The colored Jones function is q-holonomic

By Stavros Garoufalidis, Thang T. Q. Lê. 2003

244 citations recorded by the source corpus.

A function of several variables is called holonomic if, roughly speaking, it is determined from finitely many of its values via finitely many linear recursion relations with polynomial coefficients. Zeilberger was the first to notice that the abstract notion of holonomicity can be applied to verify, in a systematic and computerized way, combinatorial identities among special functions. Using a general state sum definition of the colored Jones function of a link in 3-space, we prove from first principles that the colored Jones function is a multisum of a q-proper-hypergeometric function, and thus it is q-holonomic. We demonstrate our results by computer calculations.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/b75159f2477183ddebc26f4f558b530305fd9a22

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In Applied Mathematics

In Computational Theory and Mathematics

In Discrete Mathematics and Combinatorics

Citations in the corpus

Generalized volume conjecture and the A-polynomials: The Neumann–Zagier potential function as a classical limit of the partition function (2006, background): “…ρ(λ) = ℓ * 0 1/ℓ Another geometrical aspect of the N-colored Jones polynomial J K (N; q) as a relationship with the A-polynomial is proposed as " AJ conjecture " [17]; recursion relation of the colored Jones polynomial with respect to N is conjectured to be related to the A-polynomial A K (ℓ, m).”
Gauge Theories Labelled by Three-Manifolds (2011, background): “In Chern-Simons theory, (6.25) is known as the generalized / quantum volume conjecture [8] (sometimes also called the AJ-conjecture [56, 57] in the math literature), whereas in N = 2 gauge theory it expresses Ward identities for line operators.”
DIFFERENCE EQUATION OF THE COLORED JONES POLYNOMIAL FOR TORUS KNOT (2004, background): “Motivated by “volume conjecture” [12, 18] saying that a hyperbolic volume of the knot complement dominates an asymptotic behavior of the colored Jones polynomial JK (N) at q = e2πi/N, it receives much interests toward a geometrical and topological interpretation of the quantum invariants.”
The colored Jones polynomial and the A-polynomial of Knots☆ (2004, methodology): “But in [12] the L-degree is much larger.”
Quantum Riemann Surfaces in Chern-Simons Theory (2011, methodology): “99) to the operator ÂV,31 appearing in the mathematical recursion relation [5] for the colored Jones polynomials of the trefoil.”
On the AJ conjecture for knots (2011, result): “s of the quantum torus T:= RhL±1,M±1i/(LM−t2ML), which is not commutative, but almost commutative. Let AK := {P∈T|PJK = 0}. It is a left ideal of T, called the recurrence ideal of K. It was proved in [GL] that for every knot K, the recurrence ideal AK is non-zero. Partial results were obtained earli”
On the characteristic and deformation varieties of a knot (2003, background): “Definition 1.1 We say that f is q–holonomic iff If 6= 0.”
3-Manifolds and 3d Indices (2011, background): “5) that the same “quantum Lagrangian” operators that annihilate compact [51, 52] and”