The colored Jones polynomial and the A-polynomial of Knots☆
TLDR
In this article, the relationship between the colored Jones polynomial and the A-polynomial of a knot was studied and the AJ conjecture was established for a large class of two-bridge knots.About:
This article is published in Advances in Mathematics.The article was published on 2006-12-20 and is currently open access. It has received 104 citations till now. The article focuses on the topics: Bracket polynomial & HOMFLY polynomial.read more
Citations
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The colored Jones function is q-holonomic
TL;DR: In this paper, it was shown that the colored Jones function is a multisum of a q-proper hypergeometric function, and thus it is q-holonomic.
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Quantum Riemann Surfaces in Chern-Simons Theory
TL;DR: In this article, the authors construct operators that annihilate the partition functions (or wavefunctions) of three-dimensional Chern-Simons theory with gauge groups (SU(2), SL(2,\mathbb{R}) on knot complements.
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Generalized volume conjecture and the A-polynomials: The Neumann–Zagier potential function as a classical limit of the partition function
TL;DR: In this paper, the partition function Z γ (M u ) for the cusped hyperbolic 3-manifold M u is introduced and studied, which is based on an oriented ideal triangulation of M by assigning to each tetrahedron the quantum dilogarithm function, introduced by Faddeev in his studies of the modular double of the quantum group.
Posted Content
Quantum Riemann Surfaces in Chern-Simons Theory
TL;DR: The A-hat operator as mentioned in this paper is a quantization of the knot complement's classical A-polynomial A(l,m), and it is defined by decomposing three-manifolds into ideal tetrahedra, and invoking a new, more global understanding of gluing in TQFT to put them back together.
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The Jones slopes of a knot
TL;DR: In this paper, the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement were investigated, and two knot invariants, the Jones slopes (a finite set of rational numbers) and the Jones period (a natural number), were introduced.
References
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Journal ArticleDOI
A polynomial invariant for knots via von Neumann algebras
TL;DR: In this paper, it was shown that (6, n) and (c, ra) represent the same closed braid (up to link isotopy) if and only if they are equivalent for the equivalence relation generated by Markov moves of types 1 and 2 on the disjoint union of the braid groups.
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State Models and the Jones Polynomial
TL;DR: In this article, a state model for the Jones polynomial was constructed for the bracket polynomials, which is a normalization of a regular isotopy invariant of unoriented knots and links.
MonographDOI
Quantum invariants of knots and 3-manifolds
TL;DR: In this paper, a systematic treatment of topological quantum field theories (TQFT's) in 3D is presented, inspired by the discovery of the Jones polynomial of knots, the Witten-Chern-Simons field theory, and the theory of quantum groups.
Book
An Introduction to Knot Theory
TL;DR: A selection of topics which graduate students have found to be a successful introduction to the field, employing three distinct techniques: geometric topology manoeuvres, combinatorics, and algebraic topology.