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Sphere systems in 3 manifolds and arc graphs

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Figure

Figure 2.1: All the possibilities for 3-pieces in Mlinks. The 3-piece we consider is the part outlined with grey
Figure 2.2: How to associate a 2-piece pthe black edge in the picture. The associated 2-piece is a disc if the edge boundsone square, an annulus if the edge bounds two squares, and a pair of pants if the(e) to an edge e
Figure 2.3: How to associate to a vertex vcomponents (whereif the two corresponding vertices are joined by an edge in the linkus the boundary pattern of the 3-pieceis the union of two black annuli, two red annuli and a black disc
Figure 2.4: A 4-circuit in the graph G′ and the newedges. The dotted lines representthe newedges
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