... used in the proof of Proposition 2.1 to calculate the number of (p, q, n)-dipoles that are equivalent to our underlying case, that is, the number of (p, q, n)dipoles that, when each grouping of digons ... q So there are choices for (d, b, c, a2 ), by Lemma 2.2 and the choice of a1 is xed Hence there are q (p, q, n)-dipoles of this type Similarly, there are p (p, q, n)-dipoles such 3 that the distinguished ... replacing the edges of these dipoles with face-type [62 ] with multiple non-crossing edges Because of the awkwardness of the polygonal representation of the double torus, we return to the type of diagram...