Enumeration of Branched Coverings of Nonorientable Surfaces With Cyclic Branch Points

In this paper, n-fold branched coverings of a closed nonorientable surface ${\mathcal S}$ of genus p with $r\geq 1$ cyclic branch points (that is, such that all ramification points over them are of multiplicity n) are considered. The number Np,r(n) of such coverings up to equivalence is evaluated ex...

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Bibliographic Details
Published inSIAM journal on discrete mathematics Vol. 19; no. 2; pp. 388 - 398
Main Authors Kwak, Jin Ho, Mednykh, Alexander, Liskovets, Valery
Format Journal Article
LanguageEnglish
Published Philadelphia Society for Industrial and Applied Mathematics 01.01.2005
Subjects
Applied mathematics
Computational mathematics
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Summary:In this paper, n-fold branched coverings of a closed nonorientable surface ${\mathcal S}$ of genus p with $r\geq 1$ cyclic branch points (that is, such that all ramification points over them are of multiplicity n) are considered. The number Np,r(n) of such coverings up to equivalence is evaluated explicitly in a closed form (without using any complicated functions such as irreducible characters of the symmetric groups). The obtained formulas depend on the parity of r and n. The method is based on some previous enumerative results and techniques for nonorientable surfaces. In particular, we generalize the approach developed for the counting of unbranched coverings of nonorientable surfaces and make use of the analytical method of roots-of-unity sums.
ISSN:0895-4801
1095-7146
DOI:10.1137/S0895480103424043