Uniqueness of highly representative surface embeddings

Let Σ be a (connected) surface of “complexity” κ; that is, Σ may be obtained from a sphere by adding either 1 2 κ handles or κ crosscaps. Let ρ ≥ 0 be an integer, and let Γ be a “ρ-representative drawing” in Σ; that is, a drawing of a graph in Σ so that every simple closed curve in Σ that meets the drawing in < ρ points bounds a disc in Σ. Now let Γ be another drawing, in another surface Σ of complexity κ, so that Γ and Γ are isomorphic as abstract graphs. We prove that (i) If ρ ≥ 100logκ/loglogκ (or ρ ≥ 100 if κ ≤ 2) then κ ≥ κ, and if κ = κ and Γ is simple and 3-connected there is a homeomorphism from Σ to Σ taking Γ to Γ (ii) if Γ is simple and 3-connected and Γ is 3-representative, and ρ ≥ min(320, 5logκ), then either there is a homeomorphism from Σ to Σ taking Γ to Γ, or κ ≥ κ + 10ρ.