RELAXING MULTICURVES ON THE TWICE PUNCTURED MÖBIUS BAND

Year 2023, Volume: 9 Issue: 1, 16 - 22, 26.06.2023

Abdullah Baykal , Ferihe Atalan Saadet Öykü Yurttaş

https://doi.org/10.51477/mejs.1286503

Abstract

Let N1,n (n ≥ 1) be a non-orientable surface of genus 1 with n punctures and one boundary component. Generalized Dynnikov coordinates provide a bijection between the set of multicurves in N1,n and Z2n−1 \ {0}. In this paper we restrict to the case where n = 2 and describe an algorithm to relax a multicurve in N1,2 making use of its generalized Dynnikov coordinates

Keywords

Non-orientable surface, Geneneralized Dynnikov coordinates, Multicurves

References

• Artin, E., “Theorie der Zöpfe”, Abh. Math. Sem., Univ. Hamburg, 4, 47–72, 1925.
• Artin, E., “Theory of braids”, Ann. of Math., 48(2), 101–126, 1947.
• Dynnikov, I., “On a Yang-Baxter mapping and the Dehornoy ordering”, Uspekhi Mat. Nauk, 57(3(345)), 151–152, 2002.
• Dehornoy, P., Dynnikov, I., Rolfsen, D., and Wiest, B., Ordering braids, Mathematical Surveys and Monographs, vol. 148, American Mathematical Society, Providence, RI, 2008.
• Hall, T., Yurttaş, S. Ö., “On the topological entropy of families of braids”, Topology Appl., 156(8), 1554–1564, 2009.
• Hall, T., Yurttaş, S. Ö., “Intersections of multicurves from Dynnikov coordinates”, Bulletin of the Australian Mathematical Society, 98(1), 149–158, 2018.
• Korkmaz, M., “Mapping class groups of nonorientable surfaces”, Geom. Dedicata 89, 109-133, 2002.
• Moussafir, J.O., “On computing the entropy of braids”, Funct. Anal. Other Math., 1(1), 37-46, 2006.
• Yurttaş, S. Ö., Pamuk, M., “Integral laminations on non-orientable surfaces”, Turkish Journal of Mathematics, 42, 69-82, 2018.
• Yurttaş, S. Ö., “Curves on non-orientable surfaces and crosscap transpositions”, Mathematics, 10(9), 1–33, 2022.