The spin-statistics connection, the Gauss-Bonnet theorem and the Hausdorff dimension of the quantum paths

We obtain an explicit expression relating the writhing number, $W[C]$, of the quantum path, $C$, with any value of spin, $s$, of the particle which sweeps out that closed curve. We consider a fractal approach to the fractional spin particles and, in this way, we make clear a deeper connection between the Gauss-Bonnet theorem with the spin-statistics relation via the concept of Hausdorff dimension, $h$, associated to the fractal quantum curves of the particles: \frac{h}{2+2s}=W[C]=\frac{1}{4\pi}\oint_{C}d x_{\alpha}\oint_{C}d x_{\beta} \epsilon^{\alpha\beta\gamma} \frac{(x-y)_{\gamma}}{|x-y|^3}.