Whitney's formulas for curves on surfaces

Yurii Burman, Michael Polyak

Research output: Contribution to journalArticlepeer-review


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 π1m, 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.

Original languageEnglish
Pages (from-to)97-106
Number of pages10
JournalGeometriae Dedicata
Issue number1
StatePublished - Apr 2011


  • Curves on surfaces
  • Rotation number
  • Self-intersections
  • Whiney formula

All Science Journal Classification (ASJC) codes

  • Geometry and Topology


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