[Paper Review] On the Colored Jones Polynomial and the Kashaev invariant

Vu Huynh, Thang T. Q. Lê|ArXiv.org|Mar 15, 2005

Advanced Combinatorial Mathematics11 references4 citations

TL;DR

This paper presents a determinant formula for the colored Jones polynomial using the quantum determinant of a matrix in the $q$-Weyl algebra, evaluated at the constant function 1. It proves the Kashaev invariant equals a special evaluation of this determinant and proposes a generalization to other simple Lie algebras via the Verma module of highest weight $-\delta$, suggesting a new $\mathfrak{g}$-volume conjecture linking quantum invariants to hyperbolic volume.

ABSTRACT

We express the colored Jones polynomial as the inverse of the quantum determinant of a matrix with entries in the $q$-Weyl algebra of $q$-operators, evaluated at the trivial function (plus simple substitutions). The Kashaev invariant is proved to be equal to another special evaluation of the determinant. We also discuss the similarity between our determinant formula of the Kashaev invariant and the determinant formula of the hyperbolic volume of knot complements, hoping it would lead to a proof of the volume conjecture.

Motivation & Objective

• To express the colored Jones polynomial as the inverse of a quantum determinant in the $q$-Weyl algebra.
• To prove that the Kashaev invariant arises as a special evaluation of this determinant formula.
• To generalize the Kashaev invariant to other simple Lie algebras using the Verma module of highest weight $-\delta$.
• To explore the structural similarity between the determinant formula of the Kashaev invariant and the Fuglede-Kadison determinant formula for hyperbolic volume.
• To suggest a potential path toward proving the volume conjecture using quantum determinant as an approximation of infinite-dimensional $L^2$-torsion.

Proposed method

• Use the quantum MacMahon Master theorem to derive a determinant formula for the colored Jones polynomial.
• Define a right-quantum matrix $S_\pm$ using $q$-operators $\hat{x}, \tau_x, \hat{u}, \tau_u$ acting on polynomial rings.
• Evaluate the quantum determinant of $I - q\rho'(\gamma)$, where $\rho'$ is a representation of the braid group.
• Apply the evaluation map $\mathcal{E}$ to the determinant, setting $u=1$ and $x=y=z$, to recover the colored Jones polynomial.
• Specialize the formula at $q = \exp(2\pi i/N)$ to obtain the Kashaev invariant.
• Define the $\mathfrak{g}$-Kashaev invariant as $J_K^{\prime}(V_{-\delta})|_{q=\exp(2\pi i/N)}$ for a simple Lie algebra $\mathfrak{g}$.

Experimental results

Research questions

• RQ1 Can the colored Jones polynomial be expressed as the inverse of a quantum determinant in the $q$-Weyl algebra?
• RQ2 Is the Kashaev invariant equal to a special evaluation of this quantum determinant formula?
• RQ3 What is the natural generalization of the Kashaev invariant for other simple Lie algebras?
• RQ4 How does the determinant formula for the Kashaev invariant relate to the Fuglede-Kadison determinant formula for hyperbolic volume?
• RQ5 Can the quantum determinant serve as an approximation to the $L^2$-torsion determinant, aiding a proof of the volume conjecture?

Key findings

• The colored Jones polynomial $J_K'(N)$ is equal to $v^{m-1-w(\beta)} \cdot \mathcal{E}_0(T)$, where $T = \widetilde{\det}_q(I - q\rho'(\gamma))^{-1}$, with $\mathcal{E}_0$ being a special evaluation at $q = \exp(2\pi i/N)$.
• The Kashaev invariant $\langle K\rangle_N$ is given by $v^{m-1-w(\beta)} \cdot \mathcal{E}_0(T)$ at $q = \exp(2\pi i/N)$, confirming the determinant formula for the invariant.
• The Kashaev invariant arises as a quantum determinant evaluation, establishing a direct link between quantum invariants and non-commutative determinants.
• The $\mathfrak{g}$-Kashaev invariant is defined as $J_K^{\prime}(V_{-\delta})|_{q=\exp(2\pi i/N)}$, where $V_{-\delta}$ is the Verma module of highest weight $-\delta$, the half-sum of positive roots.
• The paper suggests a $\mathfrak{g}$-volume conjecture: $\lim_{N\to\infty} \frac{|\langle K\rangle_N^\mathfrak{g}|}{N} = c_\mathfrak{g} \cdot \operatorname{Vol}(K)$, with $c_\mathfrak{g}$ depending only on $\mathfrak{g}$.
• The structural similarity between the quantum determinant formula for the Kashaev invariant and the Fuglede-Kadison determinant for hyperbolic volume suggests a potential approach to proving the volume conjecture via quantum determinant approximation.