On the Degree of the Colored Jones Polynomial

Kalfagianni, Efstratia; Lee, Christine Ruey Shan

doi:10.1007/s40306-014-0087-7

Cited by 16 publications

(51 citation statements)

References 31 publications

(51 reference statements)

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“…(2) When ∆ ≥ 0, there exists an essential surface S 2 with boundary slope bs(S 2 ) = 6 − 2(m + p + q) − 2( Note that in Theorem 2.4 the coefficient of the linear term of d + J K (n) is always negative. This actually verifies another conjecture from [11] for this family of Montesinos knots, which can be stated as follow. Theorem 2.7.…”

Section: Introductionsupporting

confidence: 87%

“…In [18], K. Motegi and T. Takata verify the conjecture for graph knots and prove that it is closed under taking connected sums. In [11], E. Kalfagianni and A. T. Tran prove the conjecture is closed under taking the (p, q)-cable with certain conditions on the colored Jones polynomial, and they formulate the Strong Slope Conjecture (see Conjecture 2.2(b)).…”

Section: Introductionmentioning

confidence: 99%

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A result on the Slope conjectures for 3-string Montesinos knots

Leng

Yang

Liu

2018

J. Knot Theory Ramifications

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The (Strong) Slope Conjecture relates the degree of the colored Jones polynomial of a knot to certain essential surfaces in the knot complement. We verify the Slope Conjecture and the Strong Slope Conjecture for 3-string Montesinos knots satisfying certain conditions. 2010 Mathematics Subject Classification. 57N10, 57M25 ., and [s 0 , · · · , s p ] and [t 0 , · · · , t q ] are defined similarly. Note that our conventions for Montesinos knots coincide with those of [22]. This article is a succeeding work of [4] and [23] (however, the results of this article is not strictly the generalization of that of [4] or [23] because we can not loosen the restriction m, p, q ≥ 1 in Lemma 3.4), and the goal of these articles is to provide more data and evidences to the (Strong) Slope Conjecture for Montesinos knots. The reason to choose the family of Montesinos knots is that as a generalization of 2-bridge knots, it is large and representative, and meanwhile well parameterized. Moreover, C. R. S. Lee and R. van der Veen's method [4] to deal with the colored Jones polynomial and its degree and Hatcher and Oertel's algorithm [15] to determine the incompressible surfaces of Montesinos knots pave the way for the proof. The strategy of the proof is straight-forward: we first find out the maximal degree of the colored Jones polynomial and then choose the essential surface which matches the degree by the boundary slope and the Euler characteristic provided by the Hatcher-Oertel algorithm.As we will see, for 3-string Montesinos knots M([r 0 , · · · , r m ], [s 0 , · · · , s p ], [t 0 , · · · , t q ]), the increasing of m, p, q does not cause much complexity, and like the cases in [4] and [23], r 0 , s 0 and s 0 , particularly the discriminant ∆ (see Theorem 2.4 and its proof in Section 3) still dominate the maximal degree of the colored Jones polynomial (Theorem 2.4) as well as the selection of the essential surface (Theorem 2.5). More specifically, when ∆ < 0, the degree of colored Jones polynomial is matched by a typical type I essential surface; when ∆ ≥ 0, it is matched by a type II essential surface, but this type II surface generally (when at least one of m,p and q is greater than 1) does not correspond to a Seifert surface while it does in [4] and [23]. The Slope ConjecturesLet K denote a knot in S 3 and N(K) denote its tubular neighbourhood. A surface S properly embedded in the knot exterior E(K) = S 3 − N(K) is

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Section: Introductionsupporting

confidence: 87%

Section: Introductionmentioning

confidence: 99%

A result on the Slope conjectures for 3-string Montesinos knots

Leng

Yang

Liu

2018

J. Knot Theory Ramifications

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“…At each crossing of D, we connect the pair of neighboring disks by a half-twisted band to construct a surface S σ ⊂ S 3 whose boundary is K. See Figure 2. Inequalities 7 and 8, that generalize and strengthen results of [18], have been more recently established by Lee. See [22,Theorem 2.4] or [21].…”

Section: Jones Surfaces Of Adequate Knotsmentioning

confidence: 60%

A Jones slopes characterization of adequate knots

Kalfagianni¹

2018

Indiana Univ. Math. J.

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We establish a characterization of adequate knots in terms of the degree of their colored Jones polynomial. We show that, assuming the Strong Slope conjecture, our characterization can be reformulated in terms of "Jones slopes" of knots and the essential surfaces that realize the slopes. For alternating knots the reformulated characterization follows by recent work of J. Greene and J. Howie.2010 Mathematics Classification: 57N10, 57M25.

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“…As an application of Theorem 1.2 and [10] we establish: Theorem 1.3. Every graph knot satisfies the slope conjecture.…”

Section: Introductionmentioning

confidence: 94%

“…Recently, the conjecture was verified for iterated cables of adequate knots and for iterated torus knots. [10,11].…”

Section: Introductionmentioning

confidence: 99%

The slope conjecture for graph knots

Motegi

Takata²

2016

Math. Proc. Camb. Phil. Soc.

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The slope conjecture proposed by Garoufalidis asserts that the Jones slopes given by the sequence of degrees of the colored Jones polynomials are boundary slopes. We verify the slope conjecture for graph knots, i.e. knots whose Gromov volume vanish.Comment: We found a gap in the proof of Lemma 3.1 in the last version (1501.01105v2). This lemma is replaced by Proposition 3.2 in [Kalfagianni and Tran; Knot cabling and the degree of the colored Jones polynomial (1501.01574v1)]. The title of the paper is also change

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On the Degree of the Colored Jones Polynomial

Cited by 16 publications

References 31 publications

A result on the Slope conjectures for 3-string Montesinos knots

A result on the Slope conjectures for 3-string Montesinos knots

A Jones slopes characterization of adequate knots

The slope conjecture for graph knots

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