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Some Fast Algorithms for Curves in Surfaces
1Mathematical Institute, University of Oxford, Woodstock Road, OX2 6GG Oxford, United Kingdom.
Discrete & Computational Geometry
|June 26, 2026
Summary
New algorithms offer topological insights into curves on surfaces by efficiently computing intersection numbers and determining isotopy. These methods improve upon prior work, handling closed surfaces with polynomial time complexity relative to surface triangulation size.
Area of Science:
- Computational Topology
- Differential Geometry
- Surface Theory
Background:
- Topological information about curves in surfaces is crucial for understanding complex geometric structures.
- Existing algorithms often struggle with closed surfaces or have non-polynomial time complexities.
- Efficient computation of intersection numbers and isotopy for 1-manifolds is a persistent challenge.
Purpose of the Study:
- To present novel algorithms for computing topological information of curves (1-manifolds) embedded in surfaces.
- To improve the efficiency and applicability of algorithms for analyzing curves in compact orientable surfaces.
- To determine the geometric intersection number and isotopy of two 1-manifolds within a given surface.
Main Methods:
- Development of algorithms based on surface triangulations or handle structures.
- Utilizing normal form and normal coordinates for representing 1-manifolds.
- Analysis of running time complexity as a polynomial function of surface size and logarithm of manifold weight.
Main Results:
- An algorithm computes the geometric intersection number of two 1-manifolds in a surface with polynomial time complexity.
- The algorithm efficiently handles closed surfaces, an improvement over previous methods.
- A related algorithm determines if two 1-manifolds are isotopic with similar time bounds, and an algorithm for normal form conversion is provided.
Conclusions:
- The presented algorithms provide significant advancements in computing topological invariants for curves on surfaces.
- The improved efficiency and applicability (e.g., closed surfaces) make these methods valuable for geometric analysis.
- These algorithms offer a robust framework for understanding the relationships between curves and surfaces in computational geometry.
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