Guts and Fibers
This chapter contains one of the main results of the manuscript, namely a calculation of the Euler characteristic of the guts of M A in Theorem 5.14. The calculation will be in terms of the number of essential product disks (EPDs) for M A which are comple
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Guts and Fibers
This chapter contains one of the main results of the manuscript, namely a calculation of the Euler characteristic of the guts of MA in Theorem 5.14. The calculation will be in terms of the number of essential product disks (EPDs) for MA which are complex, as in Definition 5.2, below. In subsequent chapters, we will find bounds on the number of such EPDs in terms of a diagram, for general and particular types of diagrams (Chaps. 6–8), and use this information to bound volumes, and relate other topological information to coefficients of the colored Jones polynomial (Chap. 9). Recall that we have shown in Theorem 4.4 that the I -bundle of MA is spanned by EPDs, each of which is embedded in a single polyhedron of the polyhedral decomposition. (See Definitions 4.2 and 4.3 on p. 55 to recall the terminology.) Thus to calculate the Euler characteristic of the guts, we calculate the minimal number of such a collection of spanning EPDs. We will do this by explicitly constructing a spanning set of EPDs with desirable properties (Lemmas 5.6 and 5.8). In Proposition 5.13, we will compute exactly how redundant the spanning set is. This leads to the Euler characteristic computation in Theorem 5.14. Along the way, we also give a characterization of when the link complement fibers over S 1 with fiber the state surface SA , in terms of the reduced state graph G0A , in Theorem 5.11.
5.1 Simple and Non-simple Disks By Theorem 4.4, every non-trivial component in the characteristic submanifold of MA is spanned by essential product disks in individual polyhedra. Our goal is to find and count these disks, starting with the lower polyhedra. Lemma 5.1. Let D be an A-adequate diagram of a link in S 3 . Consider a prime polyhedral decomposition of MA D S 3 nnSA . The essential product disks embedded in the lower polyhedra are in one-to-one correspondence with the 2-edge loops in the graph GA .
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 5, © Springer-Verlag Berlin Heidelberg 2013
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5 Guts and Fibers
Proof. By definition, an EPD in a lower polyhedron must run over a pair of shaded faces F and F 0 . By Lemma 2.21 on p. 29, these shaded faces correspond to state circles C and C 0 . Furthermore, every ideal vertex shared by F and F 0 corresponds to a segment of HA between C and C 0 , or equivalently, to an edge of GA between C and C 0 . Since an EPD must run over two ideal vertices between F and F 0 , it naturally defines a 2-edge loop in GA , whose vertices are the state circles C and C 0 . In the other direction, the two edges of a 2-edge loop in GA define a pair of ideal vertices shared by F and F 0 , hence an EPD. Thus we have a bijection. t u Typically, we do not need all the disks in the lower polyhedra to span the I -bundle. We will focus on choosing disks that are as simple as possible. Definition 5.2. Let P be a checkerboard-colored ideal polyhedron. An essential product disk D P is called .1/ Simple if D is the boundar
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