Applications
In this chapter, we will use the calculations of \(\mathrm{guts}({S}^{3}\setminus \setminus {S}_{A})\) obtained in earlier chapters to relate the geometry of A-adequate links to diagrammatic quantities and to Jones polynomials. In Sect. 9.1, we combine Th
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Applications
In this chapter, we will use the calculations of guts.S 3 nnSA / obtained in earlier chapters to relate the geometry of A-adequate links to diagrammatic quantities and to Jones polynomials. In Sect. 9.1, we combine Theorem 5.14 with results of Agol, Storm, and Thurston [6] to obtain bounds on the volumes of hyperbolic A-adequate links. A sample result is Theorem 9.7, which gives tight diagrammatic estimates on the volumes of positive braids with at least 3 crossings per twist region. The gap between the upper and lower bounds on volume is a factor of about 4:15. In Sect. 9.2, we apply these ideas to Montesinos links, and obtain diagrammatic estimates for the volume of those links. Again, the bounds are fairly tight, with a factor of 8 between the upper and lower bounds. In Sect. 9.3, we relate the quantity .guts.S 3 nnSA // to coefficients of the Jones and colored Jones polynomials of the link K D @SA . One sample application here is 0 Corollary 9.16: for A-adequate links, the next-to-last coefficient ˇK detects whether 3 a state surface is a fiber in S n K. Finally, in Sect. 9.4, we synthesize these ideas to obtain relations between the Jones polynomial and volume. As a result, the volumes of both positive braids and Montesinos links can be bounded above and below in terms of these coefficients.
9.1 Volume Bounds for Hyperbolic Links Using Perelman’s estimates for volume change under Ricci flow with surgery, Agol, Storm, and Thurston [6] have obtained a relationship between the guts of an essential surface S M and the hyperbolic volume of the ambient 3-manifold M . The following result is an immediate consequence of [6, Theorem 9.1], combined with work of Miyamoto [68, Proposition 1.1 and Lemma 4.1].
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 9, © Springer-Verlag Berlin Heidelberg 2013
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9 Applications
Theorem 9.1. Let M be finite-volume hyperbolic 3-manifold, and let S M be a properly embedded essential surface. Then vol.M / v8 .guts.M nnS //; where v8 D 3:6638 : : : is the volume of a regular ideal octahedron. Remark 9.2. By [6] and work of Calegari, Freedman, and Walker [16], the inequality of Theorem 9.1 is an equality precisely when S is totally geodesic and M nnS is a union of regular ideal octahedra. We will not need this stronger statement. In general, it is hard to effectively compute the quantity .guts.M nnS // for infinitely many pairs .M; S /. To date, there have only been a handful of results computing the guts of essential surfaces in an infinite family of manifolds: see e.g. [3, 57, 58]. In particular, Lackenby’s computation of the guts of checkerboard surfaces of alternating links [58, Theorem 5] enabled him to estimate the volumes of these link complements directly from a diagram. See [58, Theorem 1] and [6, Theorem 2.2]. In the A-adequate setting, we have the following volume estimate. Theorem 9.3. Let D D D.K/ be a prime A-adequate diagram of a hyperbolic link K. Th
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