Combinatorics of simple closed curves on the twice punctured torus

In the standard enumeration of homotopy classes of curves on a surface as w ords in a generating set for the fundamental group it is st very hard probl em to discern those that are simple. In this paper we describe how the comp lex of simple closed curves on a twice punctured torus Sigma may be given a strikingly simple description by representing them as homotopy classes of paths in a groupoid with two base points. Our starting point are the pi(1)- train tracks developed by Birman and Series. These are weighted train track s parameterizing the simple closed curves on Sigma similar to Thurston's, b ut they are defined relative to a fixed presentation of pi(1)(Sigma). We ap proach the problem by cutting the surface into two disjoint "cylinders"; th is decomposes the pi(1)-train tracks into two disjoint parts, relative to w hich all patterns and relations become much more transparent, each part red ucing essentially to the well-known case of a once punctured torus. We obta in global coordinates, called pi(1,2)-weights, for simple closed loops. The se coordinates can be easily identified with Thurston's projective measured lamination space S-3. We also solve the problem which originally motivated this work by proving a simple relationship between the leading terms of tr aces of simple loops in a holomorphic family of representations rho: pi(1)( Sigma) --> PSL(2, C) (corresponding to the Maskit embedding of the twice pu nctured torus) and the pi(1,2)-weights.