RELAXING MULTICURVES ON THE TWICE PUNCTURED MÖBIUS BAND

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

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

9. Yurttaş, S. Ö., Pamuk, M., “Integral laminations on non-orientable surfaces”, Turkish Journal of Mathematics, 42, 69-82, 2018.
10. Yurttaş, S. Ö., “Curves on non-orientable surfaces and crosscap transpositions”, Mathematics, 10(9), 1–33, 2022.