Dynnikov Coordinates and 𝝅𝟏–Train Tracks

Abstract

A well known way to coordinatize multicurves on a given surface is to use train tracks. In the case where the surface is the 𝑛– punctured disk 𝐷𝑛 an alternative and efficient way to coordinatize multicurves is achieved by the Dynnikov coordinate system which gives an explicit bijection between the set of multicurves on 𝐷𝑛 and ℤ 2𝑛−4 \{0}. In this paper we introduce.

Keywords

• Multicurves,Dynnikov coordinates,𝜋1–train track coordinates,geometric intersection number

Dynnikov Koordinatları ve 𝝅𝟏–Train Track Grafikleri

Abstract

Verilen bir yüzeyde tanımlı çoklu eğrileri koordinatlandırmanın alışılmış bir yolu train track grafiklerini kullanmaktır. Yüzeyin sonlu noktası çıkarılmış 𝐷𝑛 diski olması durumunda ise çoklu eğrilerin kümesi ile ℤ 2𝑛−4 \{0} arasında birebir ve örten bir dönüşüm veren Dynnikov koordinat sistemi çoklu eğrileri koordinatlandırmak için alternatif ve etkili bir yol sunar. Bu çalışmada, 𝐷𝑛’ de verilen bir çoklu eğrinin belirli tipten bir train track grafiği olan 𝜋1–train track grafiği koordinatlarını Dynnikov koordinatlarına bağlayan geçiş formülleri sunulmuştur. 

Keywords

• Çoklu eğriler,Dynnikov koordinatları,𝜋.–train track koordinatları

References

1. KAYNAKÇA
2. [1] Artin, E, "Theory of braids ", Ann. of Math., vol 48, no 2, pp. 101–126, 1947.
3. [2] Bestvina, M. and Handel, M. "Train-tracks for surface homeomorphisms", Topology, vol 34, no 1, pp. 109–140, 1995.
4. [3] Dehornoy, P., "Efficient solutions to the braid isotopy problem", Discrete Appl. Math., vol 156, no 16, pp. 3091–3112, 2008.
5. [4] Dehornoy, P., Dynnikov, I., Rolfsen, D. and Wiest, "Ordering braids", Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2008
6. [5] Dynnikov, I., "On a Yang-Baxter mapping and the Dehornoy ordering" Uspekhi Mat. Nauk, vol 57, no 3, pp. 151–152, 2002.
7. [6] Dynnikov, I. and Wiest, B., "On the complexity of braids", J. Eur. Math. Soc. (JEMS), vol 9, no 4, pp. 801–840, 2007.
8. [7] Fathi, A., Laudenbach, F. and Poenaru, V. Travaux de Thurston sur les surfaces, volume 66 of Astérisque. Société Mathématique de France, Paris, Séminaire Orsay, 1979.

9. [8] Finn, M.D. and Thiffeault, J.L. "Topological entropy of braids on the torus", SIAM J. Appl. Dyn. Syst., vol 6, no 1, pp. 79–98. 2007.
10. [9] Gover, P., Ross, S.D., Stremler and M.A., Kumar, P. Topological chaos, braiding and bifurcation of almost cyclic sets. Chaos, 22(4) 16, 2012.
11. [10] Hall, T. and Yurttaş, S.Ö., "On the topological entropy of families of braid", Topology Appl., vol 156, no 8, pp. 1554–1564, 2009.
12. [11] Hamidi-Tehrani, Hessam. and Chen, Zong-He., "Surface diffeomorphisms via train-tracks", Topology Appl., vol 73, no 2, pp. 141–167, 1996
13. [12] Menzel, C., Parker and J. R., "Pseudo-Anosov diffeomorphisms of the twice punctured torus", Recent Advances in Group Theory and Low-Dimensional Topology, vol 27, pp. 141–154, 2003.
14. [13] Moussafir, J., "On computing the entropy of braids", Funct.Anal. OtherMath., vol 1, no 1, pp. 37–-46, 2006.
15. [14] Penner, R. C. and Harer, J. L., Combinatorics of train tracks, Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 125, 1992.
16. [15] Schaefer, M., Sedgwick, and E., Stefankovic, D., "Computing Dehn twists and geometric intersection numbers in polynomial time", Proceedings of the 20th Canadian Conference on Computational Geometry (CCCG2008), 111-114, 2008.
17. [16] Thurston, W.P., "On the geometry and dynamics of diffeomorphisms of surfaces", Bull. Amer. Math. Soc. (N.S.), vol 19, no 2, pp. 417–431, 1988.
18. [17] Yurttaş, S.Ö., Geometric intersection of curves on punctured disks. Journal of the Mathematical Society of Japan, vol 65, no 4, pp. 1554–1564, 2013.
19. [18] Yurttaş, S.Ö. "Dynnikov and train track transition matrices of pseudo-Anosov braids ", Discrete Contin. Dyn. Syst., vol 36, no 1, pp. 541–570, 2016.
20. [19] Yurttaş, S.Ö. and Hall, T., "Counting components of an integral lamination", Manuscripta Math., vol 153, no 1, pp. 263–278, 2017.
21. [20] Yurttaş, S.Ö. and Hall, T., "Intersections of multicurves from Dynnikov coordinates", Bull. Aust. Math. Soc. vol 98, no 1, pp. 149–158, 2018.