Geom Dedicata (2007) 130:25–41 DOI 10.1007/s10711-007-9203-3 ORIGINAL PAPER Intersection-number operators and Chebyshev polynomials IV: non-planar cases Stephen P. Humphries Received: 19 March 2007 / Accepted: 9 October 2007 / Published online: 1 November 2007 © Springer Science+Business Media B.V. 2007 Abstract We define operators on a ring that allow one to determine the geometric inter- section 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, C). Keywords Geometric intersection number · Operator · Mapping class group · Character ring · Trace Mathematics Subject Classification (2000) 57M50 · 16S99 1 Introduction In [14] (see also [15]) we proved the following result: Theorem 1.1 Let R be an associative ring with identity and fix r ∈ Z(R), where Z(R) is the centre of R. Define polynomials p n = p n (r) recursively by p 0 =-2, p 1 = r, p n =-(rp n-1 + p n-2 ). Let σ : R → R be a ring homomorphism and assume that σ(r) = r . Define operators A n = A n (σ,r) : R → R,n ≥ 0, by A 0 = σ - 1 and for n> 0, we let A n = σ 2 + p n σ + 1. Now define operators B n = B n (σ,r) : R → R,n ≥ 0, by B n = A 0 A 1 ... A n . S. P. Humphries (B ) Department of Mathematics, Brigham Young University, Provo, UT 84602, USA e-mail: steve@math.byu.edu 123