UNIQUENESS OF HIGHLY REPRESENTATIVE SURFACE EMBEDDINGS

Let Sigma be a (connected) surface of ''complexity'' kappa; that is, S igma may be obtained from a sphere by adding either 1/2 kappa handles of kappa crosscaps. Let rho greater than or equal to 0 be an integer, and let Gamma be a ''rho-representative drawing'' in Sigma; that is, a drawing of a graph in Sigma so that every simple closed curve in Sigm a that meets the drawing in < rho points bounds a disc in Sigma. Now l et Gamma' be another drawing, in another surface Sigma' of complexity kappa', so that Gamma and Gamma' are isomorphic as abstract graphs. We prove that (i) If rho greater than or equal to 100 log kappa/log log kappa (or rho greater than or equal to 100 if kappa less than or equal to 2) then kappa'greater than or equal to kappa, and if kappa' = kapp a and Gamma is simple and 3-connected there is a homeomorphism from Si gma to Sigma' taking Gamma to Gamma', and (ii) if Gamma is simple and 3-connected and Gamma' is 3-representative, and rho greater than or eq ual to min (320, 5 log kappa), then either there is a homeomorphism fr om Sigma to Sigma' taking Gamma to Gamma', or kappa greater than or eq ual to kappa + 10(-4)rho(2). (C) 1996 John Wiley & Sons, Inc.