The tail of a quantum spin network

Hajij, Mustafa

doi:10.1007/s11139-015-9705-9

Cited by 23 publications

(25 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We call such k v the extension of the value k v at v along the edge e * . Since the graph D * is connected, and k v ∞ = 0, we see that there is at most one solution λ ∈ 0 of (38). Now let us look at the existence of solution of (38).…”

Section: Proposition 2 (A)mentioning

confidence: 99%

“…After the papearance of our paper on the arxiv in the late 2011, a number of papers have since been posted. Among them, Hajij gives a skein-theory proof of zero stability for alternating links and some quantum spin networks [37,38]. Motivated by the q-series of Nahm type, Andrews proves some Rogers-Ramanujan type identities [39].…”

Section: Follow-up Workmentioning

confidence: 99%

“…Since the graph D * is connected, and k v ∞ = 0, we see that there is at most one solution λ ∈ 0 of (38). Now let us look at the existence of solution of (38). Given v ∈ V(D * ), a ∈ R, and a path α of the graph * connecting v to v ∈ V( * )…”

Section: Proposition 2 (A)mentioning

confidence: 99%

See 2 more Smart Citations

Nahm sums, stability and the colored Jones polynomial

Garoufalidis

2015

Res Math Sci

114

View full text Add to dashboard Cite

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

show abstract

Section: Proposition 2 (A)mentioning

confidence: 99%

Section: Follow-up Workmentioning

confidence: 99%

See 1 more Smart Citation

Nahm sums, stability and the colored Jones polynomial

Garoufalidis

2015

Res Math Sci

114

View full text Add to dashboard Cite

show abstract

“…Here the idempotent f (n) is depicted by the box with n strands coming from the upper part of the box and n strands leaving the lower part. For other applications of the Jones-Wenzl projector see [3,9,[11][12][13]. We will also need the following equation:…”

Section: )mentioning

confidence: 99%

On Rational Knots and Links in the Solid Torus

2018

Self Cite

View full text Add to dashboard Cite

We introduce the notion of rational links in the solid torus. We show that rational links in the solid torus are fully characterized by rational tangles, and hence by the continued fraction of the rational tangle. Furthermore, we generalize this by giving an infinite family of ambient isotopy invariants of colored diagrams in the Kauffman bracket skein module of an oriented surface.2000 Mathematics Subject Classification. Primary 57M27; Secondary 57M25, 11A55 .

show abstract

“…We eventually found various applications for the bubble element. In fact, in [4] we use the bubble element to compute q-series associated with the colored Jones polynomial of certain knots that were unknown, and also to prove two sets of q-series identities: Andrews-Gordon identities involving the Ramanujan theta function and a similar set of identities involving the Ramanujan false theta function.…”

Section: Introductionmentioning

confidence: 99%

The Bubble skein element and applications

Hajij

2014

J. Knot Theory Ramifications

Self Cite

View full text Add to dashboard Cite

We study a certain skein element in the relative Kauffman bracket skein module of the disk with some marked points, and expand this element in terms linearly independent elements of this module. This expansion is used to compute and study the head and the tail of the colored Jones polynomial and in particular we give a simple q−series for the tail of the knot 85. Furthermore, we use this expansion to obtain an easy determination of the theta coefficients.1.2. Acknowledgements. I would like to thank Oliver Dasbach whose guidance made this possible. I would also like to thank Pat Gilmer for teaching me skein theory and for his helpful discussions, Cody Armond for many valuable conversations, Khaled Bataineh and Kyle Istvan for reading early drafts carefully and pointing out many corrections.

show abstract

The tail of a quantum spin network

Cited by 23 publications

References 25 publications

Nahm sums, stability and the colored Jones polynomial

Nahm sums, stability and the colored Jones polynomial

On Rational Knots and Links in the Solid Torus

The Bubble skein element and applications

Contact Info

Product

Resources

About