By Costantino F., Thurston D.

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If H is empty, the lemma is trivial. If H has no vertices, we can create two vertices by a 0 → 2 move. Encircle each vertex of H by a closed curve: this set of n curves intersects H at most 4n times and decomposes S into n blocks of the ﬁrst type and a surface S whose Euler characteristic is χ(S) − n. If H = H ∩ S contains parallel edges, we apply O(n) 0 → 2-moves in order to replace each set of parallel edges by a single edge branching only in the neighborhood of ∂S . Then we add one curve per component of ∂S in order to enclose all these trivalent vertices in annular regions.

GT/0411016. 16. S. King, ‘Polytopality of triangulations’, PhD Thesis, Universit´e Louis Pasteur, Strasbourg, June 2001. 17. R. Kirby and P. Melvin, ‘Evaluations of the 3-manifold invariants of Witten and Reshetikhin–Turaev for sl(2, C)’, Geometry of low-dimensional manifolds, 2, Durham, 1990 (ed. S. K. Donaldson and C. B. Thomas), London Mathematical Society Lecture Notes Series 151 (Cambridge University Press, Cambridge, 1990) 101–114. 18. R. Kirby and P. Melvin, ‘Local surgery formulas for quantum invariants and the Arf invariant’, Proceedings of the Casson Fest (ed.

References 1. J. F. Brock, ‘The Weil–Petersson metric and volumes of 3-dimensional hyperbolic convex cores’, J. Amer. Math. Soc. 16 (2003) 495–535. 744 FRANCESCO COSTANTINO AND DYLAN THURSTON 2. J. F. Brock, ‘Weil–Petersson translation distance and volumes of mapping tori’, Comm. Anal. Geom. 11 (2003) 987–999. 3. O. Burlet and G. de Rham, ‘Sur certaines applications g´en´eriques d’une vari´et´e close `a 3 dimensions dans le plan’, Enseign. Math. (2) 20 (1974) 275–292. 4. F. Costantino, ‘Shadows and branched shadows of 3 and 4-manifolds’, PhD Thesis, Scuola Normale Superiore, Pisa, Italy, May 2004.