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

In this subsection, we review the difinition of the $A$-polynomial following [1]. Let $K$ be a knot, and let $M_{K}$ be its complement. The set of all $SL(2;\mathbb{C})$-representations $R(M_{K})$ is an affine algebraic variety. Then, we restrict our attention to

where $\mu$ is the meridian, and $\lambda$ is the preferred longitude of $K$. We can define the eigenvalue map $\xi:R_{U}\to\mathbb{C}^{2}$ by $\xi(\rho)=(l,\alpha)$, where

For an algebraic component $C$ of $R_{U}$, the Zariski closure $\overline{\xi(C)}$ of $\xi(C)$ is an algebraic subset of $\mathbb{C}^{2}$. If $\overline{\xi(C)}$ is a curve, there exists a defining polynomial. The $A$-polynomial of a knot $K$ is the product of all such defining polynomials.

On the other hand, we provided the geometric meaning of derivatives of a potential function of the colored Jones polynomial. Let us recall some facts on the potential function briefly. See [11, 16] for details.

The potential function of the colored Jones polynomial evaluated at the root of unity is obtained by approximating $R$-matrices with continuous functions. Let $n$ and $n^{\prime}$ be odd numbers. The summands of the $R$-matrix $\mathbb{C}^{n}\otimes\mathbb{C}^{n^{\prime}}\to\mathbb{C}^{n^{\prime}}\otimes\mathbb{C}^{n}$ of the colored Jones polynomial that is assigned to a crossing $c$ are as follows [11]:

Here, $m$ and $m^{\prime}$ are integers satisfying $n=2m+1$ and $n^{\prime}=2m^{\prime}+1$, $q$ is an indeterminate, and $(q)_{k}=(1-q)\cdots(1-q^{k})$. We also note that we assign $R_{c}^{+}$ to a positive crossing, and $R_{c}^{-}$ to a negative crossing. On the other hand, corresponding potential functions $\Phi_{c}^{\pm}$ are as follows:

Here, $\operatorname{Li_{2}}$ is the dilogarithm function, $\alpha$ corresponds to $\xi_{N}^{m}$, $\beta$ corresponds to $\xi_{N}^{m^{\prime}}$, and $w_{j_{i}}$, with $i=1,2,3,4$, corresponds to $\xi_{N}^{k_{j_{i}}}$, where $N$ is an integer, and $\xi_{N}$ is the $N$-th root of unity $\exp(2\pi\sqrt{-1}/N)$. These indices $k_{j_{i}}$ or parameters $w_{j_{i}}$ are labeled to regions of a link diagram. A potential function $\Phi(\boldsymbol{\alpha},w_{1},\ldots,w_{\nu})$ of the colored Jones polynomial for a $k$-component hyperbolic link $\mathcal{L}=\mathcal{L}_{1}\cup\cdots\cup\mathcal{L}_{k}$ evaluated at $\xi_{N}$ is the summation of $\Phi_{c}^{\operatorname{sgn}(c)}$. Here, $\boldsymbol{\alpha}=(\alpha_{1},\ldots,\alpha_{k})$ is a $k$-tuple of the parameters corresponding to the colors, and $\operatorname{sgn}(c)$ is a signature of the crossing $c$, and $c$ runs over all crossings. This potential function essentially coincides with the generalized potential function in [16]¹¹1In [16], Yoon defined the generalized potential function and established the relationship between the gluing equation. The author appreciates Yoon’s valuable comment at the 18th East Asian Conference on Geometric Topology.. We will review some facts on the potential function. From the saddle point of the potential function $\Phi(\boldsymbol{\alpha},w_{1},\ldots,w_{\nu})$, we can obtain a hyperbolic structure of the complement $M_{\mathcal{L}}$ of the link $\mathcal{L}$ that is not necessarily complete. This is because

coincides with the gluing equation of an ideal triangulation of $M_{\mathcal{L}}$. Here, we choose the saddle point $(\sigma_{1}(\boldsymbol{\alpha}),\ldots,\sigma_{\nu}(\boldsymbol{\alpha}))$ so that the imagaginary part $\operatorname{Im}\Phi$ of the potential function $\Phi$ satisfies

where $\operatorname{Vol}(\mathcal{L})$ is the hyperbolic volume of the link $\mathcal{L}$. Then, the following equality holds:

where $l_{j}(\boldsymbol{\alpha})$ is the dilation component of the action of the preferred longitude of the link component $\mathcal{L}_{j}$. In other words, if

has a solution, then $M_{\mathcal{L}}$ has a corresponding hyperbolic structure. Since each equation is equivalent to a polynomial equation, we can determine the condition of whether the system of the equations has a solution by elimination. In fact, the $A$-polynomial is conjectured to be obtained by the potential function of the colored Jones polynomial [6, 15].

For a knot $K$, its colored Jones polynomial $J_{K}(n)=J_{n}(K;q)$ has a nontrivial linear recurrence relation [4]

where $c_{j}$ is an polynomial with integer coefficients. For a discrete function $f:\mathbb{N}\to\mathbb{Q}(q)$, we define operators $Q$ and $E$ by

Then, we can restate the recurrence relation (3.1) as

This polynomial $\sum c_{j}(q,Q)E^{j}$ is an element in the non-commutative algebra

To define the $A_{q}$-polynomial, we have to localize the algebra $\mathcal{A}$. Let $\sigma$ be the automorphism of the field $\mathbb{Q}(q,Q)$ given by

The Ore algebra $\mathcal{A}_{\mathrm{loc}}=\mathbb{Q}(q,Q)[E,\sigma]$ is defined by

where the multiplication of monomials is given by $aE^{i}\cdot bE^{j}=a\sigma^{i}(b)E^{i+j}$. The $A_{q}$-polynomial $A_{q}(K)(E,Q)$ is a generator of the recursion ideal of $J_{K}(n)$

with the following properties:

• •

  $A_{q}(K)$ has the smallest $E$-degree and lies in $\mathcal{A}$.

• •

  $A_{q}(K)$ is of the form $A_{q}(K)=\sum_{k}a_{k}E^{k}$, where $a_{k}\in\mathbb{Z}[q,Q]$ are coprime.

Garoufalidis proposed the following conjecture on the $A_{q}$-polynomial:

These operators generate the algebra

with relations

where $Q_{0}=Q,\ E_{0}=E,\ Q_{i}=\tilde{Q}_{i}$, and $E_{i}=\tilde{E}_{i}$ for $i\in\{1,\ldots,\nu\}$. Hereafter, we put $\boldsymbol{k}=(k_{1},\ldots,k_{\nu})$.

We especially deal with a proper $q$-hypergeometric function.

Proper $q$-hypergeometric functions satisfy the following theorem:

We put

where $\tilde{E}^{\boldsymbol{j}}=\tilde{E}_{1}^{j_{1}}\cdots\tilde{E}_{\nu}^{j_{\nu}}$. The expansion of $P$ around $\tilde{E}_{i}=1$, with $i=1,\ldots,\nu$, is

where $P_{0}(E,Q)=P(E,Q,1,\ldots,1)$, and $R_{i}$ is a polynomial in $\mathbb{Q}[q,Q]\langle E,\tilde{E}_{\boldsymbol{k}}\rangle$. Here, $\tilde{E}_{\boldsymbol{k}}=(\tilde{E}_{1},\ldots,\tilde{E}_{\nu})$. Putting $G_{i}=R_{i}F$, we have

Summing up this equality, we verify that $G(n):=\sum_{\boldsymbol{k}}F(n,\boldsymbol{k})$ satisfies

where $K_{i}$ and $k^{0}_{i}$ are fixed parameters. Since $F$ is a proper $q$-hypergeometric function, $G_{i}$ is a sum of proper $q$-hypergeometric functions. That means $P_{0}(E,Q)G(n)$ is a sum of multisums of proper $q$-hypergeometric functions with one variable less. Repeating this process, we obtain a polynomial $P_{1}(E,Q)$ such that

The annihilating polynomial (3.2) of the summand $F(n,\boldsymbol{k})$ is an element in $\mathrm{Ann}(F)\cap\mathbb{Q}[q,Q]\langle E,\tilde{E}_{\boldsymbol{k}}\rangle$. Moreover, defining the polynomials $R,S,R_{i}$, and $S_{i}\in\mathbb{Z}[q,Q_{1},\ldots,Q_{\nu}]$ by

it is known that the annihilating ideal of $F$ is generated by

Therefore, we would be able to obtain an annihilating poynomial $P(E,Q,\tilde{E}_{1},\ldots,\tilde{E}_{\nu})$ of $F(n,\boldsymbol{k})$ by eliminating $\tilde{Q}_{1},\ldots,\tilde{Q}_{\nu}$ from

From the observation above, we would also be able to obtain $\varepsilon P_{0}(E,Q)$ by eliminating $\tilde{Q}_{1},\ldots,\tilde{Q}_{\nu}$ from