@article{94e0f549a6934271b934f03a93de988f,
title = "Whitney's formulas for curves on surfaces",
abstract = "The classical Whitney formula relates the number of times an oriented plane curve cuts itself to its rotation number and the index of a base point. In this paper we generalize Whitney's formula to curves on an oriented punctured surface Σm, n, obtaining a family of identities indexed by elements of π1(Σm, n). To define analogs of the rotation number and the index of a base point of a curve γ, we fix an arbitrary vector field on Σm, n. Similar formulas are obtained for non-based curves.",
keywords = "Curves on surfaces, Rotation number, Self-intersections, Whiney formula",
author = "Yurii Burman and Michael Polyak",
note = "Funding Information: Yurii Burman was supported by the Scientific foundation of the HSE project 09-01-0015 “Hurwitz generating functions and embedded graphs”, by the CRDF grant RUM1-2895-MO-07 “Rational Cherednik algebras, inverse Macaulay systems, and complete integrability”, by the RFBR grant 08-01-00110 “Geometry and combinatorics of mapping spaces of real and complex curves”, by the joint RFBR and HSE grant 09-01-12185-ofi-m “ Combinatorial aspects of integrable models of the mathematical physics”, by the HSE Mathematical Laboratory grant TZ-62.0 (2010) “Studies in low-dimensional topology, algebraic geometry and representations theory”, by the Russian Federation grant “New methods of study of integrable systems and moduli spaces in geometry, topology and mathematical physics”, and by the N.Sh.-8462.2010.1 grant (“Scientific school of V. I. Arnold”). Michael Polyak was partially supported by the ISF grant 1261/05.",
year = "2011",
month = apr,
doi = "10.1007/s10711-010-9521-8",
language = "الإنجليزيّة",
volume = "151",
pages = "97--106",
journal = "Geometriae Dedicata",
issn = "0046-5755",
publisher = "Springer Netherlands",
number = "1",
}