ under explicit comparison conventions, with remaining discrepancies explained at the level of global conventions. Beyond the published range, we compute projection and diagram data for crossing numbers up to $N=8$ and provide a public reference implementation together with machine-readable datasets. Via the standard correspondence between virtual knots and knots in thickened surfaces, this yields a canonical and fully reproducible genus-one framework for virtual knot tabulation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_15512 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Maps on Surfaces as a Structural Framework for Genus-One Virtual Knot Classification Omelchenko, Alexander Combinatorics Geometric Topology 57K10, 05C10, 05A15 We develop a purely combinatorial framework for the systematic enumeration of knot and link diagrams supported on the thickened torus $T^2\times I$. Using the theory of maps on surfaces, cellular $4$--regular torus projections are encoded by permutation pairs $(α,σ)$, and unsensed projection classes are enumerated completely and without duplication via canonical representatives. For a fixed projection, crossing assignments are encoded by bit data, and an immediate Reidemeister~II reduction supported by a bigon face is characterized directly in terms of these bits. The genus-one generalized Kauffman-type bracket is then evaluated as a state sum entirely within the permutation model, without drawing diagrams in a fundamental polygon. The implementation is validated against published genus-one classifications for $N\le 5$ under explicit comparison conventions, with remaining discrepancies explained at the level of global conventions. Beyond the published range, we compute projection and diagram data for crossing numbers up to $N=8$ and provide a public reference implementation together with machine-readable datasets. Via the standard correspondence between virtual knots and knots in thickened surfaces, this yields a canonical and fully reproducible genus-one framework for virtual knot tabulation. |
| title | Maps on Surfaces as a Structural Framework for Genus-One Virtual Knot Classification |
| topic | Combinatorics Geometric Topology 57K10, 05C10, 05A15 |
| url | https://arxiv.org/abs/2601.15512 |