Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Potential Function of the Colored Jones Polynomial and the AJ conjecture

Shun Sawabe

TL;DR

The paper investigates how the potential function associated to the colored Jones polynomial encodes information about the $A$-polynomial and the AJ conjecture. It defines a polynomial $a_F(l,\alpha)$ from saddle-point equations of summands and relates it to nonabelian $A$-polynomials, and shows that the $A_q$-polynomial recovers $A$ via elimination with creative telescoping. A key result is that for solvable summands, $a_F(l,\alpha)$ divides $\varepsilon P_F^0(l,\alpha^2)$, establishing a link between the potential-function approach and the AJ conjecture. The framework provides a unified view connecting quantum invariants to hyperbolic geometry, with explicit discussion and an illustrative figure-eight knot example.

Abstract

The $A$-polynomial is conjectured to be obtained from the potential function of the colored Jones polynomial by elimination. The AJ conjecture also implies the relationship between the $A$-polynomial and the colored Jones polynomial. In this paper, we connect these conjectures from the perspective of the parametrized potential function.

On the Potential Function of the Colored Jones Polynomial and the AJ conjecture

TL;DR

The paper investigates how the potential function associated to the colored Jones polynomial encodes information about the -polynomial and the AJ conjecture. It defines a polynomial from saddle-point equations of summands and relates it to nonabelian -polynomials, and shows that the -polynomial recovers via elimination with creative telescoping. A key result is that for solvable summands, divides , establishing a link between the potential-function approach and the AJ conjecture. The framework provides a unified view connecting quantum invariants to hyperbolic geometry, with explicit discussion and an illustrative figure-eight knot example.

Abstract

The -polynomial is conjectured to be obtained from the potential function of the colored Jones polynomial by elimination. The AJ conjecture also implies the relationship between the -polynomial and the colored Jones polynomial. In this paper, we connect these conjectures from the perspective of the parametrized potential function.
Paper Structure (15 sections, 7 theorems, 125 equations, 2 figures)

This paper contains 15 sections, 7 theorems, 125 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Potential Functions and $A$-polynomials
  3. Colored Jones polynomials
  4. Potential functions
  5. $A$-polynomial
  6. $A_q$-polynomial and AJ conjecture
  7. $A_q$-polynomial
  8. Computation of $A_q$-polynomial
  9. $A_q$-polynomial and eliminations
  10. Polynomial obtained from a potential function
  11. Comparison with the saddle point equation
  12. Example calculation for the figure-eight knot
  13. Potential function and the $A$-polynomial
  14. Annihilating polynomials of $J_n(4_1,q)$
  15. Comparison of the derivatives and the $q$-differences

Key Result

Proposition 1.7

The system of equations coincides with under the correspondences $w_j = Q_j$, $\alpha = Q_{(0)}$, and $l = E$.

Figures (2)

  • Figure 2.1: Let $j_i$ be an index assigned to an edge $E_i$, $k_l$ be an index assigned to the left region $R_l$ of the edge $E_i$, and$k_r$ be an index assigned to the right region $R_r$ of the edge $E_i$. These indices satisfy $j_i = k_l -k_r$.
  • Figure 2.2: The indices labeled to the regions around a crossing.

Theorems & Definitions (24)

  • Conjecture 1.1: Volume Conjecture MM
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Conjecture 1.6: the AJ conjecture Ga
  • Proposition 1.7
  • Theorem 1.8
  • Definition 2.1
  • Proposition 2.2
  • Remark 2.3
  • ...and 14 more