††institutetext: Department of Physics and Center for Field Theory and
Particle Physics, Fudan University, 220, Handan Road, 200433 Shanghai, China

# On knots, complements, and $6j$-symbols

Hao Ellery Wang  Yuanzhe Jack Yang  Hao Derrick Zhang  and Satoshi Nawata
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

This paper investigates the relation between colored HOMFLY-PT and Kauffman
homology, $\operatorname{SO}(N)$ quantum $6j$-symbols and $(a,t)$-deformed
$F_{K}$. First, we present a simple rule of grading change which allows us to
obtain the $[r]$-colored quadruply-graded Kauffman homology from the
$[r^{2}]$-colored quadruply-graded HOMFLY-PT homology for thin knots. This
rule stems from the isomorphism of the representations
$(\mathfrak{so}_{6},[r])\cong(\mathfrak{sl}_{4},[r^{2}])$. Also, we find the
relationship among $A$-polynomials of $\operatorname{SO}$ and
$\operatorname{SU}$-type coming from a differential on Kauffman homology.
Second, we put forward a closed-form expression of $\operatorname{SO}(N)(N\geq
4)$ quantum $6j$-symbols for symmetric representations, and calculate the
corresponding $\operatorname{SO}(N)$ fusion matrices for the cases when
representations $R=$ ,  . Third, we conjecture closed-form expressions of
$(a,t)$-deformed $F_{K}$ for the complements of double twist knots with
positive braids. Using the conjectural expressions, we derive $t$-deformed ADO
polynomials.

###### Keywords:

Knot homology, Super-$A$-polynomial, $6j$-symbols, $F_{K}$, ADO polynomial

##  1 Introduction

Witten’s celebrated paper Witten:1988hf  has shown that the Chern-Simons
theory provides a natural framework for quantum invariants of 3-manifolds and
knots. The Chern-Simons path integral on a 3-manifold $M_{3}$ with the action

$S=\frac{k}{4\pi}\int_{M_{3}}\text{Tr}\left(A\wedge dA+\frac{2}{3}A\wedge
A\wedge A\right)~{},$

is a quantum invariant of $M_{3}$, known as the Witten-Reshetikhin-Turaev
(WRT) invariant Witten:1988hf ; Reshetikhin:1991tc . If a Wilson loop is
included along a knot in $S^{3}$, the expectation value of the Wilson loop is
a quantum invariant of the knot:

$\overline{J}^{G}_{R}(K;q):=\left<W_{R}(K)\right>=\frac{\int\left[\mathcal{D}A\right]e^{iS}W_{R}(K)}{\int\left[\mathcal{D}A\right]e^{iS}}~{},$

where the parameter $q$ is expressed by the level $k$ and the dual Coxeter
number $h^{\lor}$ of the gauge group $G$ as

$q=\exp\left(\frac{\pi i}{k+h^{\lor}}\right)~{}.$ (1)

Since the Chern-Simons functional integral on a three-manifold with boundary
is an element of the Hilbert space on the boundary which is isomorphic to the
space of WZNW conformal blocks, quantum knot invariants could be constructed
by using braiding and fusion operations on WZNW conformal blocks Moore:1989vd
; Kaul:1991np ; Kaul:1992rs ; Kaul:1993hb ; Ramadevi:1992np . When
$G=\operatorname{SU}(N)$, it corresponds to a two-variable colored HOMFLY-PT
polynomial $P_{R}(K;a,q)$, where $R$ is a representation of
$\operatorname{SU}(N)$ and $a=q^{N}$. When $G=\operatorname{SO}(N)$, it
corresponds to colored Kauffman polynomial $F_{R}(K;a,q)$, where $R$ is a
representation of $\operatorname{SO}(N)$ and $a=q^{N-1}$.111 In this paper, a
representation specified by a Young diagram $(r_{1}\geq r_{2}\geq\cdots)$ is
denoted by $R=[r_{1},r_{2},\cdots]$. In particular, we write $[r^{s}]$ for an
$r\times s$ rectangular Young diagram. The quantum knot invariants have a
beautiful property that they are polynomials of $q$ with integer coefficients.
This property led to an important development in knot theory proposed by
Khovanov Khovanov:1999qla , i.e. the categorifications of quantum knot
invariants. Khovanov constructed a bi-graded homology which itself is a knot
invariant, and its $q$-graded Euler characteristic is the Jones polynomial
$J(K;q)$ of a knot $K$.222We define the normalized quantum invariant
$J^{G}_{R}(K;q)=\frac{\overline{J}^{G}_{R}(K;q)}{\overline{J}^{G}_{R}({\raisebox{-2.84544pt}{\includegraphics[width=11.38092pt]{unknot}}};q)}~{}.$
In this paper, we focus only on knot invariants normalized by the unknot so
that HOMFLY-PT $P_{R}(K;a,q)$ and Kauffman $F_{R}(K;a,q)$ polynomials are also
normalized in a similar manner.

In khovanov2008matrix , M. Khovanov and L. Rozansky constructed a triply-
graded homology of a knot whose graded Euler characteristic is the HOMFLY-PT
polynomial. And in Dunfield:2005si , it was conjectured that the HOMFLY-PT
homology is endowed with structural properties. Similarly, the existence and
structural properties of Kauffman homology were conjectured in Gukov:2005qp
although the rigorous definition of a triply-graded homology theory
categorifying the Kauffman polynomials has not been given yet.

This line of studying knot homology has been pursued in the colored cases. In
Gukov:2011ry ; Gorsky:2013jxa , quadruply-graded colored HOMFLY-PT homology
was proposed and various structures and symmetries were uncovered. In
Nawata:2013mzx , it was further conjectured that there exist the quadruply-
graded colored Kauffman homologies with rich structural properties. The first
part of this paper investigates the relationship between the colored HOMFLY-PT
homology and Kauffman homology. In §2.1, we show a rule of changing variables
that transforms the Poincaré polynomial of $[r^{2}]$-colored HOMFLY-PT
homology into that of $[r]$-colored Kauffman homology for thin knots.

As another important development, the volume conjecture Kashaev:1996kc ;
murakami2001colored  gives a remarkable relationship between quantum
invariants of a knot $K$ and classical geometry of the knot complement
$S^{3}\backslash K$. It states that the large color asymptotic behavior of
colored Jones polynomial $J_{[r]}(K;q)$ of a hyperbolic knot provides the
hyperbolic volume of its knot complement. This was further generalized in
Gukov:2003na ; Gukov:2008jlx , which gives the relation between the large
color limit of colored Jones polynomials and the $A$-polynomial for the knot
complement. The zero locus of the $A$-polynomial $A(K;x,y)$ of a knot $K$
determines the character variety of
$\operatorname{SL}(2,\mathbb{C})$-representation of the knot group
Cooper:1994pca  so that the generalized volume conjecture paves a way to
complex Chern-Simons theory. There are further generalizations of the volume
conjecture to incorporate HOMFLY-PT and Kauffman polynomials colored by
symmetric representations and their categorifications Fuji:2012pm ;
Aganagic:2012jb ; Fuji:2012nx ; Nawata:2013mzx , leading to a notion of
super-$A$-polynomials. Since there is a differential that relates colored
Kauffman homology to HOMFLY-PT homology, we investigate the influence of the
differential on super-$A$-polynomials in §2.2. This brings about Conjecture
2.1 that provides an intriguing relationship among $A$-polynomials of
$\operatorname{SU}$ and $\operatorname{SO}$-type.

As briefly mentioned, quantum invariants can be evaluated with braiding and
fusion matrices in the WZNW models. When $q$ is a root of unity (1), the
fusion matrices of the WZNW models are equivalent to quantum $6j$-symbols of
the corresponding quantum groups (up to normalizations (37)) Moore:1988qv ;
Moore:1989vd ; kazhdan1993tensor ; kazhdan1993tensor2 ; kazhdan1994tensor ;
kazhdan1994tensor2 . The $6j$-symbols were introduced by Wigner 1965Quantum ;
wigner1993matrices  and Racah Racah:1942gsc  in the study of angular momenta,
and they are beautifully expressed by a hypergeometric series ${}_{4}F_{3}$.
In the case of $\mathfrak{sl}_{2}$, the explicit expression for quantum
$6j$-symbols was obtained in Kirillov:1991ec , which is a natural quantization
with a $q$-hypergeometric series ${}_{4}\varphi_{3}$. Nonetheless, it is
notoriously involved to obtain explicit expressions for $6j$-symbols from the
definition with the Clebsch-Gordan coefficients ($3j$-symbols), in particular,
for higher ranks Gu:2014nva . However, the information of quantum $6j$-symbols
with all symmetric representations was extracted from quantum knot invariants
for $U_{q}(\mathfrak{sl}_{N})$ Nawata:2013ooa , and it was formulated in terms
of the $q$-hypergeometric series ${}_{4}\varphi_{3}$ in Alekseev:2019nzw . In
§3, we study the $\operatorname{SO}(N)$ quantum $6j$-symbols with symmetric
representations along this line. The formulae for the classical
$\operatorname{SO}(N)$ $6j$-symbols with all representations symmetric were
obtained by Ališauskas alisauskas1987some ; Alisauskas_2002 . We quantize
these formulae and compute $\operatorname{SO}(N)$ fusion matrices of the WZNW
models for the case of $R=$ ,  . We check that the fusion matrices reproduce
the colored Kauffman polynomials of non-torus knots. This gives a piece of
evidence that the natural quantization of the Ališauskas’ formulae will be
valid for the $\operatorname{SO}(N)$ quantum $6j$-symbols.

The developments of knot invariants and volume conjectures are inseparably
bound up with the study of invariants of 3-manifolds. Chern-Simons quantum
invariants of 3-manifolds can be computed by a surgery formula of quantum
invariants of links in $S^{3}$ Reshetikhin:1991tc  when $q$ is a root of unity
(1). In the recent study of the 3d/3d correspondence, an analytic continuation
$\widehat{Z}$ of the WRT invariant has been proposed as a BPS $q$-series
Gukov:2016gkn ; Gukov:2017kmk . This invariant can be thought of as a
generating series of BPS states in a 3d $\mathcal{N}=2$ theory $T[M_{3}]$.
Also it can be regarded as a partition function of complex Chern-Simons theory
on $M_{3}$ defined on a unit disk $|q|<1$ whose radial limit $q\searrow e^{\pi
i/(k+h^{\vee})}$ becomes the WRT invariant. Its generalization to a knot
complement, denoted as $F_{K}(x,q)=\widehat{Z}(S^{3}\backslash K)$, is given
in Gukov:2019mnk  to make surgeries and TQFT method work. It is observed that
the asymptotic expansion of $F_{K}$ agrees with the Melvin-Morton-Rozansky
expansion melvin1995 ; Stavros1996 ; Rozansky:1994qe ; Rozansky98theuniversal
of the colored Jones polynomials. Moreover, $F_{K}$ is conjectured to be
annihilated by the quantum $A$-polynomial MR2172488 ; garoufalidis2005colored
of the corresponding knot $K$,

$\displaystyle\hat{A}(K;\hat{x},\hat{y},q)F_{K}(x,q)=0~{}.$ (2)

Based on willetts2020unification , it is proposed in Gukov:2020lqm  that the
limit $q\to\zeta_{p}=e^{2\pi i/p}$ of $F_{K}$ leads to the $p$-th ADO
polynomials Akutsu:1991xc .

The development of $F_{K}$ is currently following that of quantum knot
invariants although its definition is not given yet. In Park:2019xey , an
extension of $\widehat{Z}$ and $F_{K}$ to arbitrary gauge group $G$ was
studied. Analogous to HOMFLY-PT polynomials and superpolynomials,
$(a,t)$-deformation of $F_{K}$ are put forth in Ekholm:2020lqy .

Following this path, in §4.1, we provide a closed-form expression of the
$(a,t)$-deformed $F_{K}$ for double twist knots with positive braids in this
paper. Also, in §4.2, we derive closed-form formulae of $t$-deformed ADO
polynomials.

##  2 Knot homology and super-$A$-polynomial

The categorifications of quantum knot invariants shed new light on knot theory
not only because knot homologies are more powerful than quantum invariants but
also because they are functorial rasmussen2010khovanov . Moreover, they are
endowed with rich structural properties Wedrich:2016smm , and various
differentials give remarkable relationships among themselves Dunfield:2005si ;
Gukov:2011ry ; Gorsky:2013jxa ; rasmussen2016some . In particular, some
relations between HOMFLY-PT and Kauffman homology have been uncovered in
Gukov:2005qp ; Nawata:2013mzx . The rich structural properties also help us
find closed-form expressions of Poincaré polynomials of cyclotomic type
Habiro:2008  for various knots and links.

In this section, we will show a new relationship between colored HOMFLY-PT and
Kauffman homology for thin knots. Also, we investigate the effects of
differentials on super-$A$-polynomials of $\operatorname{SU}$ and
$\operatorname{SO}$-type.

###  2.1 Isomorphism between colored HOMFLY-PT and Kauffman homology for thin
knots

It has been apparent that the rich structural properties become manifest if we
introduce quadruple-gradings to HOMFLY-PT and Kauffman homology. The
quadruply-graded HOMFLY-PT homology $(\mathscr{H}^{\text{HOMFLY-
PT}}_{R}(K))_{i,j,k,l}$ with $(a,q,t_{r},t_{c})$-gradings was introduced in
Gorsky:2013jxa . With this grading, we can associate a $\delta$-grading to
every generator $x$ of the $[r]$-colored HOMFLY-PT homology defined as,

$\delta(x):=a(x)+\frac{q(x)}{2}-\frac{t_{r}(x)+t_{c}(x)}{2}.$

A knot $K$ is called homologically-thin if all generators of
$\mathscr{H}^{\text{HOMFLY-PT}}_{[r]}(K)$ have the same $\delta$-grading equal
to $\frac{r}{2}S(K)$ , where $S(K)$ is the Rasmussen $s$-invariant of $K$
rasmussen2010khovanov . Otherwise, it is called homologically-thick. In
general, thick knots possess more intricate structures Dunfield:2005si ;
Gukov:2011ry .

In a similar manner, the structral properties of quadruply-graded Kauffman
homology $(\mathscr{H}^{\text{Kauffman}}_{R}(K))_{i,j,k,l}$ were studied in
Nawata:2013mzx . In the following discussion in this section, we also use the
tilde versions, $\widetilde{\mathscr{H}}^{\text{HOMFLY-PT}}_{[r^{s}]}(K)$
Gorsky:2013jxa , $\widetilde{\mathscr{H}}^{\text{Kauffman}}_{[r^{s}]}(K)$
Nawata:2013mzx , and their corresponding Poincaré polynomials

$\displaystyle\widetilde{\mathscr{P}}_{[r^{s}]}(K;a,Q,t_{r},t_{c})$
$\displaystyle:=\sum_{i,j,k,l}a^{i}Q^{j}t_{r}^{k}t_{c}^{l}\text{dim}\left(\widetilde{\mathscr{H}}^{\text{HOMFLY-
PT}}_{[r^{s}]}(K)\right)_{i,j,k,l},$ (3)
$\displaystyle\widetilde{\mathscr{F}}_{[r^{s}]}(K;a,Q,t_{r},t_{c})$
$\displaystyle:=\sum_{i,j,k,l}a^{i}Q^{j}t_{r}^{k}t_{c}^{l}\text{dim}\left(\widetilde{\mathscr{H}}^{\text{Kauffman}}_{[r^{s}]}(K)\right)_{i,j,k,l}~{},$
(4)

where the gradings of $\mathscr{H}$ and $\widetilde{}\mathscr{H}$ are related
by

$\displaystyle\mathscr{F}_{[r^{s}]}(K;a,q,t_{r},t_{c})$
$\displaystyle:=\widetilde{\mathscr{F}}_{[r^{s}]}(K;a,q^{s},t_{r}q^{-1},t_{c}q)~{},$
(5) $\displaystyle\mathscr{P}_{[r^{s}]}(K;a,q,t_{r},t_{c})$
$\displaystyle:=\widetilde{\mathscr{P}}_{[r^{s}]}(K;a,q^{s},t_{r}q^{-1},t_{c}q)~{}.$
(6)

It turns out that there are differentials, called universal and diagonal in
Nawata:2013mzx , that relate $[r]$-colored Kauffman and HOMFLY-PT homology.
Furthermore, the well-known isomorphism of the representations Nawata:2013mzx

$\left(\mathfrak{so}_{6},[r]\right)\simeq\left(\mathfrak{sl}_{4},[r^{2}]\right),$

leads to an isomorphism between bi-graded homologies

$\mathscr{H}_{\mathfrak{so}_{6},[r]}(K)\cong\mathscr{H}_{\mathfrak{sl}_{4},[r^{2}]}(K)~{}.$
(7)

For a thin knot $K_{\text{thin}}$, this can be stated in terms of the Poincaré
polynomials (Nawata:2013mzx, , (4.57))

$\widetilde{\mathscr{F}}_{[r]}(K_{\text{thin}};a\\!=\\!q^{5},Q\\!=\\!q,t_{r}\\!=\\!q^{-1},t_{c}\\!=\\!qt)=\widetilde{\mathscr{P}}_{[r^{2}]}(K_{\text{thin}};a\\!=\\!q^{4},Q\\!=\\!q^{2},t_{r}\\!=\\!q^{-1},t_{c}\\!=\\!qt)~{}.$
(8)

$-4$$-2$024$Q$468$a$$0$$0$$8$$4$$6$$2$$4$$2$$5$$3$$7$$3$$9$$5$$11$$5$$12$$6$$t_{r}$$t_{c}$$-2$$-1$012$Q$2345$a$$0$$3$$2$$2$$3$$4$$5$$5$$6$
Figure 1: An isomorphism between $[1^{2}]$-colored HOMFLY-PT homology (left)
and uncolored Kauffman homology (right) for the trefoil. Generators with the
same color are identified under the grading change (9).

Moreover, we find an isomorphism between $[1^{2}]$-colored HOMFLY-PT homology
and the uncolored Kauffman homology of the trefoil and the figure-eight. For
example, there is a one-to-one correspondence between their generators with
the same color, shown in Figure 1. This motivates us to uplift (7) to an
isomorphism between HOMFLY-PT and Kauffman homologies in the case of a thin
knot $K_{\text{thin}}$ with grading change

$\displaystyle\left(\widetilde{\mathscr{H}}^{\text{Kauffman}}_{[r]}(K_{\text{thin}})\right)_{i,j,k,l}\cong\left(\widetilde{\mathscr{H}}^{\text{HOMFLY-
PT}}_{[r^{2}]}(K_{\text{thin}})\right)_{\frac{3}{2}i+\frac{3}{4}j-\frac{1}{2}k,-\frac{5}{2}i-\frac{5}{4}j+\frac{3}{2}k,i+\frac{1}{2}j,l},$
(9)

up to some overall grading shifts proportional to $\frac{rS(K)}{2}$ in
$(a,Q,t_{r})$ degrees. This can be expressed in terms of the Poincaré
polynomials

$\widetilde{\mathscr{F}}_{[r]}(K_{\text{thin}};a,Q,t_{r},t_{c})=\left(\frac{Q^{3}}{at_{r}^{2}}\right)^{\frac{rS(K)}{2}}\widetilde{\mathscr{P}}_{[r^{2}]}(K_{\text{thin}};a^{\frac{3}{2}}Q^{-\frac{5}{2}}t_{r},a^{\frac{3}{4}}Q^{-\frac{5}{4}}t_{r}^{\frac{1}{2}},a^{-\frac{1}{2}}Q^{\frac{3}{2}},t_{c})~{}.$
(10)

We would like to emphasize that the isomorphism holds only for thin knots, but
it is not true for thick knots. For thick knots, the dimensions of
$[r^{2}]$-colored quadruply-graded HOMFLY-PT homology and $[r]$-colored
quadruply-graded Kauffman homology are different (Gukov:2011ry, , Appendix B).

Remark that the relation $(\ref{kauffcov2})$ only provides the method to
obtain the Poincaré polynomial of $[r]$-colored Kauffman homology from that of
$[r^{2}]$-colored HOMFLY-PT homology for a thin knot, but it does not allow us
to use this relation in the opposite direction since the grading shifts of
variables in (10) are not linearly independent.

In Kononov:2016cwp ; Kameyama:2019mdf , the Poincaré polynomial of quadruply-
graded HOMFLY-PT homology of the double twist knot $K_{m,n}$ colored by
rectangular Young tableau $[r^{s}]$ has been given. (See Figure 2 for the
double twist knot.) Since the double twist knots are known to be
homologically-thin, we can obtain the expression of the Poincaré polynomial of
$[r]$-colored Kauffman homology for the double-twist knots thanks to the
relation (10). To verify the relations (9) and (10), we have checked that the
colored Kauffman homology obtained by this substitution satisfies all the
differentials and properties Nawata:2013mzx  of the colored Kauffman homology
for double twist knots. We also attached the Mathematica files that show these
properties on the arXiv page.

###  2.2 Differential on super-$A$-polynomials of SO-type

The study of the large color behaviors of colored Jones polynomials led us to
the volume conjecture murakami2001colored ; Gukov:2003na . The generalization
of the volume conjecture to the large color behavior of the Poincaré
polynomial led to the so-called super-$A$-polynomial Fuji:2012nx ;
Nawata:2013mzx . As briefly mentioned above, there are the differentials that
relate $[r]$-colored Kauffman homology and HOMFLY-PT homology. In this
subsection, we will investigate the effect of the differential on
super-$A$-polynomials.

To this end, let us first recall the generalized volume conjecture
Gukov:2003na ; Gukov:2008jlx  and the super-$A$-polynomials Fuji:2012nx . Here
we use Poincaré polynomials of triply-graded homology
$\mathscr{Q}_{[r]}(K;a,q,t)$ only with $t_{r}$-grading, also called the
superpolynomial, defined as

$\mathscr{Q}_{[r]}(K;a,q,t):=\mathscr{Q}_{[r]}(K;a,q,t_{r}=t,t_{c}=1)~{},$
(11)

where $\mathscr{Q}=\mathscr{P}\textrm{ or }\mathscr{F}$ represents Poincaré
polynomials of either HOMFLY-PT or Kauffman homology. If a knot $K$ satisfies
the exponential growth property Gukov:2011ry ; Wedrich:2016smm  (such as thin
knots and torus knots)

$\mathscr{Q}_{[r]}(K;a,q=1,t)=\Bigl{[}\mathscr{Q}_{[1]}(K;a,q=1,t)\Bigr{]}^{r}~{},$

it is conjectured (Kucharski:2017ogk, , Conjecture 4.4) that the
superpolynomial colored by a symmetric representation can be expressed as

$\mathscr{Q}_{[r]}(K;a,q,t)=\sum_{l_{1}+l_{2}+\cdots+l_{n}=r}\left[\begin{array}[]{c}r\\\
l_{1},l_{2},\ldots,l_{n}\end{array}\right]_{q^{2}}a^{\sum_{i=1}^{n}a_{i}l_{i}}t^{\sum_{i=1}^{n}t_{i}l_{i}}q^{\sum_{i,j=1}^{n}Q_{i,j}l_{i}l_{j}+\sum_{i=1}^{n}q_{i}l_{i}}~{},$
(12)

where $(a_{i},q_{i},t_{i})$ are constants and $Q_{i,j}$ is a quadratic form.
Here the $q$-multibinomial is defined by

$\left[\begin{array}[]{c}r\\\
l_{1},l_{2},\ldots,l_{n}\end{array}\right]_{q}:=\frac{\left(q;q\right)_{r}}{\left(q;q\right)_{l_{1}}\left(q;q\right)_{l_{2}}\cdots\left(q;q\right)_{l_{n}}}~{}.$

In the large color limit

$\displaystyle q=e^{\hbar}\rightarrow 1,\quad r\rightarrow\infty,\quad
a=\text{fixed},\quad t=\text{fixed},\quad x=q^{2r}=\text{fixed}~{},$ (13)

a superpolynomial asymptotes to the form

$\mathscr{Q}_{[r]}(K;a,q,t)\overset{r\to\infty,\hbar\to
0}{\underset{x=q^{2r}}{\sim}}\exp\left(\frac{1}{2\hbar}\int\log
y\frac{dx}{x}+\cdots\right)~{},$ (14)

where the integral is carried out on the zero locus of classical super-A-
polynomial

$\mathscr{A}(K;x,y,a,t)=0.$

The ellipsis in (14) represents the subleading terms that possess regular
behaviors under the limit $\hbar\to 0$.

If $\mathscr{Q}_{[r]}(K;a,q,t)$ is written as a summation only over $n$
variables $l_{1},\cdots,l_{n}$ as in (12), then its behavior under the large
color limit could be approximated as

$\mathscr{Q}_{\left[r\right]}(K;a,q,t)\sim\int
e^{\frac{1}{2\hbar}\left(\widetilde{\mathscr{W}}(K;z_{1},\cdots,z_{n};x,a,t)+\mathcal{O}(\hbar)\right)}dx\prod_{i=1}^{n}dz_{i}~{},$
(15)

where $z_{i}=q^{2l_{i}}$, $x=q^{2r}$. The leading asymptotic behavior (14)
with respect to $\hbar$ comes from the saddle points

$\exp\left(z_{i}\frac{\partial\widetilde{\mathscr{W}}(K;z_{1},\cdots,z_{n};x,a,t)}{\partial
z_{i}}\right)\bigg{|}_{z_{i}=z_{i}^{*}}=1~{}.$ (16)

and the zero locus of the classical super-A-polynomial is determined by

$\exp\left({x\frac{\partial\widetilde{\mathscr{W}}(K;z_{1}^{*},\cdots,z_{n}^{*};x,a,t)}{\partial
x}}\right)=y~{}.$ (17)

Now let us consider the relationship between super-$A$-polynomials of
$\operatorname{SU}$-type and $\operatorname{SO}$-type that comes from the
universal differential. According to (Nawata:2013mzx, , §4.5), the universal
differential $d^{\text{univ}}_{\to}$ has $(a,q,t)$-degree (0,2,1), which
relates the $[r]$-colored Kauffman homology and the $[r]$-colored HOMFLY-PT
homology as

$\mathscr{F}_{[r]}(K;a,q,t)\\!=\\!q^{rS(K)}\mathscr{P}_{[r]}(K;aq^{-1},q,t)+(1+q^{2}t)f(a,q,t)~{}.$
(18)

This implies

$\mathscr{F}_{[r]}(K;a,q,-q^{-2})=q^{rS(K)}\mathscr{P}_{[r]}(K;aq^{-1},q,-q^{-2})~{}.$
(19)

For a knot satisfying the exponential growth property, the superpolynomial
admits an expression (12). Therefore, it is straightforward to verify that the
leading order of the large color asymptotic behavior of the right hand side of
(19) is the same as that of the colored HOMFLY-PT polynomial although the
change of variables $a\to aq^{-1},t=-q^{-2}$ is non-trivial. As a result, we
have the large color asymptotic behavior

$\mathscr{F}_{[r]}(K;a,q,-q^{-2})\sim\int
e^{\frac{1}{2\hbar}\left(\widetilde{\mathcal{W}}^{\operatorname{SU}}\left(K;z_{i};x,a\right)+\mathcal{O}(\hbar)\right)}dxdz_{i}~{}.$
(20)

Thus, it encodes the information about the $a$-deformed $A$-polynomial
$A^{\textrm{SU}}(K;x,y,a)$ of SU-type. On the other hand, the
decategorification limit $t=-1$ is just the colored Kauffman polynomial
$\mathscr{F}_{[r]}(K;a,q,-1)$ and its large color limit yields $a$-deformed
$A$-polynomial $A^{\textrm{SO}}(K;x,y,a)$ of SO-type. Therefore, the
super-$A$-polynomial $\mathscr{A}^{\textrm{SO}}(K;x,y,a,t)$ of SO-type
contains both $A^{\textrm{SO}}(K;x,y,a)$ and $A^{\textrm{SU}}(K;x,y,a)$ at the
$t=-1$ specialization.

######  Conjecture 2.1.

For a knot satisfying the exponential growth property, the zero locus of the
super-$A$-polynomial of SO-type at $t=-1$

$\mathscr{A}^{\textrm{SO}}(K;x,y,a,t=-1)=0$

includes two branches $A^{\textrm{SO}}(K;x,y,a)=0$ and
$A^{\textrm{SU}}(K;x,y,a)=0$.

Let us take the trefoil as an example in which the super-$A$-polynomial of
$\operatorname{SO}$-type can be calculated. The colored superpolynomial of the
trefoil (Nawata:2013mzx, , (5.8)) and the corresponding twisted superpotential
(Nawata:2013mzx, , (6.15)) can be written as

$\displaystyle\mathscr{F}_{[r]}({\bf 3_{1}};a,q,t)$ (21)
$\displaystyle=\sum_{k=0}^{r}\sum_{j=0}^{k}\sum_{i=0}^{r-k}a^{i-k+3r}q^{3k-2j(1+r)+r(2r-3)+i(2j+2r-1)}t^{2(i-j+r)}\left[\begin{array}[]{c}r\\\
k\end{array}\right]_{q^{2}}\left[\begin{array}[]{c}k\\\
j\end{array}\right]_{q^{2}}\left[\begin{array}[]{c}r-k\\\
i\end{array}\right]_{q^{2}}$ $\displaystyle\hskip
85.35826pt\times(-a^{2}t^{3}q^{2r-2};q^{2})_{j}(-q^{2}t;q^{2})_{r-k}(-aq^{-1}t;q^{2})_{i}~{}.$

and

$\displaystyle\widetilde{\mathscr{W}}^{\textrm{SO}}({\bf 3_{1}};x,w,v,z,a,t)$
$\displaystyle=\log\left(\tfrac{wx^{3}}{z}\right)\log a-\log v\log x+(\log
x)^{2}+\log w\log(vx)$ (22) $\displaystyle+2\log\left(\tfrac{wx}{v}\right)\log
t-\tfrac{3\pi^{2}}{6}-\operatorname{Li}_{2}(x)+\operatorname{Li}_{2}(v)+\operatorname{Li}_{2}(zv^{-1})$
$\displaystyle+\operatorname{Li}_{2}(w)+\operatorname{Li}_{2}(xz^{-1}w^{-1})+\operatorname{Li}_{2}(-a^{2}t^{3}x)-\operatorname{Li}_{2}(-a^{2}t^{3}xv)$
$\displaystyle+\operatorname{Li}_{2}(-t)-\operatorname{Li}_{2}(-txz^{-1})+\operatorname{Li}_{2}(-at)-\operatorname{Li}_{2}(-atw)~{},$

where we introduce the variables as $w=q^{2i}$, $v=q^{2j}$ and $z=q^{2k}$.
Obviously, the $t=-1$ specialization leads to the large color limit of colored
Kauffman polynomial

$\widetilde{\mathscr{W}}^{\textrm{SO}}({\bf
3_{1}};x,w,v,z,a,t=-1)=\widetilde{\mathcal{W}}^{\textrm{SO}}({\bf
3_{1}};x,w,v,z,a)~{}.$ (23)

On the other hand, at the $t=-q^{-2}$ specialization, the expression
(LABEL:triple-Kauffman-trefoil) collapses to $k=r$ and $i=0$ due to the term
$(-q^{2}t;q^{2})_{r-k}$ in the summand so that it is written by the summation
only over $j$. Correspondingly, the twisted superpotential encodes the large
color limit of colored HOMFLY-PT polynomial

$\widetilde{\mathscr{W}}^{SO}({\bf
3_{1}};x,w=1,v,z=x,a,t=-1)=\widetilde{\mathcal{W}}^{\textrm{SU}}({\bf
3_{1}};x,v,a)~{}.$ (24)

The saddle point equations (15) and (16) yield the classical
super-$A$-polynomial of $\operatorname{SO}$-type of the trefoil
(Nawata:2013mzx, , (6.19)), and its $t=1$ specialization can be consequently
written as

$\displaystyle\mathscr{A}^{\operatorname{SO}}(\mathbf{3_{1}};x,y,a,t=-1)=(1-ax)A^{\operatorname{SO}}(\mathbf{3_{1}};x,y,a)A^{\operatorname{SU}}(\mathbf{3_{1}};x,y,a)~{},$
(25)

where

$\displaystyle A^{\operatorname{SO}}(\mathbf{3_{1}};x,y,a)$
$\displaystyle=a^{7}x^{7}-a^{7}x^{6}-a^{2}xy+y~{},$ (26) $\displaystyle
A^{\operatorname{SU}}(\mathbf{3_{1}};x,y,a)$
$\displaystyle=y^{2}-ya^{2}+xya^{2}-2x^{2}ya^{2}-xy^{2}a^{2}-x^{3}a^{4}+x^{4}a^{4}+2x^{2}ya^{4}+x^{3}ya^{4}-x^{4}ya^{6}.$

(For $A^{\operatorname{SU}}(\mathbf{3_{1}};x,y,a)$, see (Fuji:2012nx, ,
(2.25)) with $a\to a^{2}$.) It is evident that it satisfies Conjecture 2.1.

In principle, we can compute super-$A$-polynomials of SO-type for the double
twist knots using the result in §2.1. However, it is beyond the computational
capabilities of current desktops to solve the saddle point equations (15) and
(16) for them. However, since the colored Kauffman homology of a double twist
knot is endowed with the universal differential $d^{\text{univ}}_{\to}$, we
can easily check the relations corresponding to (23) and (24) at the level of
the twisted superpotentials for the double twist knots. In this way, we can
confirm Conjecture 2.1 for the double twist knots.

Note that the diagonal differentials (Nawata:2013mzx, , §4.6) also relate
colored Kauffman and HOMFLY-PT homology. However, they do not bring us any
interesting results once we take large color limits.

##  3 $\operatorname{SO}(N)$ quantum $6j$-symbols

$6j$-symbols are ubiquitous in physics and mathematics landau1958quantum ;
1965Quantum ; biedenharn1981angular ; butler1981point ;
varshalovich1988quantum , and they reveal very rich symmetries. In particular,
quantum knot invariants can be evaluated by using quantum $6j$-symbols
Kirillov:1991ec ; Moore:1988qv ; Moore:1989vd . Despite their importance and
versatility, it is difficult to evaluate them in higher ranks. Obtaining a
closed-form expression is moreover indispensable to study their symmetries as
in 1965Quantum ; Alekseev:2019nzw .

Kauffman polynomials (homology) colored by symmetric representations for non-
torus knots obtained in §2.1 in principle encode the information about
$\operatorname{SO}(N)$ quantum $6j$-symbols. Hence, we attempt to extract the
information in this section. First, we quantize $\operatorname{SO}(N)$
$6j$-symbols with all symmetric representations obtained by Ališauskas
alisauskas1987some ; Alisauskas_2002 . Next, we give some evidence to its
validity by using representation theory and computing Kauffman polynomials.

###  3.1 $\operatorname{SO}(N)$ quantum $6j$-symbols for symmetric
representations

Multiplicity-free $6j$-symbols of a Lie group $G$ take the form

$\begin{Bmatrix}R_{1}&R_{2}&R_{12}\\\ R_{3}&R_{4}&R_{23}\end{Bmatrix}_{G}~{},$

where the arguments are representations of $G$, and they obey the fusion rule,
$R_{12}\in(R_{1}\otimes R_{2})\cap(\overline{R}_{3}\otimes\overline{R}_{4})$,
$R_{23}\in(R_{2}\otimes R_{3})\cap(\overline{R}_{1}\otimes\overline{R}_{4})$.

Ališauskas has obtained the closed-form expression for classical
$\operatorname{SO}(N)$ $6j$-symbols with all representations symmetric
alisauskas1987some ; Alisauskas_2002 . Kirillov and Reshetikhin naturally
quantized the classical $6j$-symbols of $\operatorname{SU}(2)$ Kirillov:1991ec
by bringing the hypergeometric series ${}_{4}F_{3}$ to ${}_{4}\varphi_{3}$.
Therefore, we conjecture that we can naturally quantize Ališauskas’ results to
a closed-form expression of quantum $\operatorname{SO}(N)$ ($N\geq 4$)
$6j$-symbols for all symmetric representations,

$\displaystyle\begin{Bmatrix}a&b&e\\\
d&c&f\end{Bmatrix}_{\operatorname{SO}(N)}=$
$\displaystyle\left(\frac{[2c+N-2]_{q}[2d+N-2]_{q}[2e+N-2]_{q}}{[2]_{q}^{3}\dim_{q}[c]\dim_{q}[d]\dim_{q}[e]}\right)^{1/2}$
(27) $\displaystyle\times\begin{pmatrix}c&d&e\\\
0&0&0\end{pmatrix}_{N}^{-1}\sum_{l^{\prime}}(-1)^{(c+d-e)/2+f+N+l^{\prime}}\left[2l^{\prime}+N-3\right]_{q}$
$\displaystyle\times\begin{Bmatrix}\frac{1}{2}b&\frac{1}{2}f+\frac{1}{4}N-1&\frac{1}{2}d+\frac{1}{4}N-1\\\
\frac{1}{2}f+\frac{1}{4}N-1&\frac{1}{2}(b+N)-2&l^{\prime}+\frac{1}{2}N-2\end{Bmatrix}_{\operatorname{SU}(2)}$
$\displaystyle\times\begin{Bmatrix}\frac{1}{2}a&\frac{1}{2}f+\frac{1}{4}N-1&\frac{1}{2}c+\frac{1}{4}N-1\\\
\frac{1}{2}f+\frac{1}{4}N-1&\frac{1}{2}(a+N)-2&l^{\prime}+\frac{1}{2}N-2\end{Bmatrix}_{\operatorname{SU}(2)}$
$\displaystyle\times\begin{Bmatrix}\frac{1}{2}a&\frac{1}{2}b+\frac{1}{4}N-1&\frac{1}{2}e+\frac{1}{4}N-1\\\
\frac{1}{2}b+\frac{1}{4}N-1&\frac{1}{2}(a+N)-2&l^{\prime}+\frac{1}{2}N-2\end{Bmatrix}_{\operatorname{SU}(2)}$
$\displaystyle\times\left(\frac{[l^{\prime}]_{q}![N-3]_{q}!}{\left[l^{\prime}+N-4\right]_{q}!}\right)^{1/2},$

where the curly braces on the right hand side are the quantum $6j$-symbols of
$\operatorname{SU}(2)$ Kirillov:1991ec . Since we are only dealing with all
symmetric representations, $a,b,c,d,e,f$ are the number of boxes in the
corresponding Young tableau of one row. Here a quantum number is defined as

$[x]_{q}=\frac{q^{x}-q^{-x}}{q-q^{-1}}~{}.$ (28)

Also, a $q$-factorial is defined as the product of quantum numbers

$[x]_{q}!=[x]_{q}[x-1]_{q}\cdots[x-\lfloor x\rfloor]_{q}~{},$ (29)

where $\lfloor x\rfloor$ is the floor of $x$ so that the definition of
$q$-factorial works for both integer and half-integer arguments. The quantum
dimension of the $\operatorname{SO}(N)$ symmetric representation $[l]$ is
given by

$\dim_{q}[l]=\frac{[l+N-3]_{q}!}{[l]_{q}![N-2]_{q}!}\left([l+N-2]_{q}+[l]_{q}\right),$
(30)

and the special quantum $3j$-symbols are defined as

$\displaystyle\begin{pmatrix}l_{1}&l_{2}&l_{3}\\\ 0&0&0\end{pmatrix}_{N}=$
$\displaystyle\frac{1}{[N/2-1]_{q}!}\left(\frac{[J+N-3]_{q}!}{[N-3]_{q}![J+N/2-1]_{q}!}\right.$
(31)
$\displaystyle\left.\times\prod_{i=1}^{3}\frac{\left[2l_{i}+N-2\right]_{q}\left[J-l_{i}+N/2-2\right]_{q}!}{[2]_{q}\dim_{q}[l_{i}]\left[J-l_{i}\right]_{q}!}\right)^{1/2},$

where $J=\frac{1}{2}(l_{1}+l_{2}+l_{3})$.

Note that $R_{12}$, $R_{23}$ can be non-symmetric representations even when
$R_{1}=R_{2}=R_{3}=R_{4}$ are symmetric representations. (Some fusion rules
can be found in (36).) Furthermore, there are several relations from
representation theory, which would lead to identities among the quantum
$6j$-symbols.

  * •

The isomorphism of representations,
$(\mathfrak{so}_{6},[r])\simeq(\mathfrak{sl}_{4},[r,r])$, would lead to
equations,

$\begin{Bmatrix}[r]&[r]&[2l-j,j]\cr[r]&[r]&[2l^{\prime}-j^{\prime},j^{\prime}]\end{Bmatrix}_{\operatorname{SO}(6)}=\begin{Bmatrix}[r,r]&[r,r]&[2l,2l-j,j]\cr[r,r]&[r,r]&[2l^{\prime},2l^{\prime}-j^{\prime},j^{\prime}]\end{Bmatrix}_{\operatorname{SU}(4)},$
(32)

where $0\leq l,l^{\prime}\leq r$, $0\leq j\leq l$, $0\leq j^{\prime}\leq
l^{\prime}$. We have checked that this property holds using
$\operatorname{SU}(4)$ quantum $6j$-symbols in Mironov:2016coi , and the
result in the next subsection. Since now we only have the closed form
expression of $\operatorname{SO}(N)$ quantum $6j$-symbols for symmetric
representations, by setting $j=j^{\prime}=0$, the above equation reduces to

$\begin{Bmatrix}[r]&[r]&[2l]\cr[r]&[r]&[2l^{\prime}]\end{Bmatrix}_{\operatorname{SO}(6)}=\begin{Bmatrix}[r,r]&[r,r]&[2l,2l]\cr[r,r]&[r,r]&[2l^{\prime},2l^{\prime}]\end{Bmatrix}_{\operatorname{SU}(4)},$
(33)

where $0\leq l\leq r$. Thus we can easily obtain $\operatorname{SU}(4)$
quantum $6j$-symbols of such type, using our closed formula (27).

  * •

Also, there is an isomorphism between Lie algebras,
$\mathfrak{so}_{4}\simeq\mathfrak{sl}_{2}\oplus\mathfrak{sl}_{2}$. Restricting
ourselves to the case where all representations are symmetric, we have

$\begin{Bmatrix}[r]&[r]&[2l]\cr[r]&[r]&[2l^{\prime}]\end{Bmatrix}_{\operatorname{SO}(4)}=\left(\begin{Bmatrix}[r]&[r]&[2l]\cr[r]&[r]&[2l^{\prime}]\end{Bmatrix}_{\operatorname{SU}(2)}\right)^{2},$
(34)

where $0\leq l,l^{\prime}\leq r$.

  * •

Although our formula only works for $\operatorname{SO}(N)$ ($N\geq 4$), it’s
still worthy to note that there is another isomorphism of representations,
$(\mathfrak{so}_{3},[r])\simeq(\mathfrak{sl}_{2},[2r])$. This leads to

$\begin{Bmatrix}[r]&[r]&[2l]\cr[r]&[r]&[2l^{\prime}]\end{Bmatrix}_{\operatorname{SO}(3)}\bigg{|}_{q\to
q^{2}}=\begin{Bmatrix}[2r]&[2r]&[4l]\cr[2r]&[2r]&[4l^{\prime}]\end{Bmatrix}_{\operatorname{SU}(2)},$
(35)

where $0\leq l,l^{\prime}\leq r$.

###  3.2 $\operatorname{SO}(N)$ fusion matrices for symmetric representations

Since the fusion rule of $\operatorname{SO}(N)$ symmetric representations is
given by

$[r]\otimes[r]=\bigoplus_{l=0}^{r}\bigoplus_{j=0}^{l}[2l-j,j]~{},$ (36)

we need full $\operatorname{SO}(N)$ fusion matrices including the case that
$R_{12}$ and $R_{23}$ are non-symmetric to evaluate Kauffman polynomials
colored by symmetric representations,. At a root of unity $q$, the WZNW fusion
matrix and the corresponding quantum $6j$-symbol are related by

$a_{R_{12}R_{23}}\left[\begin{array}[]{cc}R_{1}&R_{2}\\\
R_{3}&R_{4}\end{array}\right]=\epsilon_{\left\\{R_{i}\right\\}}\sqrt{\operatorname{dim}_{q}{R_{12}}\operatorname{dim}_{q}{R_{23}}}\begin{Bmatrix}R_{1}&R_{2}&R_{12}\\\
R_{3}&R_{4}&R_{23}\end{Bmatrix},$ (37)

where $\epsilon_{\\{R_{i}\\}}=\pm 1$ and $\operatorname{dim}_{q}R$ is the
quantum dimension of the representation $R$. Since $\operatorname{SO}(N)$
representations are real, for colored knot invariants, we only need to
consider fusion matrix of the type
$a_{ts}\begin{bmatrix}\begin{smallmatrix}R&R\\\
R&R\end{smallmatrix}\end{bmatrix}$, where $t,s\in R\otimes R$.

To obtain full fusion matrices, we can make use of their properties and
symmetries 1965Quantum

  * •

The fusion matrix is symmetric,

$a_{ts}\begin{bmatrix}R&R\\\ R&R\end{bmatrix}=a_{st}\begin{bmatrix}R&R\\\
R&R\end{bmatrix}.$ (38)

  * •

The fusion matrix is orthonormal,

$\sum_{s}a_{ts}\begin{bmatrix}R&R\\\
R&R\end{bmatrix}a_{sm}\begin{bmatrix}R&R\\\ R&R\end{bmatrix}=\delta_{tm}.$
(39)

  * •

The entries in the row and column, which corresponds to the trivial
representation, are known,

$a_{\emptyset s}\begin{bmatrix}R&R\\\
R&R\end{bmatrix}=\epsilon_{s}^{R}\frac{\sqrt{\operatorname{dim}_{q}s}}{\operatorname{dim}_{q}R},$
(40)

where $\emptyset$ is the trivial representation.

Note that $6j$-symbols enjoy other symmetries, such as the Racah identity and
the pentagon (Biedenharn-Elliot) identity, which we do not use in this paper.
We already know the entries when $t$ and $s$ are symmetric representations
(27), and the entries corresponding to the trivial representation (40). Then
we can solve the rest entries by using the fact that the fusion matrix is
symmetric and orthonormal. In this paper, we will present our results of the
$\operatorname{SO}(N)$ fusion matrices
$a_{ts}\begin{bmatrix}\begin{smallmatrix}R&R\\\
R&R\end{smallmatrix}\end{bmatrix}$ for $R=~{}$ ,  .

1\. $R$ =

$\begin{Bmatrix}R&R&t\\\ R&R&s\end{Bmatrix}=\frac{1}{\operatorname{dim}_{q}R}$
$\left({\begin{tabular}[]{c|c c
c}{}\hfil&$s=\emptyset\quad$&${\vbox{\vbox{\offinterlineskip\vbox{\hrule
height=0.3pt\hbox{\vrule height=4.32pt,width=0.3pt,depth=1.07997pt\hbox
to5.4pt{\hfil}\vrule height=4.32pt,width=0.3pt,depth=1.07997pt} \hrule
height=0.3pt}\vspace{-\y@linethick} \vbox{\hrule height=0.3pt\hbox{\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt} \hrule
height=0.3pt}\vspace{-\y@linethick}
}}}$&${\vbox{\vbox{\offinterlineskip\vbox{\hrule height=0.3pt\hbox{\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt} \hrule
height=0.3pt}\vspace{-\y@linethick} }}}$\\\
\hline\cr{$t=\emptyset$}&$1$&$1$&$1$\\\
{${\vbox{\vbox{\offinterlineskip\vbox{\hrule height=0.3pt\hbox{\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt} \hrule
height=0.3pt}\vspace{-\y@linethick} \vbox{\hrule height=0.3pt\hbox{\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt} \hrule
height=0.3pt}\vspace{-\y@linethick}
}}}$}&$1$&$\frac{[N]_{q}}{[N-1]_{q}([N-2]_{q}+[2]_{q})}$&$-\frac{1}{[N-1]_{q}}$\\\
{${\vbox{\vbox{\offinterlineskip\vbox{\hrule height=0.3pt\hbox{\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt} \hrule
height=0.3pt}\vspace{-\y@linethick}
}}}$}&$1$&$-\frac{1}{[N-1]_{q}}$&$\frac{[N-2]_{q}}{[N-1]_{q}([N]_{q}+[2]_{q})}$\\\
\end{tabular}}\right)$ ,

where the quantum dimension $\operatorname{dim}_{q}R={[N-1]_{q}+1}$.

2\. $R$ =

$\begin{Bmatrix}R&R&t\\\ R&R&s\end{Bmatrix}=\frac{1}{\operatorname{dim}_{q}R}$
$\left({\begin{tabular}[]{c|c c c c c
c}{}\hfil&$s=\emptyset$&${\vbox{\vbox{\offinterlineskip\vbox{\hrule
height=0.3pt\hbox{\vrule height=4.32pt,width=0.3pt,depth=1.07997pt\hbox
to5.4pt{\hfil}\vrule height=4.32pt,width=0.3pt,depth=1.07997pt} \hrule
height=0.3pt}\vspace{-\y@linethick} \vbox{\hrule height=0.3pt\hbox{\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt} \hrule
height=0.3pt}\vspace{-\y@linethick}
}}}$&${\vbox{\vbox{\offinterlineskip\vbox{\hrule height=0.3pt\hbox{\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt} \hrule
height=0.3pt}\vspace{-\y@linethick}
}}}$&${\vbox{\vbox{\offinterlineskip\vbox{\hrule height=0.3pt\hbox{\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt} \hrule
height=0.3pt}\vspace{-\y@linethick} \vbox{\hrule height=0.3pt\hbox{\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt} \hrule
height=0.3pt}\vspace{-\y@linethick}
}}}$&${\vbox{\vbox{\offinterlineskip\vbox{\hrule height=0.3pt\hbox{\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt} \hrule
height=0.3pt}\vspace{-\y@linethick} \vbox{\hrule height=0.3pt\hbox{\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt} \hrule
height=0.3pt}\vspace{-\y@linethick}
}}}$&${\vbox{\vbox{\offinterlineskip\vbox{\hrule height=0.3pt\hbox{\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
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height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt} \hrule
height=0.3pt}\vspace{-\y@linethick} }}}$\\\
\hline\cr{$t=\emptyset$}&$1$&$1$&$1$&$1$&$1$&$1$\\\
{${\vbox{\vbox{\offinterlineskip\vbox{\hrule height=0.3pt\hbox{\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
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height=0.3pt}\vspace{-\y@linethick} \vbox{\hrule height=0.3pt\hbox{\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt} \hrule
height=0.3pt}\vspace{-\y@linethick}
}}}$}&$1$&$b_{11}$&$b_{12}$&$b_{13}$&$b_{14}$&$b_{15}$\\\
{${\vbox{\vbox{\offinterlineskip\vbox{\hrule height=0.3pt\hbox{\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
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height=4.32pt,width=0.3pt,depth=1.07997pt} \hrule
height=0.3pt}\vspace{-\y@linethick}
}}}$}&$1$&$b_{21}$&$b_{22}$&$b_{23}$&$b_{24}$&$b_{25}$\\\
{${\vbox{\vbox{\offinterlineskip\vbox{\hrule height=0.3pt\hbox{\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
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height=4.32pt,width=0.3pt,depth=1.07997pt} \hrule
height=0.3pt}\vspace{-\y@linethick} \vbox{\hrule height=0.3pt\hbox{\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
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height=4.32pt,width=0.3pt,depth=1.07997pt} \hrule
height=0.3pt}\vspace{-\y@linethick}
}}}$}&$1$&$b_{31}$&$b_{32}$&$b_{33}$&$b_{34}$&$b_{35}$\\\
{${\vbox{\vbox{\offinterlineskip\vbox{\hrule height=0.3pt\hbox{\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
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height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt} \hrule
height=0.3pt}\vspace{-\y@linethick} \vbox{\hrule height=0.3pt\hbox{\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt} \hrule
height=0.3pt}\vspace{-\y@linethick}
}}}$}&$1$&$b_{41}$&$b_{42}$&$b_{43}$&$b_{44}$&$b_{45}$\\\
{${\vbox{\vbox{\offinterlineskip\vbox{\hrule height=0.3pt\hbox{\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt\hbox to5.4pt{\hfil}\vrule
height=4.32pt,width=0.3pt,depth=1.07997pt} \hrule
height=0.3pt}\vspace{-\y@linethick}
}}}$}&$1$&$b_{51}$&$b_{52}$&$b_{53}$&$b_{54}$&$b_{55}$\\\
\end{tabular}}\right)$ ,

where the quantum dimension
$\operatorname{dim}_{q}R=[N-1]_{q}+[N-1]_{q}[N]_{q}/[2]_{q}$, and

$\displaystyle
b_{11}=\frac{-q^{4}-q^{N}-q^{2+N}+q^{4+N}-q^{2+2N}+q^{4+2N}+q^{6+2N}+q^{2+3N}}{(1+q^{2})(-1+q^{N})(q^{4}+q^{N}+q^{2N}+q^{4+N})},$
$\displaystyle b_{12}=b_{21}=\frac{q^{N+2}+1}{q^{N+2}+q^{N}+q^{2}+1},$ (41)
$\displaystyle
b_{13}=b_{31}=\frac{q^{N}\left(q^{2}-1\right)\left(q^{N+2}+1\right)}{q^{3N}+q^{2N+4}-q^{N}-q^{4}},$
(42) $\displaystyle
b_{14}=b_{41}=\frac{\left(q^{2}-1\right)^{2}q^{N}}{q^{2N}+q^{N+4}+q^{N}+q^{4}},$
(43) $\displaystyle
b_{15}=b_{51}=\frac{\left(q^{4}-1\right)q^{N-2}}{1-q^{2N}},$ (44)
$\displaystyle
b_{22}=\frac{-q^{2N}\\!+\\!q^{N+2}\\!+\\!3q^{N+4}\\!-\\!2q^{N+6}\\!-\\!q^{N+8}\\!-\\!2q^{2N+2}\\!+\\!3q^{2N+4}\\!+\\!q^{2N+6}\\!-\\!2q^{2N+8}\\!+\\!q^{3N+6}\\!-\\!2q^{N}\\!+\\!q^{2}}{\left(q^{2}+1\right)\left(q^{N}+1\right)\left(q^{N}-q^{2}\right)\left(q^{N+4}-1\right)},$
(45) $\displaystyle
b_{23}=b_{32}=-\frac{\left(q^{2}-1\right)q^{N}}{\left(q^{N}+1\right)\left(q^{N}-q^{2}\right)},$
(46) $\displaystyle
b_{24}=b_{42}=\frac{q^{N}\left(q^{2}-1\right)^{2}\left(q^{N+2}+1\right)}{\left(q^{2}-q^{N}\right)\left(q^{N+4}+q^{2N+4}-q^{N}-1\right)},$
(47) $\displaystyle
b_{25}=b_{52}=\frac{\left(q^{4}-1\right)q^{N}}{q^{N+4}+q^{2N+4}-q^{N}-1},$
(48) $\displaystyle
b_{33}=\frac{q^{2N}\left(q^{2}\\!-\\!1\right)^{2}\left(q^{2}\\!+\\!1\right)\left(q^{2N}\\!-\\!q^{N\\!+\\!2}\\!+\\!q^{N\\!+\\!6}\\!-\\!q^{2N\\!+\\!2}\\!-\\!2q^{2N\\!+\\!4}\\!-\\!q^{2N\\!+\\!6}\\!+\\!q^{2N\\!+\\!8}\\!+\\!q^{3N\\!+\\!2}\\!-\\!q^{3N\\!+\\!6}\\!+\\!q^{4N\\!+\\!4}\\!+\\!q^{4}\right)}{\left(q^{N}\\!-\\!1\right)\left(q^{N}\\!+\\!1\right)\left(q^{N}\\!-\\!q^{2}\right)\left(q^{N}\\!+\\!q^{4}\right)\left(q^{2N}\\!-\\!q^{6}\right)\left(q^{2N+2}\\!-\\!1\right)},$
(49) $\displaystyle
b_{34}=b_{43}=-\frac{q^{2N}\left(q^{2}-1\right)^{2}\left(q^{2}+1\right)\left(q^{N+2}+1\right)}{\left(q^{N}+1\right)\left(q^{2N}-q^{N+2}+q^{N+4}-q^{6}\right)\left(q^{2N+2}-1\right)},$
(50) $\displaystyle
b_{35}=b_{53}=\frac{\left(q^{2}-1\right)^{2}\left(q^{2}+1\right)q^{2N-2}}{\left(q^{2N}-1\right)\left(q^{2N+2}-1\right)},$
(51) $\displaystyle
b_{44}=\frac{q^{2N}\left(q^{2}-1\right)^{2}\left(q^{2}+1\right)\left(2q^{N+2}-2q^{N+4}+q^{N+6}+q^{2N+4}-q^{N}-q^{2}\right)}{\left(q^{2N}-q^{N+2}+q^{N+4}-q^{6}\right)\left(q^{2N+2}-1\right)\left(q^{N+4}+q^{2N+4}-q^{N}-1\right)},$
(52) $\displaystyle
b_{45}=b_{54}=-\frac{\left(q^{2}-1\right)^{2}\left(q^{2}+1\right)q^{2N}}{\left(q^{2N+2}-1\right)\left(q^{N+4}+q^{2N+4}-q^{N}-1\right)},$
(53) $\displaystyle
b_{55}=\frac{q^{2N+2}\left(q^{2}-1\right)^{2}\left(q^{2}+1\right)\left(q^{N}-q^{2}\right)}{\left(q^{N+6}-1\right)\left(q^{2N+2}-1\right)\left(q^{N+4}+q^{2N+4}-q^{N}-1\right)}.$

Using these two $\operatorname{SO}(N)$ fusion matrices and braiding operator
eigenvalues, we have computed colored Kauffman polynomials for double twist
knots when $R\\!=$ ,  . The results coincide with
$\mathscr{F}_{[r]}(K;a=q^{N-1},q,t=-1)$ obtained in §2.1, which comes from the
Poincaré polynomial of $[r^{2}]$-colored HOMFLY-PT homology (Kameyama:2019mdf,
, (3.4)) by using the relation (10).

##  4 Deformed $F_{K}$ and ADO polynomials

In the study of the 3d/3d correspondence, a new invariant $\widehat{Z}$ of a
closed 3-manifold was proposed as a BPS $q$-series Gukov:2016gkn ;
Gukov:2017kmk . To incorporate TQFT methods and surgery formula, an analogous
series $F_{K}$ for a knot complement $S^{3}\backslash K$ Gukov:2019mnk  was
introduced. Although a mathematical definition of $\hat{Z}$ or $F_{K}$ has not
been fleshed out yet, there are many approaches to understand them. A recipe
to compute $\widehat{Z}$ for a plumbed 3-manifold with a negative definite
linking matrix is given in Gukov:2017kmk . We can also use resurgence to
calculate $\widehat{Z}$ or $F_{K}$ Gukov:2016njj ; Gukov:2019mnk . Also, for a
positive braid knot $K$, $F_{K}(x,q)$ can be computed using the $R$-matrix for
Verma modules Park:2020edg . In particular, closed-form expressions of $F_{K}$
for the double twist knots with positive braids are given in Park:2020edg ,
based on the formulae of colored Jones polynomials lovejoy2017colored ;
lovejoy2019colored . The expressions are compatible with Habiro’s cyclotomic
expansion Habiro:2008 . In this section, we will generalize the results of
Park:2020edg  to the $(a,t)$-deformed case Ekholm:2020lqy , and we will see by
explicit computations that the ADO invariants Akutsu:1991xc ;
willetts2020unification ; Gukov:2020lqm  are compatible with the cyclotomic
expansions.

###  4.1 Deformed $F_{K}$ of double twist knots

We study the double twist knots, $K_{s,t}~{}(s,t\in\frac{1}{2}\mathbb{Z})$,
whose notation and the corresponding knots in Rolfsen table are shown in
figure 2. Among them, two classes possess only positive braids.

2s2t

$\begin{array}[]{c | c}\textrm{Knot}&\textrm{Name}\\\ \hline\cr{K_{1,1}}&{3_{1}}\\\ {K_{2,1}}&{5_{2}}\\\ {K_{3,1}}&{7_{2}}\\\ {K_{\frac{3}{2},-2}}&{7_{3}}\\\ {K_{2,2}}&{7_{4}}\\\ {K_{4,1}}&{9_{2}}\\\ {K_{\frac{3}{2},-3}}&{9_{3}}\\\ {K_{\frac{5}{2},-2}}&{9_{4}}\\\ {K_{3,2}}&{9_{5}}\end{array}$

Figure 2: Double twist knot $K_{s,t}$ and corresponding knots in Rolfsen
table.

### Double twist knot $K_{m,n}~{}(m,n\in\mathbb{Z}_{+})$

The cyclotomic expansions of the $[r]$-colored superpolynomials for the double
twist knots $K_{m,n}~{}(m,n\in\mathbb{Z}_{+})$ are given in Nawata:2012pg ;
Gukov:2015gmm . In fact, by substitution $q^{2r}=x$ and changing the upper
limit from $r$ to $\infty$, we can obtain the corresponding $(a,t)$-deformed
$F_{K}$:

$F_{K_{m,n}}(x,a,q,t)=(-tx)^{\log_{q}(-at)-1}\sum_{k=0}^{\infty}g_{K_{m,n}}^{(k)}(x,a,q,t),$
(54)

where

$\displaystyle g_{K_{m,n}}^{(k)}(x,a,q,t)=~{}$
$\displaystyle\sum_{j=0}^{k}\frac{\left(x;q^{-1}\right)_{k}\left(-at/q;q\right)_{k}\left(-axt^{3};q\right)_{j}}{(q;q)_{k}}$
(55)
$\displaystyle\times(-1)^{j}\,x^{k-j}\,a^{-j}\,q^{k}\,t^{-3j+2k}\begin{bmatrix}k\\\
j\end{bmatrix}_{q}{\rm Tw}_{K_{m,n}}^{(j)}(a,q,t).$ (56)

The twist factor is defined as

${\rm
Tw}_{K_{m,n}}^{(j)}(a,q,t)=\sum_{l=0}^{j}(-1)^{l}\,q^{\frac{1}{2}l(l+1)-jl}\,{\rm
tw}_{m}^{(l)}(a,q,t)\,{\rm tw}_{n}^{(l)}(a,q,t)\begin{bmatrix}j\\\
l\end{bmatrix}_{q},$ (57)

with

${\rm tw}_{m}^{(l)}(a,q,t)=\sum_{0\leq b_{1}\leq\cdots\leq b_{m-1}\leq
b_{m}=l}\prod_{i=1}^{m-1}\left(a^{b_{i}}q^{b_{i}(b_{i}-1)}t^{2b_{i}}\right){b_{i+1}\brack
b_{i}}_{q}.$ (58)

Note that, to connect the standard notation of $F_{K}$ and the ADO invariants,
we rescale the variables $a\to a^{1/2}$ and $q\to q^{1/2}$ from the
superpolynomial (11)

$F_{K_{m,n}}(x=q^{r},a,q,t)=\mathscr{P}_{[r]}(K_{m,n};a^{1/2},q^{1/2},t)~{}.$

When we write the $(a,t)$-deformed $F_{K}$ as a four-variable series
explicitly, we would usually omit the prefactor $(-tx)^{\log_{q}(-at)-1}$.
However, we have to keep it to verify the properties (65), (66), (67) of
$(a,t)$-deformed $F_{K}$. If we take $a=-t^{-1}q^{N}$, the prefactor simply
reduces to $(-tx)^{N-1}$.

### Double twist knot $K_{m+\frac{1}{2},-n}~{}(m,n\in\mathbb{Z}_{+})$

Similarly, we can find the cyclotomic expansions of the $[r]$-colored
superpolynomial for double twist knot
$K_{m+\frac{1}{2},-n}~{}(m,n\in\mathbb{Z}_{+})$ by using the structural
properties of HOMFLY-PT homology. Then, we obtain the $(a,t)$-deformed $F_{K}$
for it:

$F_{K_{m+\frac{1}{2},-n}}(x,a,q,t)=(-tx^{n})^{\log_{q}(-at)-1}\sum_{k=0}^{\infty}g_{K_{m+\frac{1}{2},-n}}^{(k)}(x,a,q,t),$
(59)

where

$\displaystyle g^{(k)}_{K_{m+\frac{1}{2},-n}}(x,a,q,t)=~{}$
$\displaystyle\sum_{j=0}^{k}\frac{\left(x;q^{-1}\right)_{k}\left(-at/q;q\right)_{k}\left(-axt^{3};q\right)_{j}}{(q;q)_{k}}$
(60) $\displaystyle\times x^{k-j}\,\begin{bmatrix}k\\\
j\end{bmatrix}_{q}\,q^{\frac{1}{2}j(j-1)+k}\,t^{-j+2k}\,{\rm
tw}_{m}^{(j)}(a,q,t)\,\\!\mathbbm{t}\\!\mathbbm{w}_{n}^{(k)}(x,q,t).$ (61)

A new twist factor is defined as

$\displaystyle\\!\mathbbm{t}\\!\mathbbm{w}_{n}^{(k)}(x,q,t)=$
$\displaystyle\sum_{0\leq b_{1}\leq\cdots\leq b_{n-1}\leq
b_{n}=k}\prod_{i=1}^{n-1}\left(x^{2b_{i}}\,q^{-b_{i}(b_{i+1}-1)}\,t^{2b_{i}}\right){b_{i+1}\brack
b_{i}}_{q}.$ (62)

The prefactor becomes simply $(-tx^{n})^{N-1}$ when taking $a=-t^{-1}q^{N}$.

### Properties of $(a,t)$-deformed $F_{K}$

The formulae above for $F_{K}$ of the double twist knots become the usual
$F_{K}$ in Park:2020edg  up on the specialization $a=q^{2},t=-1$. If we
further take $a=q^{N},t=-1$, it reduces to
$F_{K}^{\operatorname{SU}(N),sym}(x,q)$ Park:2019xey , which is the $F_{K}$
with gauge group $G=\operatorname{SU}(N)$ associated with symmetric
representations.

It is conjectured in Ekholm:2020lqy  that the $(a,t)$-deformed $F_{K}$ has the
following property,

$\hat{A}(K;\hat{x},\hat{y},a,q,t)F_{K}(x,a,q,t)=0.$ (63)

We have studied the classical super-$A$-polynomial in §2.2, and
$\hat{A}(K;\hat{x},\hat{y},a,q,t)$ is its quantum version Nawata:2012pg ,
which annihilates the colored superpolynomial of $K$. The operators in the
quantum super-$A$-polynomial act on $(a,t)$-deformed $F_{K}$ as

$\displaystyle\hat{x}F_{K}(x,a,q,t)=xF_{K}(x,a,q,t),\qquad\hat{y}F_{K}(x,a,q,t)=F_{K}(xq,a,q,t).$
(64)

In the case of $3_{1},5_{2},7_{2}$, the conjectural $(a,t)$-deformed $F_{K}$
can be annihilated by the corresponding super-$A$-polynomial in Nawata:2012pg
, the detail can be seen in the Mathematica file.

It is proposed in Ekholm:2020lqy  that $(a,t)$-deformed $F_{K}$ satisfy the
following properties:

$\displaystyle F_{K}(x,-t^{-1},q,t)$ $\displaystyle=\Delta_{K}(x,t),$ (65)
$\displaystyle F_{K}(x,-t^{-1}q,q,t)$ $\displaystyle=1,$ (66)
$\displaystyle\lim_{q\to 1}F_{K}(x,-t^{-1}q^{N},q,t)$
$\displaystyle=\frac{1}{\Delta_{K}(x,t)^{N-1}},$ (67)

where $\Delta_{K}(x,t)$ is the $t$-deformed Alexander polynomial introduced in
Ekholm:2020lqy , which can be obtained from the superpolynomial,

$\Delta_{K}(x,t)=\mathscr{P}_{{\vbox{\offinterlineskip\vbox{\hrule
height=0.3pt\hbox{\vrule height=3.52pt,width=0.3pt,depth=0.87997pt\hbox
to4.4pt{\hfil}\vrule height=3.52pt,width=0.3pt,depth=0.87997pt} \hrule
height=0.3pt}\vspace{-\y@linethick} }}}(K;a^{2}=-t^{-1},~{}q^{2}=x,~{}t).$
(68)

We have checked that our closed formulae of the $(a,t)$-deformed $F_{K}$ for
the double twist knots satisfy (65). Also, we give an analytical proof of (66)
and (67) in Appendix A.

###  4.2 $t$-deformed ADO polynomials

As studied in Gukov:2020lqm , at the limit
$q\rightarrow\zeta_{p}=e^{\frac{2\pi i}{p}}$, $F_{K}(x,q)$ is equal to the
$p$-th ADO polynomial Akutsu:1991xc  up to the Alexander polynomial
$\Delta_{K}(x^{p})$. For $(a,t)$-deformed $F_{K}$ at radial limit, it is
natural to consider a $t$-deformation of the $p$-th ADO polynomial. The formal
definition is given in Ekholm:2020lqy  as,

$\operatorname{ADO}_{K}(p;x,t)=\Delta_{K}(x^{p},-(-t)^{p})\lim_{q\to\zeta_{p}}F_{K}(x,-t^{-1}q^{2},q,t).$
(69)

In this subsection, we derive closed-form expressions of $t$-deformed ADO
polynomials of the double twist knots. As in Appendix A, it turns out that the
closed formulae (54) and (59) of cyclotomic type are suitable to compute the
ADO polynomial up on the limit $q\rightarrow\zeta_{p}=e^{\frac{2\pi i}{p}}$ in
which an infinite sum truncates to a finite one Anna:2020non . (See also
Banerjee:2020dqq  for a similar topic.)

First, we will obtain a closed-form expression of $t$-deformed Alexander
polynomials. Although $\Delta_{K}(x,t)$ can be easily calculated from the
superpolynomial by using (68), we compute it from (54) and (59) via the
substitution (65). Notice that both the formulae share a common term
$(-at/q;q)_{k}$, where $k$ runs from $0$ to $\infty$. Upon specialization
$a=-t^{-1}$, it becomes $(q^{-1};q)_{k}$ and vanishes for $k>1$. Therefore,
the infinite summation becomes a rather simple finite one. Subsequently, they
are given by

$\displaystyle\Delta_{K_{m,n}}(x,t)$
$\displaystyle=-\frac{1}{t}+\left(t+\frac{1}{t}-tx-\frac{1}{tx}\right)S_{m}(-t)S_{n}(-t)~{},$
(70) $\displaystyle\Delta_{K_{m+\frac{1}{2},-n}}(x,t)$
$\displaystyle=t^{n}[n]_{tx}-t^{n-1}[n+1]_{tx}-t^{n}[n]_{tx}\left(t+\frac{1}{t}-tx-\frac{1}{tx}\right)S_{m}(-t)~{},$
(71)

where $S_{l}(x)=\sum\limits_{i=0}^{l-1}x^{i}$, and the definition of quantum
number $[n]_{tx}$ is given in (28). We present our formulae this way, so that
the Weyl symmetry Ekholm:2020lqy  of the $t$-deformed Alexander polynomial
becomes manifest,

$\Delta_{K}(x^{-1},t)=\Delta_{K}(t^{-2}x,t)~{}.$ (72)

After detailed derivation shown in Appendix A, we obtain closed-from
expressions of the $t$-deformed ADO polynomials for double twist knots
$K_{m,n}$ and $K_{m+\frac{1}{2},-n}$,

$\displaystyle\operatorname{ADO}_{K_{m,n}}(p;x,t)$
$\displaystyle=(-tx)^{1-p}\sum_{k=0}^{p-1}\lim_{q\to\zeta_{p}}g_{K_{m,n}}^{(k)}(x,-t^{-1}q^{2},q,t),$
(73) $\displaystyle\operatorname{ADO}_{K_{m+\frac{1}{2},-n}}(p;x,t)$
$\displaystyle=(-tx^{n})^{1-p}\sum_{k=0}^{p-1}\lim_{q\to\zeta_{p}}g_{K_{m+\frac{1}{2},-n}}^{(k)}(x,-t^{-1}q^{2},q,t).$
(74)

Note that there is no dependence on $t$-deformed Alexander polynomial because
the radial limit of $(a,t)$-deformed $F_{K}$ is proportional to the inverse of
the factor $\Delta_{K}(x^{p},-(-t)^{p})$, which cancels the same factor in
(69). Especially, we can analytically derive that these formulae give rise to

$\displaystyle\operatorname{ADO}_{K}(p=1;x,t)$ $\displaystyle=1,$ (75)
$\displaystyle\operatorname{ADO}_{K}(p=2;x,t)$
$\displaystyle=\Delta_{K}(x,t),$ (76)

which implies that the $t$-deformed ADO polynomial is a generalization of
$t$-deformed Alexander polynomial.

In Table 1, 2, 3, we summarize the results of $t$-deformed ADO polynomials for
the trefoil, $5_{2}$ knot and $7_{2}$ knot.

$p$ | $\operatorname{ADO}_{3_{1}}(p;x,t)$
---|---
1 | 1
2 | $\left(-tx\right)+t+\left(-tx\right)^{-1}$
3 | $\zeta_{3}^{2}\left[\left(t\zeta_{3}x\right)^{2}+t\left(t\zeta_{3}x\right)+\left(t^{2}-\zeta_{3}^{-1}\right)+t\left(t\zeta_{3}x\right)^{-1}+\left(t\zeta_{3}x\right)^{-2}\right]$
Table 1: $t$-deformed ADO polynomials for the left-handed trefoil $p$ | $\operatorname{ADO}_{5_{2}}(p;x,t)$
---|---
1 | 1
2 | $(1-t)(-tx)-1+t-t^{2}+(1-t)(-tx)^{-1}$
3 | $\zeta_{3}^{2}\big{[}\left(1+t\zeta_{3}+t^{2}\right)\left(t\zeta_{3}x\right)^{2}+\left(-1+2t+t^{2}\zeta_{3}+t^{3}\right)\left(t\zeta_{3}x\right)+2+\zeta_{3}+(-1+\zeta_{3})t$
|
$+2t^{2}+t^{3}\zeta_{3}+t^{4}+\left(-1+2t+t^{2}\zeta_{3}+t^{3}\right)\left(t\zeta_{3}x\right)^{-1}+\left(1+t\zeta_{3}+t^{2}\right)\left(t\zeta_{3}x\right)^{-2}\big{]}$
Table 2: $t$-deformed ADO polynomials for the $5_{2}$ knot $p$ | $\operatorname{ADO}_{7_{2}}(p;x,t)$
---|---
1 | 1
2 | $(1-t+t^{2})(-tx)-1+2t-t^{2}+t^{3}+(1-t+t^{2})(-tx)^{-1}$
3 | $\zeta_{3}^{2}\big{[}\left(1+t\zeta_{3}+t^{3}\zeta_{3}+t^{4}\right)\left(t\zeta_{3}x\right)^{2}+\left(-1+(1-\zeta_{3})t+2\zeta_{3}t^{2}+t^{3}+\zeta_{3}t^{4}+t^{5}\right)\left(t\zeta_{3}x\right)$
|
$+2+\zeta_{3}+(-1+2\zeta_{3})t+(1-\zeta_{3})t^{2}+2\zeta_{3}t^{3}+t^{4}+\zeta_{3}t^{5}+t^{6}$
|
$+\left(-1+(1-\zeta_{3})t+2\zeta_{3}t^{2}+t^{3}+\zeta_{3}t^{4}+t^{5}\right)\left(t\zeta_{3}x\right)^{-1}+\left(1+t\zeta_{3}+t^{3}\zeta_{3}+t^{4}\right)\left(t\zeta_{3}x\right)^{-2}$
]
Table 3: $t$-deformed ADO polynomials for the $7_{2}$ knot

More generally, the higher rank $t$-deformed ADO is defined as Ekholm:2020lqy

$\operatorname{ADO}_{K}^{\operatorname{SU}(N)}(p;x,t)=\Delta_{K}(x^{p},-(-t)^{p})^{N-1}\lim_{q\to\zeta_{p}}F_{K}(x,-t^{-1}q^{N},q,t).$
(77)

It is also feasible to derive the closed-form expression for the $t$-deformed
ADO${}^{\operatorname{SU}(N)}_{K}(p;x,t)$ from (54) and (59)

$\displaystyle\operatorname{ADO}_{K_{m,n}}^{\operatorname{SU}(N)}(p;x,t)=$
$\displaystyle~{}\frac{1}{p}\sum_{l=0}^{p-1}(-tx)^{l+1-p}\Delta_{K_{m,n}}(x^{p},-(-t)^{p})^{\frac{(N-2)(p-1)+l}{p}}S_{p}(\zeta_{p}^{N-l-2})$
(78)
$\displaystyle\times\sum_{k=0}^{p-1}\lim_{q\rightarrow\zeta_{p}}g^{(k)}_{K_{m,n}}(x,-t^{-1}q^{N},q,t),$
(79)
$\displaystyle\operatorname{ADO}_{K_{m+\frac{1}{2},-n}}^{\operatorname{SU}(N)}(p;x,t)=$
$\displaystyle~{}\frac{1}{p}\sum_{l=0}^{p-1}(-tx^{n})^{l+1-p}\Delta_{K_{m+\frac{1}{2},-n}}(x^{p},-(-t)^{p})^{\frac{(N-2)(p-1)+l}{p}}$
(80) $\displaystyle~{}\times
S_{p}(\zeta_{p}^{N-l-2})\sum_{k=0}^{p-1}\lim_{q\rightarrow\zeta_{p}}g^{(k)}_{K_{m+\frac{1}{2},-n}}(x,-t^{-1}q^{N},q,t).$
(81)

The detailed derivation is also shown in the Appendix A. Especially, we can
analytically verify that they become

$\displaystyle\operatorname{ADO}_{K}^{\operatorname{SU}(N)}(1;x,t)=1,$ (82)
$\displaystyle\mathrm{ADO}_{K}^{\operatorname{SU}(N)}(2;x,t)=\frac{1}{2}\sum_{l,j=0}^{1}(-1)^{j(N-l)}\left[\Delta_{K}(x^{2},-(-t)^{2})\right]^{\frac{l}{2}+\frac{N}{2}-1}\left[\Delta_{K}(x,t)\right]^{1-l}~{},$
(83)

and they obey the recursion relation Ekholm:2020lqy

$\displaystyle\operatorname{ADO}^{\operatorname{SU}(N+p)}(p;x,t)=\Delta_{K}(x^{p},-(-t)^{p})^{p-1}\operatorname{ADO}^{\operatorname{SU}(N)}(p;x,t)~{}.$
(84)

Note that colored Jones polynomials of cyclotomic type Habiro:2008 ;
lovejoy2017colored ; lovejoy2019colored  at a root of unity $q=\zeta_{p}$
provide the corresponding ADO invariants Anna:2020non  even if a closed-form
expression of $F_{K}$ is not known. Therefore, one can evaluate the ADO
invariants from colored Jones polynomials in a similar fashion for arbitrary
double twist knots such as $K_{m,n}$ with $m,-n\in\mathbb{Z}_{+}$ Anna:2020non
.

##  5 Conclusion and discussion

In this paper, we presented a simple rule of grading change that would allow
us to obtain $[r]$-colored quadruply-graded Kauffman homology from
$[r^{2}]$-colored quadruply-graded HOMFLY-PT homology for thin knots. We check
the grading change by consistency with differentials and symmetries for
colored Kauffman homology of double twist knots. We also find from the
universal differential that the super-$A$-polynomials of SO-types contains
both $a$-deformed $A$-polynomials of SO and SU-type at $t=-1$. With the
natural quantization of $\operatorname{SO}(N)(N\geq 4)$ $6j$-symbols for
symmetric representations given by Ališauskas, we calculate
$\operatorname{SO}(N)$ fusion matrices for $R=~{}$ ,  and compute Kauffman
polynomials for the corresponding colors. We check the validity of the
quantization from representation theory and Kauffman polynomials. We also
conjecture a closed-form expression of the $(a,t)$-deformed $F_{K}$ of the
double twist knots $K_{m,n}$ and $K_{m+\frac{1}{2},-n}$ from the corresponding
superpolynomial. Using the $(a,t)$-deformed $F_{K}$ we proposed, we also gave
closed-form expressions of the $t$-deformed Alexander and ADO polynomial.

However, there are still immediate questions that need to be solved in the
future. First, the expression (27) is rather involved and there must be a
similar formula for it. It is also desirable to find a closed-form expression
of $6j$-symbols when fusions $R_{12}$ and $R_{23}$ are non-symmetric.
Moreover, it has been found that Liu:2018jhs  the $6j$-symbol in AdS is the
Lorentzian inversion of a crossing-symmetric tree-level exchange amplitude,
and the one-loop vertex correction in $\phi^{3}$-theory in $\text{AdS}_{d+1}$
is given by a spectral integral over the $6j$-symbol for
$\operatorname{SO}(d+1,1)$. It would be interesting that the results in §3
would bring some inspiration to the calculations of these $6j$-symbols in the
AdS background.

So far the study of the volume conjecture involves only the large color limit
of symmetric representations. It is very important to formulate volume
conjectures with arbitrary colors, relating to the moduli space of flat
$\operatorname{SL}(N,\mathbb{C})$ connections over the knot complement. Since
colored knot homology is endowed with many colored differentials, we expect a
behavior similar to Conjecture 2.1 in volume conjectures of higher ranks.

We have obtained the $(a,t)$-deformed $F_{K}$ for the two classes of double
twist knots from the corresponding superpolynomials. However, It is desirable
to develop a general method to obtain a closed-form expression of $F_{K}$ for
arbitrary knots.

###### Acknowledgements.

We would like to thank Chen Yang for collaboration at the initial stage of the
project. S.N. is indebted to Bruno le Floch for collaboration and discussion
on $6j$-symbols, and he also thanks Ryo Suzuki for identifying the reference
Alisauskas_2002  about $\operatorname{SO}(N)$ $6j$-symbols. We also would like
to thank Sunghyuk Park for identifying the relationship between the
$t$-deformed ADO polynomials of $3_{1}$ and ${}^{*}3_{1}$. This work was
supported by the National Science Foundation of China under Grant No. NSFC
PHY-1748958.

##  Appendix A Derivation of the ADO polynomials

In this Appendix, we derive the $t$-deformed $p$-th ADO polynomial (69) and
its higher rank generalization (77) from $F_{K}(x,a,q,t)$ (54) and (59). We
use $F_{K}$ as a short for,

$\lim_{q\to\zeta_{p}}F_{K}(x,a=-t^{-1}q^{N},q,t)~{},$

and similarly for other functions discussed in this section. We use $q$ and
$\zeta_{p}$ interchangeably. When an integer $k$ is displayed as
$k=k_{1}p+k_{0}$, it is implied that $k_{1},k_{0}\in\mathbb{N}$, and $0\leq
k_{0}\leq p-1$.

Our closed formulae are built from $q$-binomials and $q$-Pochhammer symbols
and we will first discuss their behavior when $q$ goes to roots of unity. The
$q$-Lucas theorem olive1965generalized  states that,

${l\brack k}_{q=\zeta_{p}}={l_{1}\choose k_{1}}{l_{0}\brack
k_{0}}_{q=\zeta_{p}}~{},$ (85)

where $l=l_{1}p+l_{0},~{}k=k_{1}p+k_{0}$. For $q$-Pochhammers, we have

$(a;q)_{k}=(1-a^{p})^{k_{1}}(a;q)_{k_{0}}~{},$ (86)

where $q=\zeta_{p}$, $k=k_{1}p+k_{0}$. In the following discussion, We will
break $F_{K}(x,a,q,t)$ into parts. As we will see later, these components
enjoy similar properties. It turns out that $F_{K}$ can be written as a
product of a infinite summation over $k_{1}$ and a finite one over $k_{0}$,
where $k=k_{1}p+k_{0}$ and $q=\zeta_{p}$, and the infinite summation over
$k_{1}$ can be repackaged by the generalized binomial theorem,

$\frac{1}{(1-z)^{n}}=\sum_{i=0}^{\infty}{n+i-1\choose i}z^{i}~{}.$ (87)

As a result, an ADO polynomial can be expressed as a finite summation of over
$k_{0}$.

###  A.1 ADO polynomials of double twist knot
$K_{m,n}~{}(m,n\in\mathbb{Z}_{+})$

For double twist knots $K_{m,n}~{}(m,n\in\mathbb{Z}_{+})$, we first analyze
$(a,t)$-deformed $F_{K}$ at radial limit $q=\zeta_{p}$. Each component of (54)
behaves as follows.

  * •

The twist factor $\operatorname{tw}^{(l)}_{m}(a,q,t)$ (58) now becomes

$\operatorname{tw}_{m}^{(l)}=\sum_{0\leq b_{1}\leq\cdots\leq b_{m-1}\leq
b_{m}=l}\prod_{i=1}^{m-1}(-q^{N})^{b_{i}}q^{b_{i}(b_{i}-1)}t^{b_{i}}{b_{i+1}\brack
b_{i}}_{q}~{}.$ (88)

We decompose the summation variables as, $b_{i}=\alpha_{i}p+\beta_{i}$, and
$l=l_{1}p+l_{0}$. Using the $q$-Lucas theorem, (88) can be written as a
product of two summations, one over $\alpha_{i}$’s and another one over
$\beta_{i}$’s. The summation over $\beta_{i}$’s would give
$\mathrm{tw}_{m}^{(l_{0})}$. Performing the summation over $\alpha_{i}$’s, we
obtain

$\mathrm{tw}_{m}^{(l)}=S_{m}^{l_{1}}((-t)^{p})\,\mathrm{tw}_{m}^{(l_{0})}~{},$
(89)

where
$S_{m}^{l_{1}}(x):=\left(S_{m}(x)\right)^{l_{1}}:=\left(\sum\limits_{i=0}^{m-1}x^{i}\right)^{l_{1}}$.

  * •

The twist factor $\operatorname{Tw}_{K_{m,n}}^{(j)}(a,q,t)$ (57) becomes

$\displaystyle\operatorname{Tw}_{K_{m,n}}^{(j)}=\sum_{l=0}^{j}(-1)^{l}q^{\frac{1}{2}l(l+1)-jl}\operatorname{tw}_{m}^{(l)}\operatorname{tw}_{n}^{(l)}{j\brack
l}_{q}~{}.$ (90)

We write the summation variables as $j=j_{1}p+j_{0}$, and $l=l_{1}p+l_{0}$.
Because of the fact that $q^{p}=1$, the $q$-Lucas theorem and (89), we obtain,

$\operatorname{Tw}_{K_{m,n}}^{(j)}=\left(1-S_{m}\left(\left(-t\right)^{p}\right)S_{n}\left(\left(-t\right)^{p}\right)\right)^{j_{1}}\operatorname{Tw}_{K_{m,n}}^{(j_{0})}~{}.$
(91)

  * •

Now $g^{(k)}_{K_{m,n}}(x,a,q,t)$ in (55) becomes

$g^{(k)}_{K_{m,n}}=\sum_{j=0}^{k}(x;q^{-1})_{k}(xt^{2}q^{N};q)_{j}(xt^{2})^{k-j}q^{k-Nj}{N-2+k\brack
k}_{q}{k\brack j}_{q}\operatorname{Tw}_{K_{m,n}}^{(j)}~{},$ (92)

where we have used the fact that

$\frac{(q^{N-1};q)_{k}}{(q;q)_{k}}={N-2+k\brack k}_{q}~{}.$ (93)

We decompose the variables as $j=j_{1}p+j_{0}$, $k=k_{1}p+k_{0}$ and
$N-2=A_{1}p+A_{0}$. Plugging (85), (86) and (91) into (92), we have

$g_{K_{m,n}}^{(k)}={A_{1}+k_{1}\choose
k_{1}}\left(1-x^{p}\right)^{k_{1}}\left[1-\left(1-x^{p}t^{2p}\right)S_{m}\left((-t)^{p}\right)S_{n}\left((-t)^{p}\right)\right]^{k_{1}}g^{(k_{0})}_{K_{m,n}}~{}.$
(94)

  * •

$F_{K_{m,n}}(x,a,q,t)$ becomes

$F_{K_{m,n}}=(-tx)^{N-1}\sum_{k=0}^{\infty}g_{K_{m,n}}^{(k)}~{}.$ (95)

Given $k=k_{1}p+k_{0}$, $N-2=A_{1}p+A_{0}$, we have

$\displaystyle F_{K_{m,n}}=$
$\displaystyle~{}(-tx)^{N-1}\sum_{k_{0}=0}^{p-1}g_{K_{m,n}}^{(k_{0})}$ (96)
$\displaystyle\times\sum_{k_{1}=0}^{\infty}{A_{1}+k_{1}\choose
k_{1}}\left(1-x^{p}\right)^{k_{1}}\left[1-\left(1-x^{p}t^{2p}\right)S_{m}\left((-t)^{p}\right)S_{n}\left((-t)^{p}\right)\right]^{k_{1}}~{}.$

Using the generalized binomial theorem, we obtain

$F_{K_{m,n}}=\frac{(-tx)^{N-1}\sum\limits_{k=0}^{p-1}g^{(k)}_{K_{m,n}}}{\left[x^{p}+\left(1-x^{p}\right)\left(1-x^{p}t^{2p}\right)S_{m}\left(\left(-t\right)^{p}\right)S_{n}\left(\left(-t\right)^{p}\right)\right]^{A_{1}+1}}.$
(97)

Recall the closed formulae of $t$-deformed Alexander polynomials (70), we can
write

$F_{K_{m,n}}=\frac{(-tx)^{A_{0}+1-p}}{\Delta_{K_{m,n}}(x^{p},-(-t)^{p})^{A_{1}+1}}\sum_{k=0}^{p-1}g^{(k)}_{K_{m,n}}~{}.$
(98)

Before we jump to the ADO polynomials, let us first examine the properties of
$(a,t)$-deformed $F_{K}$. When $p=1$ which leads to $A_{1}=N-2$, $A_{0}=0$, we
have

$\lim_{q\to
1}F_{K_{m,n}}(x,-t^{-1}q^{N},q,t)=\frac{g^{(0)}_{K_{m,n}}}{\Delta_{K_{m,n}}(x,t)^{N-1}}=\frac{1}{\Delta_{K_{m,n}}(x,t)^{N-1}}~{},$
(99)

in accordance with (67). Finally, for the $t$-deformed $\operatorname{ADO}$
polynomials, we have

$\displaystyle\operatorname{ADO}_{K_{m,n}}^{\operatorname{SU}(N)}(p;x,t)$
$\displaystyle=\Delta_{K_{m,n}}(x^{p},-(-t)^{p})^{N-1}F_{K_{m,n}}$ (100)
$\displaystyle=(-tx)^{A_{0}+1-p}\Delta_{K_{m,n}}(x^{p},-(-t)^{p})^{A_{1}(p-1)+A_{0}}\sum_{k=0}^{p-1}g_{K_{m,n}}^{(k)}~{},$

where $N-2=A_{1}p+A_{0}$.

Now let us consider some simple cases. If $p=1$, then $A_{0}=0$, we have

$\operatorname{ADO}_{K_{m,n}}^{\operatorname{SU}(N)}(p=1;x,t)=g^{(0)}_{K_{m,n}}=1~{}.$
(101)

For $N=2$ and $p=2$, then $A_{0}=A_{1}=0$, we have

$\operatorname{ADO}_{K_{m,n}}^{\operatorname{SU}(2)}(p=2;x,t)=(-tx)^{-1}\left(g^{(0)}_{K_{m,n}}+g^{(1)}_{K_{m,n}}\right)=\Delta_{K_{m,n}}(x,t)~{}.$
(102)

###  A.2 ADO polynomials of double twist knots
$K_{m+\frac{1}{2},-n}~{}(m,n\in\mathbb{Z}_{+})$

Following the same procedure in the last subsection, we decompose the closed-
form expression of $F_{K_{m+\frac{1}{2},-n}}(x,a,q,t)$ (59) into parts:

  * •

The twist factor $\\!\mathbbm{t}\\!\mathbbm{w}_{n}^{(k)}(x,q,t)$ (62) now
becomes

$\\!\mathbbm{t}\\!\mathbbm{w}_{n}^{(k)}(x,q,t)=\sum_{0=b_{0}\leq
b_{1}\leq\cdots\leq b_{n-1}\leq
b_{n}=k}\prod_{i=1}^{n-1}(x^{2b_{i}}q^{-b_{i}(b_{i+1}-1)}t^{2b_{i}}){b_{i+1}\brack
b_{i}}_{q}~{}.$ (103)

We write $b_{i}=\alpha_{i}p+\beta_{i}$, $k=k_{1}p+k_{0}$, and we can obtain

$\\!\mathbbm{t}\\!\mathbbm{w}_{n}^{(k)}=S^{k_{1}}_{n}(x^{2p}t^{2p})\,\\!\mathbbm{t}\\!\mathbbm{w}_{n}^{(k_{0})}~{}.$
(104)

  * •

Now $g^{(k)}_{K_{m+\frac{1}{2},-n}}$ in (61) becomes

$\displaystyle
g^{(k)}_{K_{m+\frac{1}{2},-n}}=\sum_{j=0}^{k}(x;q^{-1})_{k}(xt^{2}q^{N};q)_{j}x^{k-j}{N-2+k\brack
k}_{q}{k\brack
j}_{q}q^{\frac{1}{2}j(j-1)+k}t^{-j+2k}\operatorname{tw}_{m}^{(j)}\\!\mathbbm{t}\\!\mathbbm{w}_{n}^{(k)}~{}.$
(105)

We write $k=k_{1}p+k_{0}$, $j=j_{1}p+j_{0}$, and $N-2=A_{1}p+A_{0}$, and
obtain

$\displaystyle g^{(k)}_{K_{m+\frac{1}{2},-n}}=$
$\displaystyle~{}\left\\{\left(1-x^{p}\right)\left(-t\right)^{p}\left[\left(-xt\right)^{p}+\left(x^{p}t^{2p}-1\right)S_{m}\left(\left(-t\right)^{p}\right)\right]S_{n}\left(x^{2p}t^{2p}\right)\right\\}^{k_{1}}$
(106) $\displaystyle\times{A_{1}+k_{1}\choose
k_{1}}g_{K_{m+\frac{1}{2},-n}}^{(k_{0})}~{}.$

  * •

$F_{K_{m+\frac{1}{2},-n}}(x,a,q,t)$ now becomes

$F_{K_{m+\frac{1}{2},-n}}=(-tx^{n})^{N-1}\sum_{k=0}^{\infty}g_{K_{m+\frac{1}{2},-n}}^{(k)}~{}.$
(107)

We write $k=k_{1}p+k_{0}$, $N-2=A_{1}p+A_{0}$. Again, with the help of the
generalized binomial theorem, we can get rid of the infinite summation over
$k_{1}$ and obtain

$F_{K_{m+\frac{1}{2},-n}}=\frac{(-tx^{n})^{A_{0}+1-p}}{\Delta_{K_{m+\frac{1}{2},-n}}(x^{p},-(-t)^{p})^{A_{1}+1}}\sum_{k_{0}=0}^{p-1}g^{(k_{0})}_{K_{m+\frac{1}{2},-n}}~{},$
(108)

where the $t$-deformed Alexander polynomial is given in (70).

When $p=1$, then $A_{1}=N-2$, $A_{0}=0$, we have

$\lim_{q\to
1}F_{K_{m+\frac{1}{2},-n}}(x,-t^{-1}q^{N},q,t)=\frac{g^{(0)}_{K_{m+\frac{1}{2},-n}}}{\Delta_{K_{m+\frac{1}{2},-n}}(x,t)^{N-1}}=\frac{1}{\Delta_{K_{m+\frac{1}{2},-n}}(x,t)^{N-1}}~{},$
(109)

in accordance with (67). Finally, for the $t$-deformed $\operatorname{ADO}$
polynomials, we have

$\displaystyle\operatorname{ADO}_{K_{m+\frac{1}{2},-n}}^{\operatorname{SU}(N)}(p;x,t)$
$\displaystyle=\Delta_{K_{m+\frac{1}{2},-n}}(x^{p},-(-t)^{p})^{N-1}F_{K_{m+\frac{1}{2},-n}}$
(110)
$\displaystyle=(-tx^{n})^{A_{0}+1-p}\Delta_{K_{m+\frac{1}{2},-n}}(x^{p},-(-t)^{p})^{A_{1}(p-1)+A_{0}}\sum_{k=0}^{p-1}g_{K_{m+\frac{1}{2},-n}}^{(k)}~{},$

where $N-2=A_{1}p+A_{0}$.

Now we consider some simple cases. When $p=1$, $A_{0}=0$, we have

$\operatorname{ADO}_{K_{m+\frac{1}{2},-n}}^{\operatorname{SU}(N)}(p=1;x,t)=g^{(0)}_{K_{m+\frac{1}{2},-n}}=1~{}.$
(111)

For $N=2$, $p=2$, we have

$\operatorname{ADO}_{K_{m+\frac{1}{2},-n}}(p=2;x,t)=(-tx^{n})^{-1}(g^{(0)}_{K_{m+\frac{1}{2},-n}}+g^{(1)}_{K_{m+\frac{1}{2},-n}})=\Delta_{K_{m+\frac{1}{2},-n}}(x,t)~{}.$
(112)

###  A.3 Final formulae

In conclusion, for gauge group $\operatorname{SU}(N)$, the $t$-deformed $p$-th
ADO polynomials are given by

$\displaystyle\operatorname{ADO}_{K_{m,n}}^{\operatorname{SU}(N)}(p;x,t)=(-tx)^{A_{0}+1-p}\Delta_{K_{m,n}}(x^{p},-(-t)^{p})^{A_{1}(p-1)+A_{0}}\sum_{k=0}^{p-1}g^{(k)}_{K_{m,n}},$
(113)
$\displaystyle\operatorname{ADO}_{K_{m+\frac{1}{2},-n}}^{\operatorname{SU}(N)}(p;x,t)=(-tx^{n})^{A_{0}+1-p}\Delta_{K_{m+\frac{1}{2},-n}}(x^{p},-(-t)^{p})^{A_{1}(p-1)+A_{0}}\sum_{k=0}^{p-1}g^{(k)}_{K_{m+\frac{1}{2},-n}},$

where $N-2=A_{1}p+A_{0}$, $A_{1},A_{0}\in\mathbb{N}$ and $0\leq A_{0}\leq
p-1$. It is easily seen that the recursion relation conjectured in
Ekholm:2020lqy  holds,

$\operatorname{ADO}^{\operatorname{SU}(N+p)}_{K}(p;x,t)=\Delta_{K}(x^{p},-(-t)^{p})^{p-1}\operatorname{ADO}^{\operatorname{SU}(N)}_{K}(p;x,t)~{}.$
(114)

Note that $\frac{1}{p}S_{m}(\zeta_{p}^{n})=1$, only when $m|n$. Otherwise it
is zero. Therefore, the $t$-deformed $p$-th ADO polynomials, can also be
written as,

$\displaystyle\operatorname{ADO}_{K_{m,n}}^{\operatorname{SU}(N)}(p;x,t)=$
$\displaystyle~{}\frac{1}{p}\sum_{l=0}^{p-1}(-tx)^{l+1-p}\Delta_{K_{m,n}}(x^{p},-(-t)^{p})^{\frac{(N-2)(p-1)+l}{p}}S_{p}(\zeta_{p}^{N-l-2})\sum_{k=0}^{p-1}g^{(k)}_{K_{m,n}}$
(115)
$\displaystyle\operatorname{ADO}_{K_{m+\frac{1}{2},-n}}^{\operatorname{SU}(N)}(p;x,t)=$
$\displaystyle~{}\frac{1}{p}\sum_{l=0}^{p-1}(-tx^{n})^{l+1-p}\Delta_{K_{m+\frac{1}{2},-n}}(x^{p},-(-t)^{p})^{\frac{(N-2)(p-1)+l}{p}}$
$\displaystyle~{}\times
S_{p}(\zeta_{p}^{N-l-2})\sum_{k=0}^{p-1}g^{(k)}_{K_{m+\frac{1}{2},-n}}.$

Although we write them as summations over $l$, there is only one non-vanishing
term, which corresponds to that $l$ is the remainder of $(N-2)/p$.

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