A "Universal" Cíass of4-Coloured Graphs

A family of 4—cohoured grapbs depending on three integers b,1,1 and on a transitive pair of permutationsa, y EE his constructed. Each asisociated topological space turns out to hea b-fohd branched covering of el- ther a 6- or a handcuff-graph, with embedding depending on 1 and 1, or a two-hridge knot or hink of type (1,1). Moreover, the monodromy map is phetehy defined by a and r. La particuhar, when 1 = 2a udt = 1, the space is homcomorphic to the (possibly singuhar) manifoid N(o, r), which is the branched covering of the Montesinos universah graph, associated to the pair a,r. This allows us to ohtain a "universal" chass of 4—coloured graphs repre- senting ah orientable3-djmensionah singular manifolds. Further, the nec~sary and sufllcient condition for the graph to represent a manifohd is ohtaiaed and a topohogical interpretation of asimilar construction of A. Cavicchiohi is given. 1991 M,thematics Subject Classification Primary 57M25, 57M12; Secondary 57M15.