On the Degree of the Colored Jones Polynomial
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On the Degree of the Colored Jones Polynomial Efstratia Kalfagianni · Christine Ruey Shan Lee
Received: 20 January 2014 / Revised: 5 March 2014 / Accepted: 10 July 2014 / Published online: 30 October 2014 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2014
Abstract The extreme degrees of the colored Jones polynomial of any link are bounded in terms of concrete data from any link diagram. It is known that these bounds are sharp for semi-adequate diagrams. One of the goals of this paper is to show the converse; if the bounds are sharp then the diagram is semi-adequate. As a result, we use colored Jones link polynomials to extract an invariant that detects semi-adequate links and discuss some applications. Keywords Colored Jones polynomial · Ribbon graph · Semi-adequate link · Essential surfaces Mathematics Subject Classification (2010) 57M27 · 57M25 · 57M50 · 57N10 · (05C10 · 05C15 · 57M15 · 57M25)
1 Introduction The Jones polynomial and the colored Jones polynomials of semi-adequate links have been studied considerably in the literature [18, 19, 21] and [1–3, 7, 13–15] and they have been shown to capture deep information about incompressible surfaces and geometric structures of link complements [8–12]. The extreme degrees of the colored Jones polynomial of any link are bounded in terms of concrete data from any link diagram. It is known that these bounds are sharp for semi-adequate diagrams. One of the goals of this paper is to show the converse; if the bounds are sharp then the diagram is semi-adequate. As an application, we extract a link invariant, out of the colored Jones polynomial of a link, that detects precisely when
E. Kalfagianni () · C. R. S. Lee Department of Mathematics, Michigan State University, East Lansing, MI, 48824 USA e-mail: [email protected] C. R. S. Lee e-mail: [email protected]
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E. Kalfagianni, C. R. S. Lee
the link is semi-adequate. We discuss how this invariant can be thought of as generalizing certain stable coefficients of the colored Jones polynomials of semi-adequate links, studied by Armond [1], Dasbach and Lin [7], and Garoufalidis and Le [13], to all links. We also discuss how, combined with the work of Futer, Kalfagianni, and Purcell [10, 12], our invariant detects certain incompressible surfaces in link complements and their geometric types. To describe the results and the contents of the paper in more detail, recall that a link is called semi-adequate if it admits a link diagram that is A-adequate or B-adequate; see Definition 2.3 for more details. The colored Jones polynomial of a link K is a sequence of Laurent polynomial invariants {JK (n + 1, q)}n such that JK (2, q) is the ordinary Jones polynomial. Let d(n) and d ∗ (n) denote the minimum and maximum degree of JK (n + 1, q) in q, respectively. It is known that for any link diagram D of K, there exists explicit functions hn (D) and h∗n (D) such that d(n) ≥ hn (D) and d ∗ (n) ≤ h∗n (D); see Section 3.1 for more details. If D is an A-adequat
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