Curves on Non-Orientable Surfaces and Crosscap Transpositions
Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub></semantics></math...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/9/1476 |
| _version_ | 1827672143055290368 |
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| author | S. Öykü Yurttaş |
| author_facet | S. Öykü Yurttaş |
| author_sort | S. Öykü Yurttaş |
| collection | DOAJ |
| description | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub></semantics></math></inline-formula> be a non-orientable surface of genus <i>g</i> with <i>n</i> punctures and one boundary component. In this paper, we describe multicurves in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub></semantics></math></inline-formula> making use of generalized Dynnikov coordinates, and give explicit formulae for the action of crosscap transpositions and their inverses on the set of multicurves in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub></semantics></math></inline-formula> in terms of generalized Dynnikov coordinates. This provides a way to solve on non-orientable surfaces various dynamical and combinatorial problems that arise in the study of mapping class groups and Thurston’s theory of surface homeomorphisms, which were solved only on orientable surfaces before. |
| first_indexed | 2024-03-10T03:56:33Z |
| format | Article |
| id | doaj.art-5ea310084f2a4cf6a0ada2aca4cf0dec |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T03:56:33Z |
| publishDate | 2022-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-5ea310084f2a4cf6a0ada2aca4cf0dec2023-11-23T08:44:54ZengMDPI AGMathematics2227-73902022-04-01109147610.3390/math10091476Curves on Non-Orientable Surfaces and Crosscap TranspositionsS. Öykü Yurttaş0Department of Mathematics, Faculty of Science, Dicle University, 21280 Diyarbakir, TurkeyLet <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub></semantics></math></inline-formula> be a non-orientable surface of genus <i>g</i> with <i>n</i> punctures and one boundary component. In this paper, we describe multicurves in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub></semantics></math></inline-formula> making use of generalized Dynnikov coordinates, and give explicit formulae for the action of crosscap transpositions and their inverses on the set of multicurves in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub></semantics></math></inline-formula> in terms of generalized Dynnikov coordinates. This provides a way to solve on non-orientable surfaces various dynamical and combinatorial problems that arise in the study of mapping class groups and Thurston’s theory of surface homeomorphisms, which were solved only on orientable surfaces before.https://www.mdpi.com/2227-7390/10/9/1476non-orientable surfacesmulticurvesgeometric intersectioncrosscap transpositionsmapping class group |
| spellingShingle | S. Öykü Yurttaş Curves on Non-Orientable Surfaces and Crosscap Transpositions Mathematics non-orientable surfaces multicurves geometric intersection crosscap transpositions mapping class group |
| title | Curves on Non-Orientable Surfaces and Crosscap Transpositions |
| title_full | Curves on Non-Orientable Surfaces and Crosscap Transpositions |
| title_fullStr | Curves on Non-Orientable Surfaces and Crosscap Transpositions |
| title_full_unstemmed | Curves on Non-Orientable Surfaces and Crosscap Transpositions |
| title_short | Curves on Non-Orientable Surfaces and Crosscap Transpositions |
| title_sort | curves on non orientable surfaces and crosscap transpositions |
| topic | non-orientable surfaces multicurves geometric intersection crosscap transpositions mapping class group |
| url | https://www.mdpi.com/2227-7390/10/9/1476 |
| work_keys_str_mv | AT soykuyurttas curvesonnonorientablesurfacesandcrosscaptranspositions |