Mathematics – Geometric Topology
Scientific paper
2011-08-22
Mathematics
Geometric Topology
6 pages, 5 figures
Scientific paper
Hopf's Umlaufsatz relates the total curvature of a closed immersed plane curve to its rotation number. While the curvature of a curve changes under local deformations, its integral over a closed curve is invariant under regular homotopies. A natural question is whether one can find some non-trivial densities on a curve, such that the corresponding integrals are (possibly after some corrections) also invariant under regular homotopies of the curve in the class of generic immersions. We construct a family of such densities using indices of points relative to the curve. This family depends on a formal parameter q and may be considered as a quantization of the total curvature. The linear term in the Taylor expansion at q=1 coincides, up to a normalization, with Arnold's J^+ invariant. This leads to an integral expression for J^+.
Lanzat Sergei
Polyak Michael
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