The Expected Genus of a Random Chord Diagram
Tóm tắt
To any generic curve in an oriented surface there corresponds an oriented chord diagram, and any oriented chord diagram may be realized by a curve in some oriented surface. The genus of an oriented chord diagram is the minimal genus of an oriented surface in which it may be realized. Let g
n
denote the expected genus of a randomly chosen oriented chord diagram of order n. We show that g
n
satisfies:
$$g_n=\frac{n}{2}-\varTheta (\ln n).$$
I.e., there exist 0
Tài liệu tham khảo
Arratia, R., Tavare, S.: The cycle structure of random permutations. Ann. Probab. 20(3), 1567–1591 (1992)
Babson, E., Hoffman, C., Kahle, M.: The fundamental group of random 2-complexes. arXiv:0711.2704
Cairns, G., Elton, D.M.: The planarity problem for signed Gauss words. J. Knot Theory Ramif. 2(4), 359–367 (1993)
Cairns, G., Elton, D.M.: The planarity problem. II. J. Knot Theory Ramif. 5(2), 137–144 (1996)
Carter, J.S.: Classifying immersed curves. Proc. Am. Math. Soc. 111(1), 281–287 (1991)
Dowker, C.H., Thistlethwaite, M.B.: Classification of knot projections. Topol. Appl. 16, 19–31 (1983)
Dunfield, N.M., Thurston, W.P.: Finite covers of random 3-manifolds. Invent. Math. 166(3), 457–521 (2006)
Gauss, C.G.: Werke 8, 271–286 (1900)
Harer, J., Zagier, D.: The Euler characteristic of the moduli space of curves. Invent. Math. 85, 457–485 (1986)
Lando, S.K., Zvonkin, A.K.: Graphs on Surfaces and Their Applications. Encyclopaedia of Mathematical Sciences, Low-Dimensional Toplogy II, vol. 141. Springer, Berlin (2004)
Linial, N., Meshulam, R.: Homological connectivity of random 2-complexes. Combinatorica 26(4), 475–487 (2006)
Lovász, L., Marx, M.L.: A forbidden substructure characterization of Gauss codes. Acta Sci. Math. (Szeged) 38, 115–119 (1976)
Manturov, V.O.: A proof of V.A. Vassiliev’s conjecture on the planarity of singular links. Izv. Math. 69(5), 1025–1033 (2005)
Meshulam, R., Wallach, N.: Homological connectivity of random k-dimensional complexes. Random Struct. Algorithms 34, 408–417 (2009)
Pippenger, N., Schleich, K.: Topological characteristics of random triangulated surfaces. Random Struct. Algorithms 28(3), 247–288 (2006)
Read, R.C., Rosenstiehl, P.: On the Gauss crossing problem. In: Combinatorics. Proc. Fifth Hungarian Colloq., Keszthely, vol. II, pp. 843–876 (1976)
Rosenstiehl, P.: Solution algébrique du problème de Gauss sur la permutation des points d’intersection d’une ou plusieurs courbes fermés du plan. C. R. Acad. Sci. Paris Sér. A–B 283, A551–A553 (1976)
Turaev, V.: Curves on surfaces, charts, and words. Geom. Dedic. 116, 203–236 (2005)