Dynnikov Koordinatları ve ??–Train Track Grafikleri

Verilen bir yüzeyde tanımlı çoklu eğrileri koordinatlandırmanın alışılmış bir yolu train track grafiklerini kullanmaktır. Yüzeyin sonlu noktası çıkarılmış ?? diski olması durumunda ise çoklu eğrilerin kümesi ile ℤ 2?−4 {0} arasında birebir ve örten bir dönüşüm veren Dynnikov koordinat sistemi çoklu eğrileri koordinatlandırmak için alternatif ve etkili bir yol sunar. Bu çalışmada, ??’ de verilen bir çoklu eğrinin belirli tipten bir train track grafiği olan ?1–train track grafiği koordinatlarını Dynnikov koordinatlarına bağlayan geçiş formülleri sunulmuştur.

Dynnikov Coordinates and ??–Train Tracks

A well known way to coordinatize multicurves on a given surface is to use train tracks. In the case where the surface is the ?– punctured disk ?? 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 ?? and ℤ 2?−4 {0}. In this paper we introduce transition formulae between Dynnikov coordinates and the so–called ?1–train track coordinates of a given multicurve on ??.

PDF

___

[1] Artin. E, “Theory of braids”, Ann. of Math., vol 48, no 2, pp. 101–126, 1947.
[2] Bestvina. M and Handel. M, “Train-tracks for surface homeomorphisms”, Topology, vol 34, no 1, pp. 109–140, 1995.
[3] Dehornoy. P, “Efficient solutions to the braid isotopy problem”, Discrete Appl. Math., vol 156, no 16, pp. 3091– 3112, 2008.
[4] Dehornoy. P, Dynnikov, I, Rolfsen, D and Wiest, “Ordering braids”, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2008.
[5] Dynnikov. I, “On a Yang-Baxter mapping and the Dehornoy ordering” Uspekhi Mat. Nauk, vol 57, no 3, pp. 151–152, 2002.
[6] Dynnikov. I. and Wiest. B, “On the complexity of braids”, J. Eur. Math. Soc. (JEMS), vol 9, no 4, pp. 801– 840, 2007.
[7] Fathi. A, Laudenbach. F and Poenaru. V, “Travaux de Thurston sur les surfaces”, Astérisque. Société Mathématique de France, Paris, Séminaire Orsay, 1979.
[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.
[9] Gover. P, Ross, S.D. Stremler and M.A. Kumar. P, “Topological chaos, braiding and bifurcation of almost cyclic sets”, Chaos, vol 22 no 4, pp. 15-16, 2012.
[10] Hall. T and Yurttaş. S.Ö, “On the topological entropy of families of braids”, Topology Appl., vol 156, no 8, pp. 1554–1564, 2009.
[11] Hamidi Tehrani. Hessam and Chen. Zong-He, “Surface diffeomorphisms via train-tracks”, Topology Appl., vol 73, no 2, pp. 141–167, 1996.
[12] Menzel C. Parker and Menzel. Christof, “PseudoAnosov diffeomorphisms of the twice punctured torus”, Recent Advances in Group Theory and Low-Dimensional Topology, vol 27, pp. 141–154, 2003.
[13] Moussafir, J, “On computing the entropy of braids”, Funct. Anal. Other Math., vol 1, no 1, pp. 37- 46, 2006.
[14] Penner R. C and Harer. J. L, Combinatorics of train tracks, Princeton, NJ: Princeton University Press, 1992.
[16] Thurston. W, “On the geometry and dynamics of diffeomorphisms of surfaces”, Bull. Amer. Math. Soc. (N.S.), vol 19, no 2, pp. 417–431, 1988.
[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.
[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.
[19] Yurttaş. S.Ö. and Hall. T, “Counting components of an integral lamination”, Manuscripta Math., vol 153, no 1, pp. 263–278, 2017.
[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.