Flag algebras and the stable coefficients of the Jones polynomial

Garoufalidis, Stavros; Norin, Sergey; Vuong, Thao

doi:10.1016/j.ejc.2015.05.001

Cited by 4 publications

(3 citation statements)

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“…, 3 in terms of graph countings of induced plane subgraphs of the reduced Tait graph of an alternating link [41]. Norin and the first author prove that each coefficient of q k in the zero limit is a polynomial of induced plane subgraphs of the reduced Tait graph of an alternating link [42]. Finally, in another direction, q-series of Nahm type were studied by Beem-Dimofte-Pasquetti and Kashaev and the first author; see [43,44].…”

Section: Follow-up Workmentioning

confidence: 99%

Nahm sums, stability and the colored Jones polynomial

Garoufalidis

2015

Res Math Sci

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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: Follow-up Workmentioning

confidence: 99%

Nahm sums, stability and the colored Jones polynomial

Garoufalidis

2015

Res Math Sci

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114

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“…Our proof allows us to compute the coefficient of q 3 in Φ G (q), observing a new phenomenon related to induced embeddings, and guess the coefficients of q 4 and q 5 in Φ G (q). This is discussed in a subsequent publication [GVN13].…”

Section: Introductionmentioning

confidence: 90%

Alternating knots, planar graphs, and $$q$$ q -series

Garoufalidis

Vuong

2014

Ramanujan J

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Recent advances in Quantum Topology assign q-series to knots in at least three different ways. The q-series are given by generalized Nahm sums (i.e., special qhypergeometric sums) and have unknown modular and asymptotic properties. We give an efficient method to compute those q-series that come from planar graphs (i.e., reduced Tait graphs of alternating links) and compute several terms of those series for all graphs with at most 8 edges drawing several conclusions. In addition, we give a graph-theory proof of a theorem of Dasbach-Lin which identifies the coefficient of q k in those series for k = 0, 1, 2 in terms of polynomials on the number of vertices, edges and triangles of the graph. 10 4.

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“…Armond and Dasbach [AD17] showed that the tail of an adequate link only depends on its reduced A-graph. The lowest three coefficients of T K (q) of alternating link K is described in terms of its reduced Tait graph in Garoufalidis-Norin [GNV16]. Hajij also considered the tail for some quantum spin network in [Haj16].…”

Section: Introductionmentioning

confidence: 99%

The zero stability for the one-row colored $\mathfrak{sl}_3$ Jones polynomial

Yuasa¹

2020

Preprint

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The zero stability of the colored Jones polynomial was proved by using the linear skein theory for the Kauffman bracket in Armond [Arm13]. The stability says that there exists a formal power series for a minus adequate link L such that the first n coefficients of the power series and the (n + 1)-dimensional colored Jones polynomial of L agree. In this paper, we show the zero stability of the one-row colored sl 3 Jones polynomial of a minus-adequate link by using the linear skein theory for A 2 .

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Flag algebras and the stable coefficients of the Jones polynomial

Cited by 4 publications

References 18 publications

Nahm sums, stability and the colored Jones polynomial

Nahm sums, stability and the colored Jones polynomial

Alternating knots, planar graphs, and $$q$$ q -series

The zero stability for the one-row colored $\mathfrak{sl}_3$ Jones polynomial

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