The slope conjecture for graph knots

Motegi, Kimihiko; Takata, Toshie

doi:10.1017/s0305004116000566

Cited by 7 publications

(7 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…8 Fig. 18 The sequence of choices of expansions for an adjacent set of skein elements skein elements in the expansion, and removing circles by Fig. 8.…”

Section: The Degree Of the Colored Jones Polynomial Of Pretzel Knotsmentioning

confidence: 99%

“…For non-adequate knots, it is natural to ask the extent to which similar results hold. Results extending relationships observed for adequate knots exist [2,8,9,[13][14][15]18]. However, it is difficult to study the colored Jones polynomial in complete generality, since the state sum which may be used to define the polynomial often has cancellations that are difficult to control.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Cancellations in the Degree of the Colored Jones Polynomial

Lee

Veen

2021

Acta Math Vietnam

View full text Add to dashboard Cite

show abstract

“…8 Fig. 18 The sequence of choices of expansions for an adjacent set of skein elements skein elements in the expansion, and removing circles by Fig. 8.…”

Section: The Degree Of the Colored Jones Polynomial Of Pretzel Knotsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Cancellations in the Degree of the Colored Jones Polynomial

Lee

Veen

2021

Acta Math Vietnam

View full text Add to dashboard Cite

show abstract

“…As far as the authors know, the Slope Conjecture has been proved for knots with up to 10 crossings [6], adequate knots [10], 2-fusion knots [9], some pretzel knots [4] and a family of Montesinos knots [23]. 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%

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

Leng

Yang

Liu

2018

J. Knot Theory Ramifications

View full text Add to dashboard Cite

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

show abstract

“…Kalfagianni and Lee [10] mention that the Strong Slope Conjecture for graph knots is settled in [14], but its proof was not explicitly given in [14]. The driving purpose of this note is to record an explicit proof of the Strong Slope Conjecture for graph knots.…”

Section: Introductionmentioning

confidence: 99%

“…In the proofs of the Slope Conjecture for graph knots [14] and Proposition 4.3, the behavior of the maximum degree of the colored Jones function of a knot under cabling plays a key role. This behavior was originally addressed in [11,Proposition 3.2].…”

Section: Introductionmentioning

confidence: 99%

The strong slope conjecture for cablings and connected sums

Baker,

Motegi,

Takata

2018

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

The slope conjecture for graph knots

Cited by 7 publications

References 14 publications

Cancellations in the Degree of the Colored Jones Polynomial

Cancellations in the Degree of the Colored Jones Polynomial

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

The strong slope conjecture for cablings and connected sums

Contact Info

Product

Resources

About