## The colored Jones function is q-holonomic (2005)

Citations: | 23 - 6 self |

### BibTeX

@MISC{Garoufalidis05thecolored,

author = {Stavros Garoufalidis and Thang T Q Lê},

title = {The colored Jones function is q-holonomic},

year = {2005}

}

### OpenURL

### Abstract

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.

### Citations

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Citation Context ...ger’s work. Zeilberger noticed that the abstract notion of holonomicity can be applied to verify, in a systematic and computerized way, combinatorial identities among special functions, [35] and also =-=[33, 28]-=-. A starting point for Zeilberger, the so-called operator approach, is to replace functions by the recursion relations that they satisfy. This idea leads in a natural way to noncommutative algebras of... |

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Citation Context ...tly to Zeilberger’s work. Zeilberger noticed that the abstract notion of holonomicity can be applied to verify, in a systematic and computerized way, combinatorial identities among special functions, =-=[35]-=- and also [33, 28]. A starting point for Zeilberger, the so-called operator approach, is to replace functions by the recursion relations that they satisfy. This idea leads in a natural way to noncommu... |

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Citation Context ...) for colored links. Our proof will use a state-sum definition of the colored Jones function, coming from a representation of the quantum group Uq(sl2), as was discovered by Reshetikhin and Turaev in =-=[29, 32]-=-. Suppose L is a framed, oriented link of p components. Then the colored Jones function JL: N p → Z[q ±1/4 ] = Z[v ±1/2 ] can be defined using the representations of braid groups coming from the quant... |

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Citation Context ...and implementations, see [26, 27] and [30], which we will use below. Alternative algorithms of noncommutative elimination, using noncommutative Gröbner basis, have been developed by Chyzak and Salvy, =-=[8]-=-. In order for have Gröbner basis, one needs to use the following localization of the q–Weyl algebra and Gröbner basis [8]. Br = Q(q,Q1,... ,Qr)〈E1,... ,Er〉 . (Relq) In case r = 1, B1 is a principal i... |

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Citation Context ...,k), and prove Theorem 5. Wilf–Zeilberger programmed the above proof, see [28]. As time passes the algorithms get faster and more refined. For the state-of-the-art algorithms and implementations, see =-=[26, 27]-=- and [30], which we will use below. Alternative algorithms of noncommutative elimination, using noncommutative Gröbner basis, have been developed by Chyzak and Salvy, [8]. In order for have Gröbner ba... |

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Citation Context ...on is q–holonomic 1255 of view of algebra (differential Galois theory), algebraic geometry, and category theory. For an excellent introduction on holonomic functions and their properties, see [5] and =-=[7]-=-. Our approach to the colored Jones function owes greatly to Zeilberger’s work. Zeilberger noticed that the abstract notion of holonomicity can be applied to verify, in a systematic and computerized w... |

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Citation Context ...∈ U homogeneous let us define [x,y]q := xy − v (|x|,|y|) yx. Note that, in general, [y,x]q is not proportional to [x,y]q. An important property of the PBW basis is the following commutation rule, see =-=[18]-=-. If i < j then [fi,fj]q belongs to U − [j − 1,i + 1] (which is 0 if j = i + 1). Geometry & Topology, Volume 9 (2005)1280 Stavros Garoufalidis and Thang T Q Lê It follows that U − [j,i] is an algebra... |

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Citation Context ...ch as linear q–difference equations, or q–holonomic functions, or D–modules, or maximally overdetermined systems of linear PDEs which is more common in the area of algebraic analysis, see for example =-=[24]-=-. The geometric notion of D– modules gives rise to geometric invariants of knots, such as the characteristic variety introduced by the first author in [11]. The characteristic variety is determined by... |

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Citation Context ...n 2 f p 1 | l,m,n,p ∈ N}, {fl 4 ′fm 3 ′ fn 2 ′fp 1 ′ | l,m,n,p ∈ N}, (f4,f3,f2,f1) = (Fβ,FβFα − v 2 FαFβ, FβF 2 α [2] − vFαFβFα + v2F 2 αFβ [2] (f4 ′,f3 ′,f2 ′,f1 ′) = (Fα, v2 FβF 2 α It follows from =-=[23]-=- that: [2] − vFαFβFα + F 2 αFβ [2] ,Fα) ,FαFβ − v 2 FβFα,Fβ). Geometry & Topology, Volume 9 (2005)The colored Jones function is q–holonomic 1291 Lemma A.2 The structure constants of U − , in the basi... |

37 | On the quantum sl2 invariants of knots and integral homology spheres, from
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Citation Context ... the figure eight (where it reduces to a double sum). D Bar-Natan has kindly provided us with a computerized version of Proposition 3.7, [1]. 4 The cyclotomic function of a knot is q–holonomic Habiro =-=[15]-=- proved that the colored Jones polynomial (of sl2) can be rearranged in the following convenient form, known as the cyclotomic expansion of the colored Jones polynomial: For every 0–framed knot K, the... |

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Citation Context ... Using induction one can easily prove that P ′′ n∑ (n) = R(n,k)V (k), k=1 where R(k,n) is given by R(n,k) = (−1) n−k [ {2k} {2n − 1}![2n] 2n n − k We learned this formula from Habiro [15] and Masbaum =-=[25]-=-. Since CK(n) = ∑ R(n,k)JK(k) k and R(n,k) is q–proper hypergeometric and thus q–holonomic in both variables n and k, it follows that CK is q–holonomic. ] . 5 Complexity In this section we show that T... |

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(Show Context)
Citation Context ...e colored Jones polynomial JL is exactly Jβ . Hence Theorem 1 follows. Remark 3.4 In general, JK(n) contains the fractional power q 1/4 . If K has framing 0, then JK ′(n) := JK(n)/[n] ∈ Z[q ±1 ]. See =-=[20]-=-. Geometry & Topology, Volume 9 (2005)The colored Jones function is q–holonomic 1265 Remark 3.5 There is a variant of the colored Jones function JL ′ of a colored link L ′ where one of the components... |

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Introduction to Quantum Groups, volume 110
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Citation Context ...im(g) − ℓ)/2, the number of positive roots of g. 7.1.1 The quantum group U The quantum group U = Uq(g) associated to g is a Hopf algebra defined over Q(v), where v is the usual quantum parameter (see =-=[17, 22]-=-). Here our v is the same as v of Lusztig [22] and is equal to q of Jantzen [17], while our q is v 2 . The standard generators of U are Eα,Fα,Kα for α ∈ {α1,...,αℓ}. For a full set of relations, as we... |

21 | qMultiSum—a package for proving q-hypergeometric multiple summation identities
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Citation Context ...lation for F ? Let us write A = ∑ σi,j(Q)E i E j i=1 (i,j)∈S where S is a finite set, j = (j1,... ,jr), Ej = E j1 1 ...Ejr r , and σi,j(Q) are polynomial functions in Q with coefficients in Q(q); see =-=[30]-=-. The condition Geometry & Topology, Volume 9 (2005)1270 Stavros Garoufalidis and Thang T Q Lê AF = 0 is equivalent to the equation (AF)/F = 0. Since F is q–proper hypergeometric, the latter equation... |

15 |
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Citation Context ...efoil was established). Discrete functions that satisfy a nontrivial difference recursion relation are known by another name: they are q–holonomic. Holonomic functions were introduced by IN Bernstein =-=[2, 3]-=- and M Saito. The latter coined the term holonomic, that is a function which is entirely determined by the law of its differential equation, together with finitely many initial conditions. Bernstein u... |

14 | Difference equation of the colored Jones polynomial for torus knot
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Citation Context ...somorphic to the sl2(C)–character variety of a knot, viewed from the boundary torus. This, so-called AJ Conjecture, formulated by the first author is known to hold for all torus knots (due to Hikami, =-=[19]-=-), and infinitely many 2–bridge knots (due to the second-author, [21]). Thus, there is nontrivial geometry encoded in the linear recursion relations of the colored Jones function of a knot. Geometry &... |

13 |
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Citation Context ...lso [28]. This opened an entirely new view on combinatorial identities. Sabbah extended Bernstein’s approach to holonomic functions and defined the notion of a q–holonomic function, see [31] and also =-=[6]-=-. 2.1 q–holonomicity in many variables We briefly review here the definition of q–holonomicity. First of all, we need an r–dimensional version of the q–Weyl algebra. Consider the operators Ei and Qj f... |

13 | Difference and differential equations for the colored Jones function, preprint - Garoufalidis, Le - 2003 |

10 |
Doron Zeilberger, A
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Citation Context ...ger’s work. Zeilberger noticed that the abstract notion of holonomicity can be applied to verify, in a systematic and computerized way, combinatorial identities among special functions, [35] and also =-=[33, 28]-=-. A starting point for Zeilberger, the so-called operator approach, is to replace functions by the recursion relations that they satisfy. This idea leads in a natural way to noncommutative algebras of... |

9 |
Lectures on Quantum Groups, volume 6
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Citation Context ...im(g) − ℓ)/2, the number of positive roots of g. 7.1.1 The quantum group U The quantum group U = Uq(g) associated to g is a Hopf algebra defined over Q(v), where v is the usual quantum parameter (see =-=[17, 22]-=-). Here our v is the same as v of Lusztig [22] and is equal to q of Jantzen [17], while our q is v 2 . The standard generators of U are Eα,Fα,Kα for α ∈ {α1,...,αℓ}. For a full set of relations, as we... |

9 | The colored Jones polynomial and the A-polynomial of knots
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Citation Context ...boundary torus. This, so-called AJ Conjecture, formulated by the first author is known to hold for all torus knots (due to Hikami, [19]), and infinitely many 2–bridge knots (due to the second-author, =-=[21]-=-). Thus, there is nontrivial geometry encoded in the linear recursion relations of the colored Jones function of a knot. Geometry & Topology, Volume 9 (2005)The colored Jones function is q–holonomic ... |

8 | Noncommutative trigonometry and the A-polynomial of the trefoil knot
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(Show Context)
Citation Context ... = 1. In this paper we prove that the colored Jones function of any knot satisfies a linear recursion relation, similar to the above one. For a few knots this was obtained by Gelca and his colleagues =-=[13, 14]-=-. (In [13] a more complicated 5–term recursion formula for the trefoil was established). Discrete functions that satisfy a nontrivial difference recursion relation are known by another name: they are ... |

8 | The noncommutative A-ideal of a (2, 2p + 1)-torus knot determines its Jones polynomial, J. Knot Theory and Ramifications
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Citation Context ... = 1. In this paper we prove that the colored Jones function of any knot satisfies a linear recursion relation, similar to the above one. For a few knots this was obtained by Gelca and his colleagues =-=[13, 14]-=-. (In [13] a more complicated 5–term recursion formula for the trefoil was established). Discrete functions that satisfy a nontrivial difference recursion relation are known by another name: they are ... |

6 | The Quantum MacMahon Master Theorem, arXive:math.QA/0303319 - Garoufalidis, Lê, et al. |

5 |
Modules over a ring of differential operators. An investigation of the fundamental solutions of equations with constant coefficients, Funkcional. Anal. i Priloˇzen
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Citation Context ...efoil was established). Discrete functions that satisfy a nontrivial difference recursion relation are known by another name: they are q–holonomic. Holonomic functions were introduced by IN Bernstein =-=[2, 3]-=- and M Saito. The latter coined the term holonomic, that is a function which is entirely determined by the law of its differential equation, together with finitely many initial conditions. Bernstein u... |

5 |
On the characteristic and deformation varieties of a knot, from
- Garoufalidis
(Show Context)
Citation Context ... area of algebraic analysis, see for example [24]. The geometric notion of D– modules gives rise to geometric invariants of knots, such as the characteristic variety introduced by the first author in =-=[11]-=-. The characteristic variety is determined by the colored Jones function of a knot and is conjectured to be isomorphic to the sl2(C)–character variety of a knot, viewed from the boundary torus. This, ... |

5 |
A two-line algorithm for proving q-hypergeometric identities
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Citation Context ... question: if a q–holonomic function satisfies a nontrivial recursion relation, it follows that it is uniquely determined by a finite number of initial conditions. How many? This was answered by Yen, =-=[34]-=-. If G is a discrete function which satisfies a recursion relation of order J ⋆ , consider its principal symbol σ(q,Q), that is the coefficient of the leading E–term. The principal symbol lies in the ... |

3 |
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Citation Context ...[35, 33] and also [28]. This opened an entirely new view on combinatorial identities. Sabbah extended Bernstein’s approach to holonomic functions and defined the notion of a q–holonomic function, see =-=[31]-=- and also [6]. 2.1 q–holonomicity in many variables We briefly review here the definition of q–holonomicity. First of all, we need an r–dimensional version of the q–Weyl algebra. Consider the operator... |

2 |
Mathematica program, part of “KnotAtlas
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Citation Context ...t hand-trefoil (where the multisum reduces to a single sum) and the figure eight (where it reduces to a double sum). D Bar-Natan has kindly provided us with a computerized version of Proposition 3.7, =-=[1]-=-. 4 The cyclotomic function of a knot is q–holonomic Habiro [15] proved that the colored Jones polynomial (of sl2) can be rearranged in the following convenient form, known as the cyclotomic expansion... |

2 |
A Haefliger
- Borel, Grivel, et al.
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(Show Context)
Citation Context ...s function is q–holonomic 1255 of view of algebra (differential Galois theory), algebraic geometry, and category theory. For an excellent introduction on holonomic functions and their properties, see =-=[5]-=- and [7]. Our approach to the colored Jones function owes greatly to Zeilberger’s work. Zeilberger noticed that the abstract notion of holonomicity can be applied to verify, in a systematic and comput... |

2 |
The C–polynomial of a knot, preprint
- Garoufalidis, Sun
- 2005
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Citation Context ...the number of twists) for the nth colored Jones polynomial of twist knots, for every fixed n. For computations of recursion relations of the cyclotomic function of twist knots, we refer the reader to =-=[12]-=-. 6.2 Recursion relations for the colored Jones function of the figure 8 knot The Mathematica package qMultiSum.m can compute recursion relations for q–multisums. Using this, we can compute equally ea... |

1 |
of differential operators, volume 21 of North-Holland Mathematical
- Björk, Rings
- 1979
(Show Context)
Citation Context ...ocks together to give the colored Jones function preserve q–holonomicity. IN Bernstein defined the notion of holonomic functions f : R r −→ C, [2, 3]. For an excellent and complete account, see Bjork =-=[4]-=-. Zeilberger’s brilliant idea was to link the abstract notion of holonomicity to the concrete problem of algorithmically proving combinatorial identities among hypergeometric functions, see [35, 33] a... |

1 |
A Riese, Mathematica software, available at: http://www.risc.uni-linz.ac.at/research/combinat/risc/software/qZeil
- Paule
(Show Context)
Citation Context ...,k), and prove Theorem 5. Wilf–Zeilberger programmed the above proof, see [28]. As time passes the algorithms get faster and more refined. For the state-of-the-art algorithms and implementations, see =-=[26, 27]-=- and [30], which we will use below. Alternative algorithms of noncommutative elimination, using noncommutative Gröbner basis, have been developed by Chyzak and Salvy, [8]. In order for have Gröbner ba... |

1 | Reshetikhin, VG Turaev, Ribbon graphs and their invariants derived from quantum groups - Yu - 1990 |