Geodesic
On surfaces glued of 2n-gons
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 19 (2017) no. 3, pp. 31-40.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper 2n-gons and surfaces obtained through identification of 2n-gon's sides in pairs (i.e. through sewing) are considered. As well-known, one can get surface of any genus and orientability through sewing but it's very uneasy to calculate by only the polygon and the way of sewing, because to do this one need to calculate the number of vertices appearing after identification; even for small n the problem is almost impossible if one want to do this directly. There are different ways to solve the task. The canonical variant of 4q-gon sewing (2q-gon sewing) giving an orientable (unorientable) surface of genus q is well-known, as the Harer-Zagier' numbers, that are the numbers of variants of sewing a 2n-gon to an orientable surface of gunes q. In this paper we offer a new way of Euler characteristic's of obtained surface calculation (and, hence, its genus) undepending on its orientability by means of three-colour graph and information about closed surfaces topological classification.
Keywords: 2n-gon, Ejler characteristic, orientability, sewing. 
@article{SVMO_2017_19_3_a2,
     author = {V. E. Kruglov and G. N. Talanova},
     title = {On surfaces glued of 2n-gons},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {31--40},
     publisher = {mathdoc},
     volume = {19},
     number = {3},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2017_19_3_a2/}
}
TY  - JOUR
AU  - V. E. Kruglov
AU  - G. N. Talanova
TI  - On surfaces glued of 2n-gons
JO  - Žurnal Srednevolžskogo matematičeskogo obŝestva
PY  - 2017
SP  - 31
EP  - 40
VL  - 19
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SVMO_2017_19_3_a2/
LA  - ru
ID  - SVMO_2017_19_3_a2
ER  - 
%0 Journal Article
%A V. E. Kruglov
%A G. N. Talanova
%T On surfaces glued of 2n-gons
%J Žurnal Srednevolžskogo matematičeskogo obŝestva
%D 2017
%P 31-40
%V 19
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SVMO_2017_19_3_a2/
%G ru
%F SVMO_2017_19_3_a2
V. E. Kruglov; G. N. Talanova. On surfaces glued of 2n-gons. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 19 (2017) no. 3, pp. 31-40. http://geodesic.mathdoc.fr/item/SVMO_2017_19_3_a2/

[1] G. B. Shabat, A. I. Sgibnev, “Sewings of polygons”, Kvant, 3 (2011), 17–22 (In Russ.)

[2] J. Harer, D. Zagier, “The Euler characteristic of the moduli space of curves”, Inventiones mathematicae, 85 (1986), 457–486 | DOI | MR

[3] C. Kosniowski, A First Course in Algebraic Topology, Cambridge University Press, 1980, 269 pp. | MR | Zbl

[4] A. A Oshemkov, V. V. Sharko, “About classification of Morse-Smale flows on 2-manifolds”, Matematicheskiy sbornik, 189:8 (1998), 93–140 (In Russ.) | DOI | MR | Zbl

[5] J. Milnor, A. Wallace, Differential TopologMilnorWallacey: a First Steps, Mir, M., 1972, 223 pp. (In Russ.) | MR

[6] E. A. Leontovich, A. G. Mayer, “About trajectories determining qualitative structure of sphere partition into trajectories”, Dokl. Akad. AN SSSR, 14:5 (1937), 251–257 (In Russ.)

[7] E. A. Leontovich, A. G. Mayer, “About scheme determining topological structure of partition into trajectories”, Dokl. Akad. AN SSSR, 103:4 (1955), 557–560 (In Russ.)

[8] M. Peixoto, “On the classification of flows on two manifolds”, Dynamical systems, Proc. Symp. (Univ. of Bahia, Salvador, Brasil.), 1971 | MR | Zbl