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Algorithms and topological invariants for dynamic systems. III. Algorithms for Recognition and Classification of 2-Dimensional Surfaces

Alexandr Prishlyak

TL;DR

This paper addresses the problem of recognizing and classifying 2-dimensional manifolds and related stratified 2D sets within simplicial and regular CW-complexes. It develops concrete algorithms for connectivity, local planarity, orientability, and the computation of the Euler characteristic $\chi$ and genus $g$, with extensions to higher dimensions. A combinatorial framework is introduced, including SLW-graphs, rotation systems, and chord diagrams, to encode embeddings and test isomorphisms via surface attachments. The work assembles extensive catalogs of regular cell decompositions and embedded graphs, enabling practical topological invariants and classifying data with potential applications in dynamics and Morse-Smale vector field analysis, and it situates the results in a broader literature on simplicial topology.

Abstract

We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused basic concepts of diferential topology. In the second part (arXiv:2502.00506 ) we discused the main discrete topological structures used in the topological theory of dynamic systems. In third part we construct algorithms that allow us recognise 2-manifolds and 3-manifolds in simplicial complexes and regular CW-complex and detreminate topological type for 2-manifolds (connectivity, type of orientability, genus, number of boundary component).

Algorithms and topological invariants for dynamic systems. III. Algorithms for Recognition and Classification of 2-Dimensional Surfaces

TL;DR

This paper addresses the problem of recognizing and classifying 2-dimensional manifolds and related stratified 2D sets within simplicial and regular CW-complexes. It develops concrete algorithms for connectivity, local planarity, orientability, and the computation of the Euler characteristic and genus , with extensions to higher dimensions. A combinatorial framework is introduced, including SLW-graphs, rotation systems, and chord diagrams, to encode embeddings and test isomorphisms via surface attachments. The work assembles extensive catalogs of regular cell decompositions and embedded graphs, enabling practical topological invariants and classifying data with potential applications in dynamics and Morse-Smale vector field analysis, and it situates the results in a broader literature on simplicial topology.

Abstract

We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused basic concepts of diferential topology. In the second part (arXiv:2502.00506 ) we discused the main discrete topological structures used in the topological theory of dynamic systems. In third part we construct algorithms that allow us recognise 2-manifolds and 3-manifolds in simplicial complexes and regular CW-complex and detreminate topological type for 2-manifolds (connectivity, type of orientability, genus, number of boundary component).

Paper Structure

This paper contains 14 sections, 3 theorems, 10 equations, 10 figures.

Table of Contents

  1. Algorithm for Checking Connectivity
  2. Algorithm for Checking the Connectivity of a Simplicial Complex and Finding Connected Components.
  3. Algorithm for checking 2-dimensional complex for local planarity
  4. Algorithm for Checking the Orientability of Surfaces
  5. Determination of the genus and topological type of a surface
  6. Algorithms for Higher-Dimensional Manifolds
  7. SLW-graphs of stratified sets and graph embeddings in surfaces
  8. SLW-graphs of Closed Surfaces
  9. Rotation System of the Embedded Graph
  10. Chord Diagrams
  11. List of Regular Cell Complexes
  12. List of Rotation Systems and Chord Diagrams
  13. List of Embedded Graphs
  14. Literature Review

Key Result

Theorem 1

hat02prish-modtop06 A cellular complex is connected if and only if its 1-skeleton is connected.

Figures (10)

  • Figure 1: Closed surfaces: a) sphere $S^2$, b) torus $T^2$, c) projective plane $\mathbb{R}P^2$, d) Klein bottle $Kl$.
  • Figure 2: Surfaces with boundary: a) cylinder $S^1 \times [0,1]$, b) Möbius strip $Mo$, c) sphere with three holes $F_{0,3}$, d) torus with a hole $F_{1,1}$, e) Klein bottle with a hole $N_{2,1}$, f) Möbius strip with a hole $N_{1,2}$.
  • Figure 3: Closed surfaces: a) orientable genus 2 surface $F_2$, b) non-orientable genus 2 surface 3 $N_3$.
  • Figure 4: Surfaces with boundaries.
  • Figure 5: Chord diagrams with six edges on $F_3$
  • ...and 5 more figures

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3