Recognizing Essential Product Disks
Theorem 5.14 reduces the problem of computing the Euler characteristic of the guts of M A to counting how many complex EPDs are required to span the I-bundle of the upper polyhedron. Our purpose in this chapter is to recognize such EPDs from the structure
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Recognizing Essential Product Disks
Theorem 5.14 reduces the problem of computing the Euler characteristic of the guts of MA to counting how many complex EPDs are required to span the I -bundle of the upper polyhedron. Our purpose in this chapter is to recognize such EPDs from the structure of the all-A state graph GA . The main result is Theorem 6.4, which describes the basic building blocks for such EPDs. Each corresponds to a 2-edge loop of the graph GA . The proofs of this chapter require detailed tentacle chasing arguments, and some are quite technical. To assist the reader, we break the proof of Theorem 6.4 into four steps, and keep a running outline of what has been accomplished, and what still needs to be accomplished. The tentacle chasing does pay off, for by the end of the chapter we obtain a mapping from any EPD to one of only seven possible sub-graphs of HA . By investigating the occurrence of such subgraphs, we are able to count complex EPDs in large classes of link complements. Two such classes are studied in detail in Chaps. 7 (links with diagrams without non-prime arcs) and 8 (Montesinos links). Together with the results of Chap. 5, these give applications to guts, volumes, and coefficients of the colored Jones polynomials.
6.1 2-Edge Loops and Essential Product Disks To find essential product disks in the upper polyhedron, we will convert any EPD into a normal square and use machinery developed in Chap. 4. Lemma 6.1 (EPD to oriented square). Let D be a prime, A-adequate diagram of a link in S 3 , with prime polyhedral decomposition of MA D S 3 nnSA . Suppose there is an EPD embedded in MA in the upper polyhedron. Then the boundary of the EPD can be pulled off the ideal vertices to give a normal square in the polyhedron with the following properties.
D. Futer et al., Guts of Surfaces and the Colored Jones Polynomial, Lecture Notes in Mathematics 2069, DOI 10.1007/978-3-642-33302-6 6, © Springer-Verlag Berlin Heidelberg 2013
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6 Recognizing Essential Product Disks
.1/ Two opposite edges of the square run through shaded faces, which we label green and orange.1 .2/ The other two opposite edges run through white faces, each cutting off a single vertex of the white face. .3/ The single vertex of the white face, cut off by the white edge, is a triangle, oriented such that in counter-clockwise order, the edges of the triangle are colored orange–green–white. With this convention, the two white edges of the normal square cannot lie on the same white face of the polyhedron. Proof. The EPD runs through two shaded faces, green and orange, and two ideal vertices. Any ideal vertex meets two white faces. Thus we may push an arc running over an ideal vertex slightly off the vertex to run through one of the adjacent white faces instead. Note that there are two choices of white face into which we may push the arc, giving oppositely oriented triangles. For each vertex, we choose to push in the direction that gives the triangle oriented as in the statement of the lemma. Finally, to see that the two white ed
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