Walks along braids and the colored Jones polynomial

Armond, Cody

doi:10.1142/s0218216514500072

Cited by 12 publications

(15 citation statements)

References 7 publications

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“…On the other hand, when p > 1, K p ,0 (q) is not modular of any weight, according to K. Ono. This disproves any conjectured modularity properties of 0 (q), even for 5 2 . On the other hand, …”

Section: Table 5 Twist Knotssupporting

confidence: 76%

“…The zero stability is also proven independently by Armond for all adequate links [1], which include alternating links and closures of positive braids, see also [2]. The advantage of our approach is that it proves stability to all orders and gives explicit formulas (in the form of generalized Nahm sums) for the limiting series, which in particular implies convergence in the open unit disk in the q-plane and allow for the study of their redial asymptotics.…”

Section: Introductionmentioning

confidence: 64%

“…For the one stability of f (2) n (q), use Corollary 6 to write s = ms P + s where |s| = m and p is a B-polygon with κ(P) edges. It follows that…”

Section: The Computation Of K1 (Q) In Terms Of a Planar Diagrammentioning

confidence: 99%

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Nahm sums, stability and the colored Jones polynomial

Garoufalidis

2015

Res Math Sci

114

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Nahm sums are q-series of a special hypergeometric type that appear in character formulas in the conformal field theory, and give rise to elements of the Bloch group, and have interesting modularity properties. In our paper, we show how Nahm sums arise naturally in the quantum knot theory -we prove the stability of the coefficients of the colored Jones polynomial of an alternating link and present a Nahm sum formula for the resulting power series, defined in terms of a reduced diagram of the alternating link. The Nahm sum formula comes with a computer implementation, illustrated in numerous examples of proven or conjectural identities among q-series. MSC: Primary 57N10; Secondary 57M25.Keywords: Nahm sums; Colored Jones polynomial; Links; Stability; Modular forms; Mock-modular forms; q-holonomic sequence; q-series; Conformal field theory; Thin-thick decomposition BackgroundThe colored Jones polynomial of a link is a sequence of Laurent polynomials in one variable with integer coefficients. We prove in full a conjecture concerning the stability of the colored Jones polynomial for all alternating links.A weaker form of stability (zero stability, defined below) for the colored Jones polynomial of an alternating knot was conjectured by Dasbach and Lin. The zero stability is also proven independently by Armond for all adequate links [1], which include alternating links and closures of positive braids, see also [2]. The advantage of our approach is that it proves stability to all orders and gives explicit formulas (in the form of generalized Nahm sums) for the limiting series, which in particular implies convergence in the open unit disk in the q-plane and allow for the study of their redial asymptotics.Stability was observed in some examples by Zagier, and conjectured by the first author to hold for all knots, assuming that we restrict the sequence of colored Jones polynomials to suitable arithmetic progressions, dictated by the quasi-polynomial nature of its q-degree [3,4]. Zagier asked about modular and asymptotic properties of the limiting q-series. In a similar direction, Habiro asked about zero stability of the cyclotomic function of alternating links in [5].Our generalized Nahm sum formula comes with a computer implementation (using as input a planar diagram of a link), and allows the study of its asymptotics when q

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Section: Table 5 Twist Knotssupporting

confidence: 76%

Section: Introductionmentioning

confidence: 64%

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Nahm sums, stability and the colored Jones polynomial

Garoufalidis

2015

Res Math Sci

114

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“…This can be used to prove various q-identities and it was first utilized by Armond and Dasbach in [2] where they showed that the Andrews-Gordon identity for the theta function can be proven using two methods to compute the of the tail of the (2, 2k + 1) torus knots. In particular Armond and Dasbach use R-matrices and a combinatorial version of the quantum determinant formulation of Huynh and Le [10] developed by Armond [4] to compute the colored Jones polynomial of the (2, 2k + 1) torus knot. These computations are then used to obtain two expressions of the tail associated with the (2, 2k + 1) torus knot.…”

Section: By Assumption We Have Tγmentioning

confidence: 99%

The tail of a quantum spin network

Hajij

2015

Ramanujan J

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Abstract. The tail of a sequence {Pn(q)} n∈N of formal power series in Z [[q]] is the formal power series whose first n coefficients agree up to a common sign with the first n coefficients of Pn. This paper studies the tail of a sequence of admissible trivalent graphs with edges colored n or 2n. We use local skein relations to understand and compute the tail of these graphs. We also give product formulas for the tail of such trivalent graphs. Furthermore, we show that our skein theoretic techniques naturally lead to a proof for the Andrews-Gordon identities for the two variable Ramanujan theta function as well to corresponding identities for the false theta function.

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“…In [1] it was shown that for a knot K , which can be expressed as the closure of a positive braid, T K .q/ D 1. In [3] Abhijit Champanerkar and Ilya Kofman show this for a knot expressed as a positive braid with a full twist, and also determine a sequence of coefficients beyond the first N .…”

Section: Introductionmentioning

confidence: 99%