Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Classiﬁcation of (p, q, n)-dipoles on nonorientable surfaces. | Classification of p q n -dipoles on nonorientable surfaces Yan Yang Department of Mathematics Tianjin University Tianjin yanyang0206@ Yanpei Liu Department of Mathematics Beijing Jiaotong University Beijing ypliu@ Submitted May 4 2009 Accepted Jan 30 2010 Published Feb 8 2010 Mathematics Subject Classifications 05C10 05C30 Abstract A type of rooted map called p q n -dipole whose numbers on surfaces have some applications in string theory are defined and the numbers of p q n -dipoles on orientable surfaces of genus 1 and 2 are given by Visentin and Wieler The Electronic Journal of Combinatorics 14 2007 R12 . In this paper we study the classification of p q n -dipoles on nonorientable surfaces and obtain the numbers of p q n -dipoles on the projective plane and Klein bottle. 1 Introduction A surface is a compact 2-dimensional manifold without boundary. It can be represented by a polygon of even edges in the plane whose edges are pairwise identified and directed clockwise or counterclockwise. Such polygonal representations of surfaces can also be written by words. For example the sphere is written as O0 aa- where a- is identified with the opposite direction of a on the boundary of the polygon. In general Op p q n aibia b and Nq Ị Ị aiai denote respectively an orientable surface of genus p and a nonorientable surface of genus q. Of course N1 O1 and N2 are respectively the projective plane the torus and the Klein bottle. Every surface is homeomorphic to precisely one of the surfaces Op p 0 or Nq q 1 2 5 . Supported by NNSF of China under Grant THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 N12 1 Let S be the collection of surfaces and let AB be a surface. The following topological transformations and their inverses do not change the orientability and genus of a surface TT 1 Aaa-B o AB where a AB TT 2 AabBab o AcBc where c AB and TT 3 AB o Aa a-B where AB 0. In fact what is determined under these transformations is a .