By Costantino F., Thurston D.

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Turaev, ‘Invariants of 3-manifolds via link polynomials and quantum groups’, Invent. Math. 103 (1991) 547–597. 31. J. D. Roberts, ‘Skein theory and Turaev–Viro invariants’, Topology 34 (1995) 771–787. 32. H. Rubinstein and M. Scharlemann, ‘Comparing Heegard splittings of non-Haken 3-manifolds’, Topology 35 (1996) 1005–1026. 33. O. Saeki, ‘Simple stable maps of 3-manifolds into surfaces’, Topology 35 (1996) 671–698. 34. S. GT/0407047. 35. D. columbia. pdf. 36. W. P. org/publications/books/gt3m/.

To get a section of π1 over ∂B, we / Sing(P ) need to choose how to connect the endpoints of the fi with curves in π1−1 (wi ). But wi ∈ and so π1−1 (wi ) is an oriented curve, so we connect the endpoints of fi and fi−1 through an oriented arc over wi . 6 to compute the gleam. We now construct a section B of π1 over B. To do this, let T (1) be the 1-skeleton of T and for each component Bi of B\π1 (T (1) ), choose a face Fi of T so that Bi ⊂ π1 (Fi ) and let Bi = π1−1 (Di ) ∩ Fi . One can choose the faces so that they ﬁt coherently on the edges of T projecting inside B, because by construction, these edges correspond to noncritical points.

The legs of the graph F meet in an additional point at ∞. − 21 − 12 + 14 +1 − 12 + 14 − 21 Figure 21. The polyhedron (with two vertices) is used to complete the construction of P near the codimension-2 singularities of the second type. The boundary of the polyhedron (thicker in the picture) coincides with the boundary of the polyhedron P already constructed away from the singularity. has six edges, we have v(ei ) = 6t. i Let us now count the total number of segments of singular values in R2 (that is, the number of fi ).