A Quantum Gauss-Bonnet Theorem

TL;DR

This paper extends a polynomial invariant of plane curves to curves on closed surfaces, linking it to a generalized Gauss-Bonnet theorem and providing new integral formulas for the J+ invariant.

Contribution

It introduces a new invariant for homologically trivial curves on closed surfaces, generalizing previous invariants and connecting them to a Gauss-Bonnet type formula.

Findings

Invariant applies to curves on closed surfaces with nonzero Euler characteristic

Provides an integral formula for the J+ invariant in this setting

Links polynomial invariants to classical differential geometry results

Abstract

In their paper "Integrating curvature: From Umlaufsatz to J+ invariant" Lanzat and Polyak introduced a polynomial invariant of generic curves in the plane as a quantization of Hopf's Umlaufsatz, and showed that Arnold's J+ invariant could be derived from their polynomial, leading to an integral formula for J+. Here we extend their invariant to the case of homologically trivial generic curves in closed oriented surfaces with Riemannian metric. The resulting invariant turns out to be a quantization of a new formula for the rotation number, which can be viewed as a form of the Gauss-Bonnet Theorem. We show that J+ can be calculated from the generalized invariant when the Euler characteristic of the surface is nonzero, thereby obtaining an integral formula for J+ for homologically trivial curves in oriented surfaces with nonzero Euler characteristic.