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Maps on Surfaces as a Structural Framework for Genus-One Virtual Knot Classification

Alexander Omelchenko

TL;DR

This work presents a complete, reproducible pipeline for genus-one virtual knot classification by encoding torus projections as cellular $4$-regular maps via permutation pairs $(\alpha,\sigma)$ and quotienting by unsensed surface homeomorphisms to canonical representatives. It then builds diagrams on fixed projections through crossing bits, applies an intrinsic bigon-based reduction rule, and evaluates a genus-one Kauffman-type bracket entirely in permutation form, with contractible vs. essential state circles detected through $\mathbb{F}_2$-homology. The authors validate against established genus-one tables for $N\le 5$, extend computations to $N\le 8$, and provide public code and machine-readable projection and diagram datasets that enable complete reproducibility. The approach isolates projection enumeration from diagram evaluation, offering scalable genus-one tabulation and a canonical, surface-based refinement of virtual knot data with potential extensions to higher genus. Overall, the framework delivers a canonical, checkable, and extensible path from maps on surfaces to genus-one virtual knot tables, with concrete data and tooling to support SEO-friendly dissemination and reuse in downstream analyses.

Abstract

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.

Maps on Surfaces as a Structural Framework for Genus-One Virtual Knot Classification

TL;DR

This work presents a complete, reproducible pipeline for genus-one virtual knot classification by encoding torus projections as cellular -regular maps via permutation pairs and quotienting by unsensed surface homeomorphisms to canonical representatives. It then builds diagrams on fixed projections through crossing bits, applies an intrinsic bigon-based reduction rule, and evaluates a genus-one Kauffman-type bracket entirely in permutation form, with contractible vs. essential state circles detected through -homology. The authors validate against established genus-one tables for , extend computations to , and provide public code and machine-readable projection and diagram datasets that enable complete reproducibility. The approach isolates projection enumeration from diagram evaluation, offering scalable genus-one tabulation and a canonical, surface-based refinement of virtual knot data with potential extensions to higher genus. Overall, the framework delivers a canonical, checkable, and extensible path from maps on surfaces to genus-one virtual knot tables, with concrete data and tooling to support SEO-friendly dissemination and reuse in downstream analyses.

Abstract

We develop a purely combinatorial framework for the systematic enumeration of knot and link diagrams supported on the thickened torus . Using the theory of maps on surfaces, cellular --regular torus projections are encoded by permutation pairs , 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 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 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.
Paper Structure (44 sections, 13 theorems, 32 equations, 4 tables)

This paper contains 44 sections, 13 theorems, 32 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Maps on surfaces
  3. Maps, projections, and the permutation model
  4. Equivalence, components, and loop constraints
  5. Prime projections on the torus: unsensed maps and canonical enumeration
  6. The projection class and standing constraints
  7. Unsensed equivalence in the permutation model
  8. Canonical representatives and deduplication
  9. Enumeration: completeness and non-duplication
  10. Implementation overview and reproducibility
  11. Straight-ahead components and component stratification
  12. Primeness filters (explicit combinatorial witnesses)
  13. Genus-one benchmark data
  14. Remark on higher genus
  15. From projections to diagrams on the torus: crossing data, local reductions, and state-sum invariants
  16. ...and 29 more sections

Key Result

Lemma 2.3

Let $(\alpha,\sigma)$ be a labelled orientable map and let $\varphi=\sigma\alpha$. Then and the cycles of $\varphi$ are precisely the oriented boundary walks of faces.

Theorems & Definitions (51)

  • Definition 2.1: Map on a surface
  • Definition 2.2: Permutation encoding
  • Lemma 2.3: Vertices, edges, faces
  • proof
  • Lemma 2.4: Connectedness $\Leftrightarrow$ transitivity
  • proof
  • Proposition 2.5: Sensed and unsensed equivalence
  • proof
  • Lemma 2.6
  • proof
  • ...and 41 more