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The colored Jones function is q-holonomic

Stavros Garoufalidis, Thang TQ Le

TL;DR

<3-5 sentence high-level summary> The paper proves that the colored Jones function for knots and links is q-holonomic, meaning it is determined by a finite set of linear q-difference recurrences with polynomial coefficients. It achieves this by expressing the invariant as a state-sum of q-proper hypergeometric terms and showing that the building blocks (R-matrices, braid actions) and their assembly preserve q-holonomicity; it also provides an explicit multisum representation and extends the result to the cyclotomic function and to simple Lie algebras (except possibly G2). The authors develop effective complexity bounds and computer-verified recursions for several knots, illustrating the practical computability of recurrences. This work connects quantum knot invariants with the algebraic theory of holonomic functions and suggests a broad holonomicity principle underlying quantum algebraic structures and topological quantum field theory.

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.

The colored Jones function is q-holonomic

TL;DR

<3-5 sentence high-level summary> The paper proves that the colored Jones function for knots and links is q-holonomic, meaning it is determined by a finite set of linear q-difference recurrences with polynomial coefficients. It achieves this by expressing the invariant as a state-sum of q-proper hypergeometric terms and showing that the building blocks (R-matrices, braid actions) and their assembly preserve q-holonomicity; it also provides an explicit multisum representation and extends the result to the cyclotomic function and to simple Lie algebras (except possibly G2). The authors develop effective complexity bounds and computer-verified recursions for several knots, illustrating the practical computability of recurrences. This work connects quantum knot invariants with the algebraic theory of holonomic functions and suggests a broad holonomicity principle underlying quantum algebraic structures and topological quantum field theory.

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.

Paper Structure

This paper contains 46 sections, 23 theorems, 108 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Zeilberger meets Jones
  3. Synonymous notions to holonomicity
  4. Plan of the proof
  5. Acknowledgements
  6. $q$--holonomic and $q$--hypergeometric functions
  7. $q$--holonomicity in many variables
  8. Assembling $q$--holonomic functions
  9. Examples of $q$--holonomic functions
  10. $q$--hypergeometric functions
  11. The colored Jones function for $\mathfrak{sl}_2$
  12. Proof of Theorem \ref{['thm.1']} for links
  13. A multisum formula for the colored Jones function of a knot
  14. The cyclotomic function of a knot is $q$--holonomic
  15. Complexity
  16. ...and 31 more sections

Key Result

Theorem 1

The colored Jones function of every knot is $q$--holonomic.

Figures (3)

  • Figure 1: A word $w$, the corresponding braid $\beta$, its long closure ${\mathcal{K}}'$, and a coloring of ${\mathcal{K}}'$
  • Figure 2: The twist knot $K_p$, for integers $p$. For $p=-1$, it is the Figure 8, for $p=0$ it is the unknot, for $p=1$ it is the left trefoil and for $p=2$ it is the Stevedore's ribbon knot.
  • Figure 3: $(\mathcal{B}_+)_{{\mathbf n}_1,{\mathbf n}_2}^{{\mathbf m}_1,{\mathbf m}_2}$ and $(\mathcal{B}_-)_{{\mathbf n}_1,{\mathbf n}_2}^{{\mathbf m}_1,{\mathbf m}_2}$

Theorems & Definitions (48)

  • Theorem 1
  • Theorem 2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 3
  • ...and 38 more