Graph Minors. XI. Circuits on a Surface

Let Γ be a graph or hypergraph drawn on a connected surface which is not a sphere, in such a way that every non-null-homotopic curve meets the drawing at least θ times. We show that this defines a "tangle of order θ" in Γ, in the sense of earlier papers of this series. Also, there is a natural distance function defined by the drawing, and we show that for any point c of the surface, and any θ′ > θ, the set of points within distance θ′ of c is simply-connected.