A non-commutative formula for the colored Jones function

Garoufalidis, Stavros; Loebl, Martin

doi:10.1007/s00208-006-0018-6

Cited by 7 publications

(6 citation statements)

References 24 publications

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“…• In [3], this theorem is only stated explicitly for braid closures. A very similar (though inequivalent) state sum for the colored Jones function was recently obtained in Theorem 7, [4].…”

Section: Gauss Diagram Formulation Of Jones Function Of a Knotsupporting

confidence: 67%

“…A very similar (though inequivalent) state sum for the colored Jones function was recently obtained in Theorem 7, [4].…”

Section: Theorem 1 [3]mentioning

confidence: 78%

“…Put concisely, the colored Jones function depends on two parameters, q and µ, and one can make sense of it if at least one of the following holds: (i) q is a root of unity; (ii) µ ∈ N; (iii) q = 1 + h is considered as a formal parameter with expressions as formal power series in h and µ is fixed; (iv) q is considered as a formal parameter with expressions as formal power series in h and q µ is fixed. 4 The WRT and Ohtsuki invariants 4.1 The SU (2) and SO(3) 3-manifold invariants Suppose that q is a K th root of unity. The prescription of [21] defines for a link L with c components and framing f i = ±1 on the i th component,…”

Section: Gauss Diagram Formulation Of Jones Function Of a Knotmentioning

confidence: 99%

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On Habiro's Cyclotomic Expansions of the Ohtsuki Invariant

Lawrence

Ron

2006

J. Knot Theory Ramifications

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We give a self-contained treatment of Le and Habiro's approach to the Jones function of a knot and Habiro's cyclotomic form of the Ohtsuki invariant for manifolds obtained by surgery around a knot. On the way we reproduce a state sum formula of Garoufalidis and Le for the colored Jones function of a knot. As a corollary, we obtain bounds on the growth of coefficients in the Ohtsuki series for manifolds obtained by surgery around a knot, which support the slope conjecture of Jacoby and the first author.

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Section: Gauss Diagram Formulation Of Jones Function Of a Knotsupporting

confidence: 67%

“…A very similar (though inequivalent) state sum for the colored Jones function was recently obtained in Theorem 7, [4].…”

Section: Theorem 1 [3]mentioning

confidence: 78%

Section: Gauss Diagram Formulation Of Jones Function Of a Knotmentioning

confidence: 99%

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On Habiro's Cyclotomic Expansions of the Ohtsuki Invariant

Lawrence

Ron

2006

J. Knot Theory Ramifications

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“…More precisely, the flares, outside of the central clusters, contain only knots with equal signature. This insight relates with the Garoufalidis conjecture, building on the following theorem [16,18,19]. Theorem 5.1 (Garoufalidis '03).…”

Section: Ball Mapper On Knot Polynomial Datamentioning

confidence: 99%

Mapper-type algorithms for complex data and relations

Dlotko¹,

Gurnari²,

Sazdanovic³

2021

Preprint

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This work brings methods from topological data analysis to knot theory and develops new data analysis tools inspired by this application. We explore a vast collection of knot invariants and relations between then using Mapper and Ball Mapper algorithms. In particular, we develop versions of the Ball Mapper algorithm that incorporate symmetries and other relations within the data, and provide ways to compare data arising from different descriptors, such as knot invariants. Additionally, we extend the Mapper construction to the case where the range of the lens function is high dimensional rather than a 1-dimensional space, that also provides ways of visualizing functions between high-dimensional spaces. We illustrate the use of these techniques on knot theory data and draw attention to potential implications of our findings in knot theory.

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“…Notice that there is one minor difference between the above definition and the one of [GL06]; Namely, the weights in [GL06] are assumed to be positive whereas here θ is allowed to attain the zero value.…”

Section: The Case Of An Embedded Graphmentioning

confidence: 99%

The infinitesimal multiplicities and orientations of the blow-up set of the Seiberg–Witten equation with multiple spinors

Haydys¹

2019

Advances in Mathematics

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I construct multiplicies and orientations of tangent cones to any blow-up set Z for the Seiberg-Witten equation with multiple spinors. This is used to prove that Z determines a homology class, which is shown to be equal to the Poincaré dual of the first Chern class of the determinant line bundle. I also obtain a lower bound for the 1-dimensional Hausdorff measure of Z.The infinitesimal structure of the blow-up set for the Seiberg-Witten equation Definition 3. A closed nowhere dense set Z ⊂ M is called a blow-up set for the Seiberg-Witten equation with multiple spinors, if there is a solution (A, Ψ, 0) of (1) defined over M \ Z such that the following holds:(i) |Ψ| extends as a Hölder-continuous function to all of M and Z = |Ψ| −1 (0);As explained in Remark 34 below, (ii) holds automatically provided (A, Ψ, 0) is a limit of the Seiberg-Witten monopoles with τ k → 0, τ k = 0. The arguments used in this manuscript require |Ψ| to be continuous only; The Hölder continuity of |Ψ| is needed to ensure that the results of [Tau14] can be applied. In particular, a combination of [HW15, App. A] and [Tau14, Thm 1.3] yields that the Hausdorff dimension of Z is at most one. Notice also that (i) can be replaced by a weaker condition, for example [Tau14, (1.5)].I would like to stress that no extra assumptions on the Riemannian metric on M or the regularity of Z are required in Theorem 4. In particular, viewing Z as a subset of M only, the homology class of Z may be ill defined.An interpretation of Theorem 4 is that there are topological restrictions on blow-up sets for the Seiberg-Witten equation with a fixed Spin c -structure. For example, if L is non-trivial, then Z can not be empty. Although this follows immediately from Theorem 4, this statement can be proved directly by an elementary argument. Indeed, assume that for some non-trivial L there is a solution (A, Ψ, 0) of (1) such that Ψ vanishes nowhere, i.e., Z = ∅. The equation

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A non-commutative formula for the colored Jones function

Cited by 7 publications

References 24 publications

On Habiro's Cyclotomic Expansions of the Ohtsuki Invariant

On Habiro's Cyclotomic Expansions of the Ohtsuki Invariant

Mapper-type algorithms for complex data and relations

The infinitesimal multiplicities and orientations of the blow-up set of the Seiberg–Witten equation with multiple spinors

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