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We adapt some of the methods of quantum Teichm\"uller theory to construct a family of representations of the pure braid group of the sphere.
Comment: Revised and expanded with a review of the spanning tree complex. To appear in Communications in Contemporary Mathematics, special volume in memory of Xiao-Song Lin. 18 pages
Comment: 28 pages, 14 figures, final version
Comment: 14 pages, 2 figures
Comment: 12 pages, 4 figures
We show that the maximal number of singular moves required to pass between any two regularly homotopic planar or spherical curves with at most n crossings, grows quadratically with respect to n. Furthermore, this can be done with all curves along the way having at most n+2 crossings.
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