Intersection-number operators and Chebyshev polynomials IV: non-planar cases

Abstract

We define operators on a ring that allow one to determine the geometric intersection number of two simple closed curves on an oriented surface of genus g ≥ 0 with free fundamental group. A variation of these same operators was used in a previous paper to do a similar thing on a punctured disc. This result is used to give a necessary condition for two words in a free group to have the same character under all representations of the free group into \(SL(2,{\mathbb{C}})\) .