Ideal Polyhedra
Recall that \({M}_{A} = {S}^{3}\setminus \setminus {S}_{A}\) is S 3 cut along the surface S A . In the last chapter, starting with a link diagram D(K), we obtained a prime decomposition of M A into 3-balls. One of our goals in this chapter is to show that
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Ideal Polyhedra
Recall that MA D S 3 nnSA is S 3 cut along the surface SA . In the last chapter, starting with a link diagram D.K/, we obtained a prime decomposition of MA into 3-balls. One of our goals in this chapter is to show that, if D.K/ is A-adequate (see Definition 1.1 on p. 4), each of these balls is a checkerboard colored ideal polyhedron with 4-valent vertices. This amounts to showing that the shaded faces on each of the 3-balls are simply-connected, and is carried out in Theorem 3.12. Once we have established the fact that our decomposition is into ideal polyhedra, as well as a collection of other lemmas concerning the combinatorial properties of these polyhedra, two important results follow quickly. The first is Proposition 3.18, which states that all of the ideal polyhedra in our decomposition are prime. The second is a new proof of Theorem 3.19, originally due to Ozawa [76], that the surface SA is essential in the link complement if and only if the diagram of our link is A-adequate. All the results of this chapter generalize to -adequate, -homogeneous diagrams. We discuss this generalization in Sect. 3.4. The results of this chapter will be assumed in the sequel. To prove many of these results, we will use the combinatorial structure of the polyhedral decomposition of the previous chapter, in a method of proof we call tentacle chasing. This method of proof, as well as many lemmas established here using this method, will be used again quite heavily in parts of Chaps. 4, 6–8. Therefore, the reader interested in those chapters should read the tentacle chasing arguments carefully, to be prepared to use such proof techniques later. In particular, tentacle chasing methods form a crucial component in the proofs of our main results, which reside in Chaps. 5 and 9 respectively. However, a reader who is eager to get to the main theorems and their applications, and who seeks only a top-level outline of the proofs, may opt to survey the results of this chapter while taking the proofs on faith. The top-level proofs of the main results in Chap. 5 and the applications in Chap. 9 will not make any direct reference to tentacle chasing.
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 3, © Springer-Verlag Berlin Heidelberg 2013
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3 Ideal Polyhedra
Fig. 3.1 Building blocks of a shaded face: an innermost disk, a tentacle, and a non-prime switch
Fig. 3.2 Far left: A directed spine of a tentacle. Left to right: Shown is how directed tentacles connect to an innermost disk, to another tentacle, across a non-prime switch
3.1 Building Blocks of Shaded Faces To prove the main results of this chapter, first we need to revisit our construction of shaded faces for the upper 3-ball. Shaded faces in the upper 3-ball are built of one of three pieces: innermost disks, tentacles, and non-prime switches. See Fig. 3.1. Recall that a tentacle is directed, starting at the portion adjacent to the segment of HA (the head) and ending where the
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