H. Whitney [Compositio math., Groningen, 4, 276–284 (1937; JFM 63.0647.01)], studied regular closed curves \(\gamma:[0,1]\to\mathbb R^2, \gamma(0)=\gamma(1)\) in the plane and introduced the winding (rotation) number \(w(\gamma)\) of such a curve as the total angle through which the tangent \(\gamma'(t)\) turns while traversing the curve. Then he proved that two curves may be deformed into each other iff they have the same winding number and gave a method of determining the winding number by counting the algebraic number of times that the curve cuts itself. After Whitney, other mathematicians have studied such problems on surfaces.
This paper treats curves on oriented punctured surfaces \(\Sigma=\Sigma_{m,n}\) of genus \(m\) with \(n\) punctures (note that \(\Sigma_{0,1}=\mathbb R^2\) and \(\Sigma_{0,1}=T^2\)). Then \(\gamma\) is a loop in \(\Sigma,\) so \([\gamma]\in \pi_1(\Sigma,p).\) Also, for every self-intersction point \(d=\gamma(u)=\gamma(v), u<v\) we have a loop \(\gamma(t)\) with \(t\in [0,u]\cup [v,1]\) and denote by \(\tau_d(\gamma)\in \pi_1(\Sigma,p)\) the homotopy class of such a loop. V. G. Turaev [Math. USSR, Sb. 35, 229–250 (1979; Zbl 0422.57005)], constructed an element of the group ring \(\mathbb Z[\pi]\) corresponding to \(\gamma\) which is defined as \(<\gamma> = \sum_{d\in D(\gamma)} \mathrm{sgn}(d)\tau_d(\gamma),\) where \(D(\gamma)\) denotes the set of double points of \(\gamma\), \(\mathrm{sgn}(d)=\pm 1\) is the usual sign of double points defined by the orientations of \((\gamma'(u),\gamma'(v))\) and \(\Sigma.\) For \(\gamma\) in \(\mathbb R^2\) it holds that \(<\gamma> = -w(\gamma)+2\mathrm{ind}(\gamma,p).\)
This paper generalizes Whitney’s formula to \(\Sigma_{m,n}\). To define analogs of rotation number and the index of a base point of \(\gamma\), the authors fix an arbitrary vector field on \(\Sigma_{m,n}\) obtaining a family of identities indexed by elements of \(\pi_1(\Sigma,p).\) Similar formulas are obtained for non-based curves.