I was reading about Gauss-Bonnet for circles, that integrating the curvature over the circle $\gamma(t)$ leads to the number of times $\gamma'(t)$ rotates around the origin.  (The degree of the Gauss map $S^1 \to S^1$).

\\[ \frac{1}{2\pi} \int_{S^1} \kappa \\, ds = \mathrm{rot}(\Gamma)  \\]

The curvature here is kind of like the 1st chern class of the real line bundle made from the tangent vectors at each point around the curve.  Here's it's being paired with the circle.

In a paper by [Michael Polyak][1], I found a q-deformed version of Gauss-Bonnet (rather Hopf Umlaufsatz) which differs from the usual one only when there are double points:


\\[ \frac{1}{2\pi} \int_{S^1} \kappa \\, q^{\mathrm{Ind}(\gamma(s), \gamma)} \\, ds 
- \sum_{double} \theta_d \\, q^{\mathrm{Ind}(d, \gamma)}(q^{1/2} - q^{-1/2}) \\]

$\mathrm{Ind}(\gamma(s), \gamma)$ is the index, the average of the winding numbers of $\gamma$ around points inside and outside the circle, near $\gamma(t)$.  $d$ runs over the double points of $\gamma(t)$ and $\theta_d$ is the angle between the two tangents at $d$.

[![enter image description here][2]][2]

The exponent jumps around as you move around the circle. The sum over double points are a correction for smoothing the double points into the union of disjoint circles. I am wondering, does it come from a q-deformed version of the Gauss-Bonnet formula?

[![enter image description here][3]][3]


  [1]: http://www2.math.technion.ac.il/~polyak/publ/curvature.pdf
  [2]: https://i.sstatic.net/tJWX6.png
  [3]: https://i.sstatic.net/6m42W.png