Integrating curvature: from Umlaufsatz to J^+ invariant

TL;DR

This paper explores a family of curvature densities on plane curves, generalizing the total curvature and connecting it to Arnold's J^+ invariant through a quantization approach.

Contribution

It introduces a novel family of curvature densities depending on a parameter q, linking total curvature to the J^+ invariant via a quantization framework.

Findings

Constructed a family of densities invariant under regular homotopies.

Connected the linear term of the family to Arnold's J^+ invariant.

Provided an integral expression for J^+.

Abstract

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^+.