  In the context of quantum gravity for spacetimes of dimension 2+1, we
describe progress in the construction of a quantum Goldman bracket for
intersecting loops on surfaces. Using piecewise linear paths in R^2
(representing loops on the spatial manifold, i.e. the torus) and a quantum
connection with noncommuting components, we review how holonomies and Wilson
loops for two homotopic paths are related by phases in terms of the signed area
between them. Paths rerouted at intersection points with other paths occur on
the r.h.s. of the Goldman bracket. To better understand their nature we
introduce the concept of integer points inside the parallelogram spanned by two
intersecting paths, and show that the rerouted paths must necessarily pass
through these integer points.
