Curves on Non-Orientable Surfaces and Crosscap Transpositions

S. Öykü Yurttaş
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S. Öykü Yurttaş: Department of Mathematics, Faculty of Science, Dicle University, 21280 Diyarbakir, Turkey

Mathematics, 2022, vol. 10, issue 9, 1-33

Abstract: Let N g , n be a non-orientable surface of genus g with n punctures and one boundary component. In this paper, we describe multicurves in N g , n 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 N g , n 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.

Keywords: non-orientable surfaces; multicurves; geometric intersection; crosscap transpositions; mapping class group (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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