Some Fast Algorithms for Curves in Surfaces.
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| Title: | Some Fast Algorithms for Curves in Surfaces. |
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| Authors: | Lackenby, Marc1 (AUTHOR) lackenby@maths.ox.ac.uk |
| Source: | Discrete & Computational Geometry. Jul2026, Vol. 76 Issue 1, p539-588. 50p. |
| Subjects: | Algorithms, Intersection numbers, Topology, Geometric surfaces |
| Abstract: | We present some algorithms that provide useful topological information about curves in surfaces. One of the main algorithms computes the geometric intersection number of two properly embedded 1-manifolds C 1 and C 2 in a compact orientable surface S. The surface S is presented via a triangulation or a handle structure, and the 1-manifolds are given in normal form via their normal coordinates. The running time is bounded above by a polynomial function of the number of triangles in the triangulation (or the number of handles in the handle structure), and the logarithm of the weight of C 1 and C 2 . This algorithm represents an improvement over previous work, since its running time depends polynomially on the size of the triangulation of S and it can deal with closed surfaces, unlike many earlier algorithms. Another algorithm, with similar bounds on its running time, can determine whether C 1 and C 2 are isotopic. We also present a closely related algorithm that can be used to place a standard 1-manifold into normal form. [ABSTRACT FROM AUTHOR] |
| Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 194776522 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Some Fast Algorithms for Curves in Surfaces. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Lackenby%2C+Marc%22">Lackenby, Marc</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> lackenby@maths.ox.ac.uk</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Discrete+%26+Computational+Geometry%22">Discrete & Computational Geometry</searchLink>. Jul2026, Vol. 76 Issue 1, p539-588. 50p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Algorithms%22">Algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Intersection+numbers%22">Intersection numbers</searchLink><br /><searchLink fieldCode="DE" term="%22Topology%22">Topology</searchLink><br /><searchLink fieldCode="DE" term="%22Geometric+surfaces%22">Geometric surfaces</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: We present some algorithms that provide useful topological information about curves in surfaces. One of the main algorithms computes the geometric intersection number of two properly embedded 1-manifolds C 1 and C 2 in a compact orientable surface S. The surface S is presented via a triangulation or a handle structure, and the 1-manifolds are given in normal form via their normal coordinates. The running time is bounded above by a polynomial function of the number of triangles in the triangulation (or the number of handles in the handle structure), and the logarithm of the weight of C 1 and C 2 . This algorithm represents an improvement over previous work, since its running time depends polynomially on the size of the triangulation of S and it can deal with closed surfaces, unlike many earlier algorithms. Another algorithm, with similar bounds on its running time, can determine whether C 1 and C 2 are isotopic. We also present a closely related algorithm that can be used to place a standard 1-manifold into normal form. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1007/s00454-026-00845-7 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 50 StartPage: 539 Subjects: – SubjectFull: Algorithms Type: general – SubjectFull: Intersection numbers Type: general – SubjectFull: Topology Type: general – SubjectFull: Geometric surfaces Type: general Titles: – TitleFull: Some Fast Algorithms for Curves in Surfaces. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Lackenby, Marc IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 07 Text: Jul2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 01795376 Numbering: – Type: volume Value: 76 – Type: issue Value: 1 Titles: – TitleFull: Discrete & Computational Geometry Type: main |
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