Journal article
Some fast algorithms for curves in surfaces
- Abstract:
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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 C1 and C2 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 C1 and C2. 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 C1 and C2 are isotopic. We also present a closely related algorithm that can be used to place a standard 1-manifold into normal form.
- Publication status:
- Published
- Peer review status:
- Peer reviewed
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- Files:
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(Preview, Version of record, pdf, 999.7KB, Terms of use)
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- Publisher copy:
- 10.1007/s00454-026-00845-7
Authors
- Funder identifier:
- https://ror.org/0439y7842
- Grant:
- EP/Y004256/1
- Publisher:
- Springer
- Journal:
- Discrete and Computational Geometry More from this journal
- Publication date:
- 2026-04-21
- Acceptance date:
- 2026-03-19
- DOI:
- EISSN:
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1432-0444
- ISSN:
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0179-5376
- Language:
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English
- Pubs id:
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2392141
- Local pid:
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pubs:2392141
- Deposit date:
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2026-03-19
- ARK identifier:
Terms of use
- Copyright holder:
- Marc Lackenby
- Copyright date:
- 2026
- Rights statement:
- Copyright © 2026, The Author(s). This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
- Licence:
- CC Attribution (CC BY)
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