Volume Conjecture and WKB Asymptotics

Abstract

We consider $q$-difference equations for colored Jones polynomials. These polynomials are invariants for the knots and their asymptotics plays an important role in the famous Volume Conjecture (VC) for the complement of the knot to the $3d$ sphere. We study WKB asymptotic behavior of the $n$th colored Jones polynomial at the point $\exp\{2\pi i/N\}$ when $n$ and $N$ tends to infinity and limit of $n/N$ belongs to $[0,1]$. We state a Theorem on asymptotic expansion of general solutions of the $q$-difference equations. For the partial solutions, corresponding to the colored Jones polynomials, using some heuristic and numeric consideration, we suggest a conjecture on their WKB asymptotics. For the special knots under consideration, this conjecture is in accordance with the VC.

APPENDIX: PROCEDURE OF THE BASIS EXPANSION AND PROOF OF THEOREMS 1 AND 2

At first, we consider the procedure for finding the diagonalizing transformations $V_{n}(q):V_{n+1}^{-1}\mathcal{A}_{n}V_{n}\rightarrow D_{n}$, where $D_{n}$ is the diagonal matrix defined in (18), for the homogeneous problem (14) in the separation zone.

1.1 Expansion of the Diagonalizing Transformation $V_{n}$

We change variables in the matrix $\mathcal{A}_{n}(q^{n},q)$ according to (3),

$$(n,q)\to(N,t),\quad\text{ where}\quad n=Nt,\quad q=e^{2\pi i/N},\quad N=1,2,\ldots,\quad t\in(0,1],$$

and denote $\mathbf{A}(N,t):=\mathcal{A}_{n}(q^{n},q)|_{q=e^{2\pi i/N},\,n=Nt}$. We expand $\mathbf{A}(N,t)$ in the small parameter $(1/N)$:

$$\mathbf{A}(N,t)=\mathbf{A}_{0}(t)+\frac{1}{N}\,\mathbf{A}_{1}(t)+\frac{1}{N^{2}}\,\mathbf{A}_{2}(t)+\ldots,$$

(67)

where the leading term of the expansion is

$$\mathbf{A}_{0}(t)=\mathcal{A}_{n}(e^{2\pi it},1).$$

(68)

Our goal is to find the expansion in $(1/N)$ of the matrix $\mathbf{V}(N,t):=V_{n}(q^{n},q)|_{q=e^{2\pi i/N},\,n=Nt}$, i.e.,

$$\mathbf{V}(N,t)=\mathbf{V}_{0}(t)+\frac{1}{N}\mathbf{V}_{1}(t)+\frac{1}{N^{2}}\mathbf{V}_{2}(t)+\ldots.$$

(69)

To do it we introduce an additional parameter $\tau$ and a power series $\widehat{V}(N,t,\tau)$ in $\tau$ as follows

$$\widehat{V}(N,t,\tau):=\mathbf{V}_{0}(t)+\tau\,\widetilde{V}_{1}(N,t)+\tau^{2}\,\widetilde{V}_{2}(N,t)+\ldots,$$

(70)

solving (in the formal sense) the problem of transforming the matrix $\mathbf{A}$ into a diagonal one:

$$\widehat{V}(\cdot,t+\tau,\tau)^{-1}\mathbf{A}(\cdot,t)\,\widehat{V}(\cdot,t,\tau)\equiv\textrm{diag}\left[\widehat{V}(\cdot,t+\tau,\tau)^{-1}\mathbf{A}\,\widehat{V}\right].$$

The coefficients of the formal series $\widehat{V}(\cdot,t+\tau,\cdot)$ are calculated by re-expansion

$$\widetilde{V}_{j}(N,t+\tau)\,=\,\sum\limits_{i=0}^{\infty}\tau^{i}\frac{d^{i}}{i!\,dt^{i}}\widetilde{V}_{j}(N,t).$$

(71)

We seek the power series (70) using successive approximations, which give coefficients for (70). As the initial approximation $\widehat{V}^{(0)}$ we take the matrix $\mathbf{V}_{0}(t)$, $\widehat{V}^{(0)}:=\mathbf{V}_{0}$, see (22). Equality (68) implies that $[\mathbf{V}_{0}(t)]^{-1}\,\mathbf{A}_{0}(t)\mathbf{V}_{0}(t)=\textrm{diag}\left\{\lambda_{1}(z),\ldots,\lambda_{d}(z)\right\}|_{z=\exp\{2\pi it\}}$. We seek the subsequent approximations² for (70) by the rule

$$\widehat{V}^{(k+1)}(N,t,\tau)=\widehat{V}^{(k)}(I+X^{(k)}),\quad k=0,1,2,\ldots,$$

where

$$X_{ii}^{(k)}:=0,\quad X_{ij}^{(k)}:=\dfrac{\left(\left[\widehat{V}^{(k)}(\cdot,t+\tau,\cdot)\right]^{-1}\mathbf{A}(\cdot,t)\widehat{V}^{(k)}(\cdot,t,\cdot)\right)}{\lambda_{i}-\lambda_{j}},\quad i\neq j.$$

(72)

Furthermore, the series $\widehat{V}^{(l)}$ and $\widehat{V}^{(l+1)}$ coincide up to $o(\tau^{(l)})$, see [24]. This circumstance allows us to significantly simplify (72), used in [24], to calculate $X_{ij}^{(k)}$ with the required accuracy. Denote by $\textrm{coeff}(f(\tau),\tau^{k})$ the $k$th coefficient of the expansion of a function $f(\tau)$ in the power $\tau$.

Theorem 3. Define $S^{(k)}(N,t,\tau)$ , $L^{(k)}(N,t,\tau)$ and $M^{(k)}(N,t,\tau)$ by the rule

$$S^{(k)}:=\left[\widehat{V}^{(k)}(\cdot,t+\tau,\tau)\right]^{-1}\mathbf{A}(\cdot,t)\widehat{V}^{(k)}(\cdot,t,\tau)=:L^{(k)}+O(\tau^{k})=:L^{(k)}+\tau^{k}M^{(k)}+o(\tau^{k}).$$

(73)

Then

$$M^{(k)}=\mathbf{V}_{0}^{-1}(t)\textrm{coeff}\left(\mathbf{A}(\cdot,t)\widehat{V}^{(k)}(\cdot,t,\tau)-\widehat{V}^{(k)}(\cdot,t+\tau,\tau)\,L^{(k)},\,\tau^{k}\right),$$

and, denoting

$$\hat{X}_{ij}^{(k)}:=\frac{M^{(k)}_{ij}}{\lambda_{i}-\lambda_{j}},\quad i\neq j,\quad\hat{X}_{ii}^{(k)}:=-\sum_{j\neq i}\hat{X}_{ij}^{(k)},$$

we have

$$L^{(k+1)}=L^{(k)}+\textrm{diag}[M^{(k)}];$$

(74)

$$\widehat{V}^{(k+1)}=\widehat{V}^{(k)}+\tau^{k}\,\mathbf{V}_{0}^{-1}\hat{X}^{(k)}.$$

(75)

Remark. Compare (74) and (72). Now we do not need to invert $\widehat{V}^{(k)}$ on each step $k$.

Proof. Definition (73) implies

$$\mathbf{A}(\cdot,t)\widehat{V}^{(k)}(\cdot,t,\cdot)-\widehat{V}^{(k)}(\cdot,t+\tau,\cdot)\,L^{(k)}=\tau^{k}\widehat{V}^{(k)}(\cdot,t+\tau,\cdot)M^{(k)}+o(\tau^{k})=\tau^{k}\mathbf{V}_{0}M^{(k)}+o(\tau^{k}).$$

Thus, to compute the matrix $M^{(k)}$ at each $k$ step, it is not necessary to invert the matrix $\widehat{V}^{(k)}(\cdot,t+\tau,\cdot)$ each time, it suffices to invert the matrix $\mathbf{V}_{0}$ once. From this we immediately obtain expressions for $\hat{X}_{ij}^{(k)}$ if $i\neq j$ and for $L^{(k+1)}$ in (74). There is freedom to choose $\hat{X}_{ii}^{(k)}$. In our case, it is due to the matrix normalization condition $\widehat{V}^{(k)}$, which is fixed, like the matrix $\mathbf{V}_{0}$, by the equality of the elements of the first row to one:

$$\widehat{V}^{(k)}=\begin{pmatrix}1&\ldots&1\\ \lambda_{1}(z)+O(\tau)&\ldots&\lambda_{d}(z)+O(\tau)\\ \vdots&&\vdots\\ \lambda_{1}^{d-1}(z)+O(\tau)&\ldots&\lambda_{d}^{d-1}(z)+O(\tau)\\ \end{pmatrix},$$

which implies that the sums of the elements of each column $\hat{X}^{(k)}$ are equal to zero.

To prove (75), note again that $\widehat{V}^{(k)}(N,t,\tau)$ has ‘‘established’’ expansion terms up to order $\tau^{(k-1)}$ (inclusive), but for calculation $\widehat{V}^{(k+1)}(N,t,\tau)$, with the corresponding accuracy (up to order $\tau^{k}$) we need get $\widehat{V}^{(k)}(N,t+\tau,\tau)$ up to order $\tau^{k}$. Taking into account (72) and the fact that the elements of $S^{(k)}$ are of order $O(\tau^{k})$, we have

$$\widehat{V}_{n}^{(k+1)}:=\widehat{V}_{n}^{(k)}(I+X_{n}^{(k)})=\widehat{V}_{n}^{(k)}(I+O(\tau^{k})),\quad\widehat{V}_{n}^{(k+1)}=\widehat{V}_{n}^{(k)}(I+\tau^{k}\hat{X}_{n}^{(k)})+o(\tau^{k}).$$

This implies (75). Theorem 3 is proved. $\Box$

Finally, at the end of the procedure for finding the approximate value of ‘‘diagonalizer’’ $V_{n}$, we substitute $\tau=1/N$ and use partial sums

$$V_{n}=\mathbf{V}(N,t)\approx\sum\limits_{j=0}^{k+1}\widetilde{V}_{j}(N,t),\quad V_{n+1}=\mathbf{V}\left(N,t+\frac{1}{N}\right)\approx\sum\limits_{j=0}^{k+1}\widetilde{V}_{j}\left(N,t+\frac{1}{N}\right).$$

(76)

As you can note, see (71), (72), to realize this procedure we need expressions for the derivatives (with respect to the parameter $t$) of the $\{\lambda_{j}(z)\}$ branches of the spectral curve (17). These expressions can be obtained using the parametrization of the curve $\{\lambda_{j}(z)\}$. Indeed, the connection between $z$ and $t$,

$$q^{n}=z=e^{2\pi it},$$

(77)

implies the dependence of $\{\lambda_{j}(z)\}$ on $t$, which, in turn, causes the dependence of the variable $s(t)$ parameterizing the spectral curve $\lambda(z)$. Thus, the desired expressions for $\{\lambda_{j}^{\prime}(z(t))\}$ are obtained using $s^{\prime}(t)$. For example, for the $5_{2}$ knot, the following result is true.

Lemma 3. Let $r$ be defined in (57) and $R_{3}(s)$ in (54). Then

$$s^{\prime}(t)=2\pi i\,r\,\displaystyle\frac{R_{3}(s)-2}{R_{3}^{\prime}(s)}.$$

(78)

Proof. Differentiate (57) using (77) and transform it as

$$\dfrac{d}{dt}{\mathcal{Z}}=2\pi i\left(z-\frac{1}{z}\right)=\dfrac{8\pi i\,r}{r^{2}-1}=2\pi i\,r\,(R_{3}(s)-2).$$

This implies (78). It is useful to know the derivative of $r$. Differentiating (57), we have

$$\frac{d}{dt}\,r(s(t))=-\frac{2\pi i}{2}(r^{2}-1)=\frac{-4\pi i}{R_{3}(s)-2}.$$

$\Box$

1.2 Basis Expansion

We use as basis vectors the columns of the matrix $B_{n}$:

$$B_{n}:=V_{n}\prod\limits_{m=m_{0}}^{n-1}D_{m}=:V_{n}\,\Pi_{n},\quad D_{m}:=\textrm{diag}\left[V_{m+1}^{-1}\mathcal{A}_{m}V_{m}\right]=:\textrm{diag}\left\{D^{(1)},\ldots,D^{(d)}\right\}.$$

In our situation, for the ‘‘diagonalizers’’ $V_{n}$ approximation, we take the segments of the power series by $(1/N)$ from (76). Now we obtain the expansion coefficients in $(1/N)$ for $\Pi_{n}$. To do it, we start with a series for diagonal matrices $D_{n}$, for which the approximation gives the segment of the series in $(1/N)$ from (74):

$$D_{n}\approx L^{(k)}(N,t,\tau)|_{\tau=1/N}=:\textrm{diag}\left\{\lambda_{j}(z)+\sum\limits_{i=1}^{\infty}\dfrac{l_{i}^{(j)}(z)}{N^{i}}\right\}_{j=1}^{d},\quad z=e^{2\pi it}.$$

(79)

with a fixed (providing the required precision) $k$. Consider the decomposition for $\Pi_{n}=:\textrm{diag}\{\pi_{j}\}_{j=1}^{d}$:

$$\pi_{j}(z,N):=\prod\limits_{m=m_{0}}^{n-1}D^{(j)}_{m}=\exp\left\{\sum\limits_{i=-1}^{\infty}\dfrac{\varphi_{i}^{(j)}(z)}{N^{i}}\right\}.$$

(80)

For the leading expansion term from (79) we get (up to a multiplicative constant)

$$\pi_{j}=\exp\left\{N\int\limits^{t}\ln\lambda_{j}\,d\tilde{t}\right\}+O(1)=\exp\left\{\dfrac{N}{2\pi i}\int\limits^{z}\ln\lambda_{j}(\tilde{z})\frac{d\tilde{z}}{\tilde{z}}\right\}+O(1).$$

We continue for the remaining coefficients in (80). We have

$$\ln\pi_{j}(qz)=\ln\pi_{j}(z)+\frac{2\pi i}{N}\cdot z\frac{\partial}{\partial z}\left(\ln\pi_{j}(z)\right)+\frac{1}{2}\left(\frac{2\pi i}{N}\right)^{2}\left(\left(z\frac{\partial}{\partial z}\right)^{2}(\ln\pi_{j}(z))\right)+\ldots$$

Substituting (80) and re-expanding the result in powers of $N^{-1}$, we get

$$\ln\pi_{j}(qz)=\sum\limits_{i=-1}^{\infty}\dfrac{\varphi_{i}}{N^{i}}+\dfrac{2\pi i}{N}\sum\limits_{i=-1}^{\infty}\dfrac{z\varphi^{\prime}_{i}}{N^{i}}+\ldots$$

$${}=N\varphi_{-1}+\left(\varphi_{0}+2\pi iz\varphi^{\prime}_{-1}\right)+\frac{1}{N}\left(\varphi_{1}+2\pi iz\varphi^{\prime}_{0}+(2\pi i)^{2}z(z\varphi^{\prime}_{-1})^{\prime}\right)+\ldots$$

By subtracting $\ln\pi_{j}$ from the resulting expansion expression, see (80), we remove the coefficients $\varphi_{-1},\varphi_{0},\varphi_{1},\ldots$, leaving only their derivatives. Finally, comparing the remainder with the decomposition $\ln D^{(j)}=\ln\lambda_{j}+\ldots$, obtained from (79), we successively obtain expressions for these derivatives. Thus, $\varphi_{-1}=\dfrac{1}{2\pi i}\int\limits^{z}\ln\lambda(\xi)\dfrac{d\xi}{\xi}$, and all $\varphi_{k}$ are primitive.

Thus, the procedure of the basis expansion is constructed and Theorem 1 is proved.

1.3 Proof of Theorem 2

We follow [24]. Step (1): The existence and algebraicity of $V_{k}$ follow from the construction rules: the initial coefficients are algebraic functions, the algebraicity is preserved under the re-expansion, the establishment of series is shown by the same way as in [24]. Singularities arise only from division of the matrix elements on $\lambda_{i}-\lambda_{j}$. Therefore, it remains to correctly estimate the order of these singularities by induction on $k$.

Choosing a real $s$ for $z$ on the circle (univalent branch of $z(s)$), we denote $\widehat{s}:=\lim_{z{\to}z_{j}}z^{-1}(z(s))$ and note that $\lambda_{2}{-}\lambda_{3}\to 0$ as $z{\to}z_{j}$. Expressing the resulting execution of the procedure of the matrices $V_{1},V_{2},\ldots$ through the uniformizing parameter $s$, we obtain in the denominator increasing degrees of the radical $r_{1}:=\sqrt{s^{4}-2s^{3}-5s^{2}-2s+1}\sim\sqrt{s{-}\widehat{s}}$.

To prove the first estimate in (24), the inductive assumption is that for $k{\geqslant}1$ (and for all smaller numbers) the elements of $V_{k}$ contain $r_{1}^{3k-1}$ in the denominator. The induction base is verified by directly calculating $V_{1}$, see [4, Appendix].

When calculating $V_{k+1}$, the degree of $r_{1}$ increases by $2$ when differentiating (re-expansion), and $1$ more adds division by $\lambda_{i}-\lambda_{j}$. The cycle of calculating the coefficients of $V$ has the form

$$J:=\sum^{k}_{j=0}\textrm{coeff}\left({\mathcal{A}}_{n},N,j{+}1\right)V_{k-j}-\sum^{k}_{j=0}\Big{[}\Big{(}2\pi iz\frac{d}{dz}\Big{)}^{j+1}V_{k-j}\Big{]}\frac{\Lambda}{(j{+}1)!}$$

$${}-\sum^{k-1}_{l=0}\sum^{k}_{j=l}\Big{[}\Big{(}2\pi iz\frac{d}{dz}\Big{)}^{j-l}V_{k-j}\Big{]}\frac{D_{l+1}}{(j{-}l)!},$$

$$D_{k+1}=\textrm{diag}[J];\quad X_{ij}=J_{ij}/(\lambda_{i}{-}\lambda_{j})\quad(i{\neq}j);\quad X_{jj}=-\textstyle\sum\limits_{i:i\neq j}X_{ij};\quad V_{k+1}=V_{0}X.$$

The expansion coefficients of ${\mathcal{A}}_{n}$ have no singularities for $z=z_{1,2}$. Due to the inductive assumptions in the denominator $V_{j}$ ($j{>}0$) the degree of the radical $r_{1}$ is equal to $3j{-}1$, and in the denominator $D_{l+1}$ is equal to $3l{+}1$. We make sure that the induction transition is correct.

The proof of the second estimate in (24) is similar.

Step (2): Concerning the bounds (25), we note that [24] proposed a method for proving estimates of off-diagonal elements of quantities (72), which makes it possible to check satisfiability of (19) in some domain in which the formal series for $V_{n}$ and $\Pi_{n}$ can be considered as asymptotic ones.

Let $\mathcal{D}(n,z)\not\equiv 0$ be a discriminant of the characteristic polynomial of the matrix $\mathcal{A}_{n}(z)$, $\lambda_{j}$ be its eigenvalues. $\mathcal{D}(n,z)$ is a polynomial in $z$ with rational in $n$ coefficients. The denominators of $\mathcal{D}(n,z)$ depend only on $n$. Below by $\mathcal{D}(n,z)$ we mean the numerator which is a polynomial in $n$ and $z$. if $\mathcal{D}(n,z)$ is a constant then we put this constant is more than one. Write

$$\Omega:=\{(z,n)\in\mathbb{C}^{2}:\ |n|>C,\ |\mathcal{D}(n,z)|>1\},$$

$$R\equiv R(z,n_{0}):=\min_{n:(z,n)\in\partial\Omega}|n-n_{0}|,\quad\textrm{and}\quad R_{0}:=\frac{1}{3}R,$$

$$\Delta\equiv\Delta(z,n_{0},R_{0}):\quad\min_{j\neq k}|\lambda_{j}(z,n)-\lambda_{k}(z,n)|>\Delta\quad\textrm{for}\quad|n-n_{0}|<R_{0},$$

$$\widetilde{M}\equiv\widetilde{M}(z,n_{0},R_{0}):\quad||V^{(0)}_{\tilde{n}}{\mathcal{A}}_{n}V^{(0)}_{n}||<\widetilde{M}\quad\textrm{for}\quad|\tilde{n}-n_{0}|<2R_{0},\quad|n-n_{0}|<2R_{0}.$$

In the notation of [24], fixing $m$ and taking $r:=\frac{R_{0}}{m+1}$, we get $k\sim\frac{r_{1}}{R}$. The bound $\frac{\Delta}{\widetilde{M}}\leq 2$ follows from the definition. Also, $M\sim\frac{\widetilde{M}}{\Delta}$, $y\sim\frac{\widetilde{M}r_{1}}{R\Delta}$, so $k\lesssim y$. Expression for $q$ (see [24, Theorem $2^{*}$]) is

$$q\sim M(k{+}y)\sim My\sim\frac{\widetilde{M}^{2}r_{1}}{\Delta^{2}R}.$$

Now choose $r_{1}:=\theta R\frac{\Delta^{2}}{\widetilde{M}^{2}}$ for some small constant $\theta$. Then condition $r_{1}<r$ is true, $q\sim\theta$, $y\lesssim\theta$, so for a suitable $\theta$, Theorem $2^{*}$ can be used. For the off-diagonal part $\{S^{(l)}(n,\tau)\}$ we have $\{S^{(l)}(n,\tau)\}=O(\tau^{l+1})$, hence

$$\|\{S^{(l)}(n,1)\}\|\leqslant r_{1}^{-l-1}\sup\limits_{|\tau|<r_{1}}\|\{S^{(l)}(n,\tau)\}\|$$

for those $r_{1}>1$ for which Theorem $2^{*}$ from [24] can be applied. Therefore, if

$$\frac{\widetilde{M}^{2}}{\Delta^{2}R}=O(N^{-\varepsilon}),$$

(81)

then there is $l$, for which (19) is fulfilled.

In our case, in a neighborhood of points $z_{1,2}$ the first estimate in (24) implies $\widetilde{M}\sim 1$, $\Delta\sim\sqrt{|z-z_{j}|}$, $R\sim N|z-z_{j}|$, whence (81) gives the first bound in (25).

Near points $z=\pm 1$ (24) implies $\widetilde{M}\sim 1$, $\Delta\sim|z-(\pm 1)|$, $R\sim N|z-(\pm 1)|$, whence (81) gives the second estimate in (25). Theorem 2 is proved.

The numeric experiments have shown, that in fact, $V_{j}$ with the first $j$ are holomorphic at $z=\pm 1$. Therefore, there is a conjecture that the series (21) can be used in the entire separation zone up to the points $z=\pm 1$.