To each chord system of the 2-disk \(D^2\) is associated a generalized braid group, a compact orientable surface with boundary and a homomorphism from the generalized braid group to the class group of the surface. Perron and Vannier showed that for certain simple types of chord systems this homomorphism is injective, and conjectured that it is injective in general. In the present paper it is shown that for most other types of chord systems this is not true.
A chord system of the 2-disk \(D^2\) is a family of properly embedded segments (chords) \(\Sigma_1,\dots, \Sigma_n\) each two of which are either disjoint or intersect in exactly one point. The generalized braid group associated to the chord system has generators \(x_1,\dots, x_n\) and braid type relations \(x_ix_j= x_jx_i\) resp. \(x_ix_jx_i= x_jx_ix_j\) if the corresponding chords are disjoint resp. intersect. A compact orientable surface with boundary is obtained by attaching for each chord a 1-handle (or band) to the boundary of the 2-disk joining the endpoints of the chord. The Dehn twists along the simple closed curves obtained by closing each chord with the core of the corresponding handle satisfy also the above braid relations, so one obtains a homomorphism from the generalized braid group to the mapping class group of the surface. Chord systems are described by graphs, and it is shown, by explicit computations in the mapping class group, that for most types of graphs this homomorphism is not injective.