Colored Kauffman Homology and Super-A-polynomials

We study the structural properties of colored Kauffman homologies of knots. Quadruple-gradings play an essential role in revealing the differential structure of colored Kauffman homology. Using the differential structure, the Kauffman homologies carrying the symmetric tensor products of the vector representation for the trefoil and the figure-eight are determined. In addition, making use of relations from representation theory, we also obtain the HOMFLY homologies colored by rectangular Young tableaux with two rows for these knots. Furthermore, the notion of super-A-polynomials is extended in order to encompass two-parameter deformations of PSL(2,C) character varieties.

so/sp Lie algebras and any representations. As for categorifications of Kauffman polynomials, the uncolored case has been explored in [17]. The goal of this paper is to clarify the structural properties of colored Kauffman homology as in the case of the HOMFLY homology [12,13]. When we introduce quadruple-gradings in an appropriate manner, the structural features of [r]-colored Kauffman homology appear to be also evident. Interestingly, there is a similarity between [r]-colored Kauffman homology and [r, r]-colored HOMFLY homology. In particular, one can find the counterparts in [r]-colored Kauffman homology of all the differentials present in [r, r]-colored HOMFLY homology.
It is worth mentioning that refined Chern-Simons theory [18] is another way to approach the HOMFLY homology for torus knots. In fact, in the case of the symmetric representations, refined Chern-Simons invariants coincide with the Poincaré polynomials of the HOMFLY homology for torus knots [18][19][20][21] though the reason is not fully understood yet. On the other hand, for SO(N )/Sp(N ) gauge groups, the refined Chern-Simons invariants are different from the Poincaré polynomials of the Kauffman homology [22,23].
In this paper, we shall provide the Poincaré polynomials of the [r]-colored Kauffman homology and the [r, r]-colored HOMFLY homology for the trefoil and the figure-eight, which are obtained using the structural properties and representation theory. The results facilitate the study of the large color behaviors of the [r]-colored Kauffman homology, which provide two-parameter deformations of P SL(2, C) character varieties, called super-A-polynomials of SO-type. For [r]-colored HOMFLY homology, the analogous computations have been performed in [20,[24][25][26]. The super-A-polynomials play an important role in the 3d/3d correspondence [24,25].
Finally, the authors would like to remark to mathematicians that no statements except the results in §2 have been proven in this paper so that precise mathematical formulation is waiting to be given.
The organization of the paper is as follows. In §2, we briefly discuss the polynomial invariants, mainly focusing on Kauffman polynomials of torus knots. This section paves the way for the other sections. In §3, the appearance of these polynomial invariants in the context of string theory is summarized. This approach has been extended to identify the knot homology with the space of BPS states. We also discuss the mirror geometry in the B-model in the presence of an orientifold. §4 is devoted to describe the structure of the colored Kauffman homology. We give a detailed list of the properties of the quadruply-graded colored Kauffman homology. In addition, we explicitly show the degrees of all colored differentials and re-gradings in the corresponding colored isomorphisms. Furthermore, the isomorphisms of knot homologies are provided from the viewpoint of representation theory. In §5, the structural features are used to obtain the Poincaré polynomials of the [r]-colored Kauffman homology for the trefoil and the figure-eight. We also present [r, r]-colored HOM-FLY homology for these knots making use of the relations from representation theory. To determine these homological invariants, the refined exponential growth property plays an essential role. §6 contains computations for the super A-polynomials where P SL(2, C) character varieties of the knot complements are obtained by taking the appropriate limit. In §7, the relation to the 3d/3d correspondence is briefly discussed. Finally, we conclude with comments and open-problems in §8. Appendix A provides a list of the conventions and notations in this paper. In Appendix B, the non-trivial check for the integrality conjecture proposed by Marinõ for the figure-eight has been carried out by using the invariants obtained in Appendix 5. The figures for colored Kauffman homology of the trefoil that are too big for the main text are placed in Appendix C.

Skein relation
The unreduced HOMFLY polynomial, P (K; a, q) is a two-variable polynomial invariant of any oriented knot K in S 3 , defined using the following skein relations of oriented planar diagrams with normalization such that the invariant for the unknot is given by The unreduced Kauffman polynomial, F (K; λ, q) is another two-variable invariant polynomial of unoriented knots defined via the skein relations with normalization The reduced versions of the HOMFLY and Kauffman polynomials, P (K) and F (K), can be obtained by dividing the corresponding unreduced polynomials by the unknot factor P (K) = P ( )P (K) , F (K) = F ( )F (K) , (2.5) so that P ( ) = F ( ) = 1.

Chern-Simons theory and polynomial invariants
Chern-Simons gauge theory based on any compact semi-simple group G provides a natural framework for the study of knots and links [1]. The Chern-Simons action S is where A is the g-valued gauge connection and k is the coupling constant. A natural metricindependent observable in Chern-Simons theory is a Wilson loop operator W R (K) = Tr R U K along a knot K carrying the representation R of g where U K = P exp K A . It was heuristically outlined in [1] that the expectation value of the Wilson loop operator beomes a quantum invariant of the knot K. Here, the quantum parameter q is expressed by where h ∨ represents the dual Coxeter number of the gauge group. The constructions of [1] soon led to a rigorous formulation of the quantum invariants by the representation theory of quantum group U q (g) [2]. It turns out that the quantum invariant J g R (K; q) becomes a polynomial with respect to q for any representation R of g.
The evaluations of quantum invariants can be carried out by using the relation between Chern-Simons theory and the Wess-Zumino-Novikov-Witten conformal field theory [27][28][29]. In fact, the skein relations in §2.1 can be obtained by braiding operations on four point conformal blocks. On one hand, in the context of SU (N ) Chern-Simons theory, braiding operations on the four point conformal block with the fundamental representation provides the HOMFLY skein relation (2.1) by substituting a for q N [1]. On the other hand, by placing the defining representation on the conformal block, a similar method gives the Kauffman skein relation (2.3) in SO(N )/Sp(N ) Chern-Simons theory where λ = q N −1 for SO(N ) or λ = q N +1 for Sp(N ) [30][31][32].
Assigning higher rank representations R on a knot, the sl(N ) quantum invariant J sl(N ) R (K; q) and the so(N )/sp(N ) quantum invariant J so(N )/sp(N ) R (K; q) turn into the colored HOMFLY invariant and the colored Kauffman invariant, respectively, with the same changes of variables. It is appropriate to mention that a representation R of sl(N ) placed on a component of an oriented link is changed to the conjugate representation R when the orientation of that component is reversed. The change of relative orientations in a link will affect colored HOMFLY invariants of the link while it does not in the case of a knot. On the other hand, all the representations of so(N ) and sp(N ) are real R = R, implying that colored Kauffman invariants are suitable for the study of unoriented knots and links. In Chern-Simons theory, the quantum invariant of the unknot proves to be equal to the quantum dimension of the representation R living on the unknot: (2.9) Note that the quantum dimension of the representation R with highest weight Λ R is given by where α > 0 are positive roots of g and ρ is the Weyl vector. The square bracket refers to the quantum number, defined by Normalizing the unknot invariant, the reduced colored HOMFLY and Kauffman invariants of a knot are in polynomial form with respect to two variables. As for explicit computations of invariants, the colored Kauffman polynomials of torus knots can be implemented for any representation R, making use of the modular SL(2, Z) transformations [14]. To date, however, no calculations have been performed for colored Kauffman polynomials of non-torus knots by any method. In this paper, we will report some progress for the colored Kauffman polynomials of the figure-eight.

Torus knots and SL(2, Z) transformation
As explained in §2.2, there is a systematic procedure of determining quantum invariants with an arbitrary representation of the (Q, P )-torus knot. This was introduced by Rosso and Jones for SU (2) Chern-Simons invariants [33], and has been further generalized through the construction of torus knot operators for SU (N ) [34][35][36] and SO(N )/Sp(N ) Chern-Simons invariants [14]. Let us briefly review the procedure. The Chern-Simons invariant of the unknot with winding number Q carrying the representation R can be written as Tr R U Q . Using the fact that the Hilbert space on the torus in Chern-Simons theory is isomorphic to the space of conformal blocks, one can write that which is called the Adams operation. Since the (Q, P ) torus knot can be obtained by performing the modular T transformation with the fractional power P/Q to the unknots with winding number Q, the quantum invariant of the (Q, P )-torus knot can be written as It was shown in [35] that a closed form expression of the uncolored HOMFLY polynomial of the (Q, P )-torus knot can be obtained by this procedure since the Adams operation (2.25) involves only hook representations when the representation R is a single box . Therefore, let us try to obtain a closed form expression of the uncolored Kauffman polynomial of the (Q, P )-torus knot in the same manner. It follows from (2.12) that the so(N ) quantum invariant of the unknot is Therefore, the holonomy matrix can be written as Subsequently, one can convince oneself that the expectation value of the unknot with winding number m is given by where the conjugacy class of length m in the rank-m symmetric group S m is expressed by k m . From the first to the second line, we use the fact that the character χ R (k m ) of the symmetric group S m with representation R at the conjugacy class k m of length m vanishes except for the hook representations R = [m − s, 1 s ] of m boxes (s = 0, · · · , m − 1): For the framed unknot f with framing f , we can just perform the SL(2, Z) transformation The quantum dimension and the quadratic Casimir of the representation R m.s of so(N ) are given by As in the case of the HOMFLY polynomial [35], with appropriate normalization, the substitution of f = P Q , m = Q will give us the Kauffman polynomial for the (Q, P )-torus knot K Q,P : . (2.34) This expression proves to be consistent with (4.4) of [37]. Although the HOMFLY polynomials of the torus knots can be written in terms of the q-hypergeometric function 2 φ 1 , the term e πiQ + 1 in (2.34) as well as the symmetry F (K Q,P ; λ, q) = F (K P,Q ; λ, q) under the interchange of Q and P prevent us from writing it in a similar manner.

A-model description
It was shown in [38] that SU (N ) Chern-Simons theory on S 3 can be realized in the A-model topological string on the deformed conifold T * S 3 , where N A-branes wraps the Lagrangian submanifold S 3 . For SO(N )/Sp(N ) gauge groups, an orientifold has to be introduced [15] in this setting. The deformed conifold T * S 3 which can be expressed by admits the anti-holomorphic involution I : x i → x i where the set of the fixed points under the involution I is S 3 . Whether the gauge group is SO(N ) or Sp(N ) depends on the choice of orientifold action on the gauge group. Generalizing the Gopakumar-Vafa duality [39], Chern-Simons theory with SO(N )/Sp(N ) gauge groups on S 3 at large N is equivalent to closed topological string theory on the resolved conifold X = O(−1) ⊕ O(−1) → CP 1 in the presence of an orientifold [15]. To specify the anti-holomorphic involution, let us briefly recall the geometry of the resolved conifold. The resolved conifold X can be described as a toric variety X = C 4 /C * , where C 4 is parametrized by X i , i = 1, . . . , 4, with charges (1, 1, −1, −1) with respect to the C * action. In these variables, the resolved conifold X is where the size of the CP 1 is set by the FI parameter r. This is complexified by the theta angle of the gauged linear sigma model to give the complexified Kähler parameter t = r −iθ on which the A-model amplitudes depend. The action of the anti-holomorphic involution τ on the space X is defined as follows: In particular, it acts freely on X, so that the quotient space X/τ contains a 2-cycle RP 2 instead of CP 1 .
With this setting, the large N duality [15] can be concretely depicted in the following way: the free energy of Chern-Simons theory with SO(N )/Sp(N ) gauge groups on S 3 provides the closed topological string partition function on the orientifolded resolved conifold log Z Note that the variables in Chern-Simons theory are identified with the parameters in the closed topological string as wherec expresses the charge on the orientifold plane. Actually, the right hand side of (3.4) illustrates the fact that the closed topological string partition function receives contributions from an oriented sector and an unoriented sector where the factor 1/2 in the first term takes care of modding by the anti-holomorphic involution Z 2 . More specifically, the oriented sector is written in terms of g 2−2g s , and the unoriented sector contains only odd powers of g s indicating Riemann surfaces with genus g and one cross-cap.
To incorporate a Wilson loop along a knot K in Chern-Simons theory, another stack of M Lagrangian branes have to be inserted on the conormal bundle N K to the knot K in S 3 [40]. Furthermore, the large N duality can be naturally extended to open topological string on the resolved conifold by wrapping M branes on the Lagrangian submanifold L K associated to the knot K which is the geometric transition of the submanifold N K . Instead of the partition function, the insertion of the Wilson loop is captured by the Ooguri-Vafa operator where U is the holonomy matrix of the SO(N )/Sp(N ) gauge field along the knot K and V is the holonomy matrix of the SU (M ) gauge group associated to the probe branes on N K . Therefore, on the deformed conifold side, the free energy is given by the logarithm of the expectation value of the Ooguri-Vafa operator where the expectation value of Tr R (U ) provides the unreduced R-colored Kauffman polynomial F R (K; λ, q) while Tr R (V ) gives the Schur polynomial s R (v) labeled by the representation R for SU (M ). In a similar manner to (3.4), the free energy can be reformulated in terms of the open topological string partition function on the resolved conifold where the superscript c denotes the contribution from Riemann surfaces with c cross-caps [41,42]. Meanwhile, it had been hard to separate the c = 0 and c = 2 contributions, since their genus expansions are similar, whereas the c = 1 contribution can be extracted using parity argument in the variable λ [41][42][43].
In order to isolate the c = 0 contribution, it was proposed in [16] that one has to take into account two sets of Lagrangian branes on N K, and N K,− , which are related by the anti-holomorphic involution in the covering geometry. If you deform the conormal bundle N K to the fiber direction by , the anti-holomorphic involution creates the two stacks of probe branes N K, and N K,− = I(N K, ) (See Figure 9 in [16]). Since the U (N ) invariants account for the partition function of the covering geometry, the corresponding invariants are described by HOMFLY polynomials P (R,S) (K) carrying a composite representation (R, S). Here, the composite representation (R, S) can be considered as a representation with highest weight Λ R + Λ S where S is the conjugate representation of S. Using this set-up, the c = 0 contribution in (3.8) is given by [16] F c=0 These open-string topological amplitudes can be related to counting degeneracies of M2-M5 bound states in M-theory on the resolved conifold X [16,40,[44][45][46][47], where the configurations of M M5-branes are as follows: where D ∼ = R 2 is the cigar of the Taub-NUT space T N ∼ = R 4 . The M2-branes wrap a two-cycle β ∈ H 2 (X, L K ) of X and end on the stack of the M5-branes. In the orientifold background, a two-cycle can be either an orientable (c = 0) or a non-orientable (c = 1, 2) Riemann surface of genus g with h boundaries. The boundary condition is specified by the h-tuple winding number w = (w 1 , · · · , w h ) where the total number |R| of boxes in the Young diagram R for SU (M ) is equal to |R| = h i=1 w i . The Káhler parameter λ becomes fugacity for the charge β and the variable z = q − q −1 corresponds to the fugacity of the charge g in the index which counts M2-M5 bound states. Therefore, one can define the reformulated invariants by the number N c R;g,β (K) of the M2-M5 bound states g≥0 β∈Z Via the geometric transition, the reformulated invariants can be written in terms of the Chern-Simons invariants of the knot K, which is discussed in more detail in Appendix B. Most importantly, since N c R;g,β (K) is the number of the M2-M5 bound states, it is conjectured to be an integer [16,40,46]. In Appendix B, we verify this conjecture for the figure-eight with R = or .
It is conjectured in [10] that the space of the M2-M5 bound states is isomorphic to the knot homologies H BPS ∼ = H knot . More precisely, we count the BPS states weighted by the charge β as well as the charge (s, r) of the rotation group U (1) q × U (1) t of the non-compact space T N ∼ = R 4 where the q-and t-gradings correspond to the equivariant action U (1) q × U (1) t on the tangent and normal bundle of D ∼ = R 2 in T N ∼ = R 4 . With an appropriate change of basis, the space of the BPS states can be identified with the triply-graded homology, so-called Kauffman homology, which categorifies the Kauffman polynomial. The large N duality predicts that the Kauffman homology is isomorphic to so(N )/sp(N ) homology at large N [11,17]. However, when the Kähler parameter N ∼ log(λ) varies, the BPS spectrum jumps. Therefore, as argued in [12], it is anticipated that there exist differentials in the knot homology which capture jumps of the BPS spectrum. In the uncolored case [17], the structure of the Kauffman homology has been studied. Moreover, it is natural to expect that the colored Kauffman homology incorporates rich differential structure as in colored HOMFLY homology [12,13]. Hence, the main goal of this paper is to investigate the differential structure of the colored Kauffman homology.

B-model description
Mirror symmetry relates the A-model on M to B-model on the mirror manifoldM . For non-compact toric Calabi-Yau M , the mirror manifoldM [48] is given by where the spectral (holomorphic) curve H(x, y; a) = 0 with complex structure a can be viewed as the moduli space of the canonical Lagrangian brane [49]. For instance, the spectral curve for the mirror manifold of the resolved conifold [49] is expressed by where the canonical Lagrangian brane wraps the submanifold L corresponding to the unknot. It was pointed out in [50] that there is an ambiguity that preserves the geometry of the brane at infinity. It turns out that this corresponds to the mirror geometry for the configuration of the framed unknot f , where the spectral curve can be obtained [35,50] by the modular transformation T f of the curve (3.13) (3.14) Generalizing this, it was shown in [35] that the spectral curve corresponding to the configuration for the torus knot T Q,P can be derived by the SL(2, Z) transformation of the curve (3.13) as we obtain the Rosso-Jones formula (2.26): The next step is to understand the mirror geometry of the resolved conifold with the Lagrangian submanifold L T Q,P for the torus knot T Q,P in the presence of an orientifold. Mirror symmetry maps an anti-holomorphic involution of the A-model into a holomorphic involution of the B-model. The holomorphic involution on the manifolď mirror to the orientifold action on the resolved conifold X was explicitly written in [51]: This holomorphic involution can be extended to the geometry uv = H T Q,P (x, y; a) mirror to the resolved conifold with M5-branes wrapping on the Lagrangian submanifold L T Q,P associated to the torus knot T Q,P in such a way that Hence, the geometry mirror to the configuration for the torus knot T Q,P in the presence of an orientifold is uv = H T Q,P (x, y; a) with the involution (3.18).
On the other hand, in [52], the B-model description has been considered in the context of the SYZ formulation [53]. Given a Lagrangian brane whose topology is S 1 × R 2 , the moduli space receives the disc instanton corrections depending on the Lagrangian brane. Thus, even with the same resolved conifold background, the disc corrected moduli space of L K is dependent of a knot K. Furthermore, it is conjectured in [52] that the disc-corrected moduli space of L K is given by the a-deformed A-polynomial of SU -type for a knot K A SU (K; x, y; a) = 0 , (3.19) and the corresponding mirror manifoldX K (3.12) is expressed as uv = A SU (K; x, y; a) . (3.20) The detailed explanation for the a-deformed A-polynomial of SU -type will be given in §6.
Note that this conjecture encompasses any knots including non-torus knots.
Although the a-deformed A-polynomial of SU -type for the framed unknot f coincides with the spectral curve H f (x, y; a) given in (3.14) with suitable change of variables, they are no longer the same for general torus knots. For instance, the spectral curve H T 2,3 (x, y; a) = 0 for the trefoil is of genus zero while the zero locus of the a-deformed A-polynomial of SU -type for the trefoil A SU (3 1 ; x, y; a) = 0 determines a curve of genus one. Further study has to be undertaken in order to understand the relation between the two descriptions in the case of torus knots. In particular, it is important to study whether the application of the topological recursions [54] to the a-deformed A-polynomial of SUtype A SU (K; x, y; a) = 0 would provide the large color asymptotic expansion of the colored HOMFLY polynomial P [r] (K; a, q) as done in the case of colored Jones polynomials [55,56].
Following the generalized SYZ formulation [52], it would be easy to conjecture that the disc-corrected moduli space of the Lagrangian brane associated to a knot K in the presence of an orientifold is given by the zero locus of the λ-deformed A-polynomial A SO (K; x, y; λ) of SO-type. Nevertheless, the authors would like to emphasize that it is desirable to provide some support for the generalized SYZ conjecture [52] involving a non-trivial knot first in the SU context.

Review of quadruply-graded HOMFLY homology
In the case of the categorifications, the realization of knot homologies as the space of certain BPS states has given rise to various predictions on the structure of the colored HOMFLY homology. First, it was predicted in [11] that there exists a triply-graded ((a, q, t)-graded) HOMFLY homology (H HOMFLY (K)) i,j,k , whose graded Euler characteristic is given by the HOMFLY polynomial P (K; a, q). It is endowed with a set of anti-commuting differentials {d N } N ∈Z where the homology with respect to d N >0 is isomorphic to the sl(N ) homology [8], which categorifies the sl(N ) quantum invariant P sl(N ) (K; q) = P (K; a = q N , q): In the sequel, the uncolored triply-graded HOMFLY homology [9] and the differentials d N [57] were put on mathematically rigorous footing.
In [12], this approach has been extended to the colored case. Especially, the concrete study has been carried out for the HOMFLY homology carrying the symmetric and anti-symmetric representations.
where r > ≥ 0. In addition, it was realized in [12,58] that the [r]-colored HOMFLY homology for certain classes of knots, such as thin knots and torus knots, exhibits the exponential growth property  Let us note that the representation R t is the transposition of the representation R. This involution φ is called the mirror/transposition symmetry in [12]. Actually, the involution φ exchanges the positive and negative differentials when the representation R is either a symmetric or an anti-symmetric representation. In attempting to elucidate the mirror/transposition symmetry, the two homological gradings denoted by t r and t c are introduced so that colored HOMFLY homology turns into quadruply-graded (H HOMFLY R (K)) i,j,k, : (a, q, t r , t c )-gradings [13]. Although the four gradings are independent in general, one can associate a δ-grading to every generator x of the [r]-colored quadruply-graded HOMFLY homology of a thin knot K thin by where we denote the S-invariant by S(K) [59]. Moreover, it became apparent that all of the structural properties and isomorphisms become particularly elegant with the introduction of the Q-grading defined by when the representation is specified by a rectangular Young diagram [r ρ ]. While it is just a regrading of (H HOMFLY (K) due to its importance: It is the uncolored case only when the two t-gradings coincide and therefore the resulting homology is triply-graded in agreement with [11]. It simply follows from (4.7) that the qand Q-gradings of the uncolored homology are the same. The definite advantage of the quadruply-graded theory is that it makes all of the structural features and isomorphisms completely explicit. To see them, let us define the Poincaré polynomial of the quadruply-graded homology: where they are related by P [r ρ ] (K; λ, q ρ , t r q −1 , t c q) = P [r ρ ] (K; λ, q, t r , t c ) . Now, let us briefly describe the structural properties of the quadruply-graded colored HOM-FLY homology. We refer the reader to [13] for more detail.
• Self-symmetry (Conjecture 3. which can be stated at the level of the Poincaré polynomial which can be expressed in terms of the Poincaré polynomial P [ρ r ] (K; a, Q, t r , t c ) = P [r ρ ] (K; a, Q, t c , t r ) = P [r ρ ] (K; a, Q −1 t −ρ c t −r r , t c , t r ) .(4.14) This lifts the following relation between the colored HOMFLY polynomials for any representation R.
• Refined exponential growth property (Conjecture 3.8 and 3.9 [13]) Let K be either a thin knot or a torus knot. The [ρ r ]-colored quadruply-graded HOMFLY homology of the knot K obeys the refined exponential growth property (4.17) It follows immediately that The analogous statement at the polynomial level is as follows. For any knot K and an arbitrary representation R, the following identity holds [19]: where |R| is the total number of the Young diagram corresponding to the representation R.
• sl(n|m) differentials It was proposed in [13] that the colored HOMFLY homology is actually gifted with a collection of the differentials {d n|m } labeled by two non-negative integers (n, m) associated to the Lie superalgebra sl(n|m). These are generalizations of the differentials {d N } (4.1). It appears that the representation theory of sl(n|m) explains the behavior of the colored differentials. In this paper, we will not go into the detail about the sl(n|m) differentials.
• (K). These are generalizations of (4.2): The isomorphisms above involve regrading. One of the striking features of the quadruply-graded homology is that it makes the regrading very explicit.
• Universal colored differentials If the representation is specified either by [r, r] or by [2 r ], there exists yet another set of colored differentials d ↑ or d ← so that They are called universal colored differentials because they are universal in the sense that their a-degree is equal to 0.

Properties of quadruply-graded Kauffman homology
Let us now discuss about the categorifications of Kauffman polynomials. The properties of the triply-graded homology that categorifies the uncolored Kauffman polynomials have been investigated in [17,60]. Like HOMFLY homology, it is gifted with a collection of the differentials {d N } N ∈Z so that the homology with respect to the differential d N >1 is isomorphic to the so(N ) homology, while the homology with respect to the differential d N <0 is isomorphic to the sp(N ) homology. Furthermore, it was found in [17] through the analysis of the Landau-Ginzburg theory that the most characteristic property of the Kauffman homology is that it contains the HOMFLY homology. More precisely, it is endowed with the so-called universal differential d univ → ← whose homology is isomorphic with the HOMFLY homology: From the perspective of topological string theory, it is natural to think that there exists the triply-graded homology theory categorifying Kauffman polynomials colored by arbitrary representations. Especially, it is expected that the structure becomes clear if we use quadruple-gradings, as in the case of colored HOMFLY homology, when the colors are specified by rectangular Young tableaux. Hence, our goal in this section is to clarify all the structural features and isomorphisms in the colored quadruply-graded Kauffman homology.
As we have seen in §2.2, for any representation R, there is the R-colored reduced Kauffman polynomial F R (K; λ, q) of a knot K. We conjecture the existence of the finitedimensional homology H Kauffman R (K) of a knot K categorifying the R-colored reduced Kauffman polynomial F R (K; λ, q) of the knot K. In this paper, we focus on the case that the representation R is specified by a rectangular Young tableau [r ρ ]. In this case, we further conjecture that the [r ρ ]-colored Kauffman homology (H Kauffman reduces to the [r ρ ]-colored Kauffman polynomial in the following way: Although the four gradings are generally independent, one can associate a δ-grading to every generator x of the [r]-colored quadruply-graded Kauffman homology of a thin knot K thin by As in the HOMFLY homology, by introducing the Q-grading (4.7), we define the tildeversion of the [r ρ ]-colored quadruply-graded Kauffman homology and its Poincaré polynomial In terms of the Poincaré polynomials, the relation (4.26) can be rephrased by It is the uncolored case only when the two t-gradings coincide and therefore the resulting homology is triply-graded in agreement with [17]. It clearly follows from (4.7) that the qand Q-gradings of the uncolored homology are the same.
In what follows, we conjecture the structural properties of the [r ρ ]-colored Kauffman homology. Although they are very similar, we predict that there are two differences between the [r ρ ]-colored Kauffman homology and the [r ρ ]-colored HOMFLY homology. One of the difference is that the [r ρ ]-colored Kauffman homology does not enjoy the self-symmetry. The other is that there are differentials which relate the [r]-colored Kauffman homology to the [r]-colored HOMFLY homology.

• Mirror/Transposition symmetry
The [r ρ ]-colored Kauffman homology enjoys mirror/transposition symmetry which can be rephrased in terms of the Poincaré polynomial At the decategorified level, for any representation R, there is the comparable symmetry between the R-colored and the R t -colored Kauffman polynomial Only in the uncolored case can the mirror/transposition symmetry be regarded as the self-symmetry.
• Refined exponential growth property Let K be a thin knot or a torus knot. Then, the Kauffman homology carrying a rectangular Young tableaux possesses the refined exponential growth property The analogous statement at the polynomial level is as follows. For any knot K and an arbitrary representation R, the following identity holds: • so/sp differentials The colored Kauffman homology is gifted with a set of the differentials d N so that the homology of H Kauffman R (K) with respect to d N is isomorphic either to the so(N ) homology carrying the representation R for N ≥ 2 or to the sp(N ) homology carrying the representation R t for N ≤ −2 . (4.34) • Universal differentials The [r ρ ]-colored Kauffman homology is endowed with the universal differential d univ so that the homology with respect to the universal differential is isomorphic to the [r ρ ]-colored HOMFLY homology They are universal in the sense that their λ-degree is equal to 0.

• Diagonal differentials
We conjecture the existence of other differentials whose homologies in the [r ρ ]-colored Kauffman homology are isomorphic to the [r ρ ]-colored HOMFLY homology. We call them diagonal differentials, d ± diag , so that They are not universal since their λ-degree is equal to −1. They are diagonal in the sense that every generator • Colored differentials There exists a collection of colored differentials that send the colored Kauffman homology to those with the lower-rank representations.
It should be stressed that the existence of the colored differentials becomes manifest only when we use the tilde-version of the colored Kauffman homology.
• Universal colored differentials If the representation is specified either by [r, r] or by [2 r ], there exists yet another set of colored differentials d ↑ or d ← so that They are called universal colored differentials because they are universal in the sense that their λ-degree is equal to 0.
In the following subsection, we shall explicate all the differentials in detail. Since the size of the colored Kauffman homology is too large for concrete study of arbitrary rectangular Young tableaux, we restrict ourselves to the case that the representations are specified by the Young tableaux [r] and their transpositions [1 r ].
Before moving on to the next subsection, let us define the Poincaré polynomial of the homology with respect to a differential d in the HOMFLY homology and the Poincaré polynomial of the homology with respect to a differential d in the Kauffman homology

so/sp differentials
There is a set of the differentials {d N } N ∈Z inherent in the colored Kauffman homology so that the homology with respect to d N is isomorphic to the colored so/sp homology (4.34). 1 To see the isomorphism (4.34) explicitly, it is convenient to use the Poincaré polynomials in the (λ, q, t r , t c )-gradings. Specifically, in the case of the [r]-colored Kauffman homology, we have the following identities at the level of the Poincaré polynomials where the (λ, q, t r , t c )-degrees of the differential d N acting on H Kauffman . (4.43) The differential d −2 , which specializes the uncolored Kauffman homology to the sp(2) homology, acts nontrivially even on the Kauffman homology of a thin knot [12,17].
On the other hand, the (λ, q, t r , t c )-degrees of the differential d N acting on H Kauffman , so that we can see the isomorphism (4.34) in terms of the Poincaré polynomials in the following way: . (4.45)

Relations from representation theory
The so/sp specializations by using the differentials d N are useful to determine the colored Kauffman homology. In fact, there are several isomorphisms of representations which lead to nontrivial identities among the homological invariants.
• It is well-known that the vector representation of so (3) is isomorphic to the spin-1 representation of sl (2). Moreover, since this relation can be extended to the symmetric product (so (3) Particularly, since the differentials act trivially for a thin knot K thin , the naive substitutions lead to the identity • In addition, since so(4) is isomorphic to sl(2) ⊕ sl(2) as Lie algebras, we have (4.49) In particular, for a thin knot K thin , the identity holds even with the t c -gradings (4.50) • The isomorphism between sp(2) and sl(2) leads to Moreover, the differential d −2 can be evident in the (λ, Q, t r , t c )-grading whose degree (K thin ). We predict that the Poincaré polynomial of the homology H * ( H Kauffman with respect to the differential d −2 can be expressed in terms of the [r]-colored HOMFLY homology • Furthermore, the isomorphism of the representations, provides us with the isomorphism between the [r]-colored so(6) homology and the [r, r]-colored sl(4) homology in the t c -grading Specifically for a thin knot K thin , we have the following relation: (K). Furthermore, the [r]-colored Kauffman homology of a thin knot is expected to have a similar differential structure to the [r, r]-colored HOMFLY homology because there is a one-to-one correspondence between the generators of both the homologies through (4.55). Actually, in the [r]-colored Kauffman homology, one can find the counterparts of all the colored differentials inherent in the [r, r]-colored HOMFLY homology. This can be seen in Table 1, where the differentials are related by  Table 1. Comparison of the differential structure of [r]-colored Kauffman homology with that of [r, r]-colored HOMFLY homology. In the last four rows, we have the grading relation 5λ + Q − t r + t c = 4a + 2Q − t r + t c , and the t c -degrees are the same. If we reverse the direction of the universal differential d univ → , theses relations still hold in the first row.

Universal differentials
In this subsection we discuss the universal differential acting on colored Kauffman homology. The most typical feature in the differential structure of uncolored Kauffman homology is the existence of the universal differentials that relate the Kauffman homology to the HOMFLY homology [17]. It is interesting to ask if there are extensions of the universal differentials to the higher rank representations. In [12], by using the relation (4.54), the differential d + (K). Additionally, the mirror/transposition symmetry ensures that there exists the differential d univ

Diagonal differentials
The relation (4.54) predicts the existence of the differential in H Kauffman (K). In fact, it is easy to find such a differential d − diag as well as its cousin d + diag whose (λ, Q, t r , t c )-degrees on H Kauffman We call them the diagonal differentials since every generator x ∈ H * ( H Kauffman (K), d ± diag ) with respect to the diagonal differentials is isomorphic to the [r]-colored HOMFLY homology, where the precise grading changes are given by It straightforwardly follows from the mirror/transposition symmetry that the (λ, The homology H * ( H Kauffman It turns out that the diagonal differential d − diag on H Kauffman

Colored differentials
Analogous to the colored differentials in HOMFLY homology, the colored Kauffman homology is also equipped with colored differentials which send to the lower rank colored Kauffman homology: The mirror/transposition symmetry (4.29) tells us the (λ, Q, t r , t c )-degrees of the colored differentials d ± correspond to generalizations of the differential d 1 to higher rank representations. The homology with respect to the canceling differential is one-dimensional, and its grading can be written in terms of the S-invariant S(K) [59] F ( H Kauffman At general values of ∈ Z, the isomorphisms ( Here, we stress that these grading changes become clear only when we use the tilde-version of the colored Kauffman homology. On the other hand, the grading changes of the other colored differentials are not as straightforward. They are only evident at t c = 1 (for symmetric representations) or t r = 1 (for anti-symmetric representations), where (4.71)

Universal colored differentials
There exists yet another set of colored differentials, called universal colored differentials, when the color involves the Young tableaux with double boxes. The analogous differentials in HOMFLY homology are d ← and d ↑ . By using the same symbols in the Kauffman homology, the homology with respect to the universal colored differentials is isomorphic to the uncolored Kauffman homology The (λ, Q, t r , t c )-degrees of the differentials d ← and d ↑ are given by where the re-gradings in the isomorphisms (4.72) are provided by In this section, we will investigate the [r]-colored Kauffman homology and the [r, r]-colored HOMFLY homology of both the trefoil and the figure-eight in order to make the properties summarized in §4 explicit. To obtain the Poincaré polynomials, the refined exponential growth property is very helpful.

Trefoil
Before considering the colored Kauffman homology, let us recall the [r]-colored HOMFLY homology of the trefoil. The Poincaré polynomial of the quadruply-graded [r]-colored HOMFLY homology of the trefoil knot can be written as The identity of the first line (5.1) with the second line (5.2) represents the self-symmetry (4.12). Translating into the triply-graded homology P [r] (3 1 ; a, q, tq −1 , q) in the t r -grading, the expression in the first line (5.1) is equal to (3.2) in [25]. We now move on to the Kauffman homology. The uncolored Kauffman homology has been indeed obtained in [17]. Here, we write the Poincaré polynomial in the (λ, Q, t r , t c )degrees The respective homology diagram is drawn in Figure 1. In the uncolored case, the qdegree is equal to the Q-degree for each element since the t r -degree is the same as the t c -degree. In order to see the universal differential explicitly, we color the homology H * ( H Kauffman [1] (3 1 ), d univ → ) with respect to the differential d univ → with red, and the element exact under the differential d univ [2] (3 1 ; q, t) = P [4] (3 1 ; q, q 1/2 , tq −1/2 , q 1/2 ) , F [2] (3 1 ; q 3 , q, t r = t, t c = 1) = P so(4) [2] (3 1 ; q, t) = P [2] (3 1 ; q 2 , q, tq −1 , q)  that Figure 11 shows how the universal differential d ← acts on the triply-graded homology. Since the trefoil is a thin knot, it is easy to find the t c -grading in (C.1) by using δ-grading (4.25), where the S-invariant S(3 1 ) of the trefoil is 2. Then, through (4.7), one can write the tilde-version of the [2]-colored quadruply-graded Kauffman homology with the (λ, Q, t r , t c )-gradings (C.6).
Proceeding further, we will try to obtain the [r]-colored Kauffman homology, making use of the refined exponential growth property (4.32). Actually, the refined exponential growth property is so powerful that it specifies the form of the [r]-colored quadruply-graded Kauffman homology at the t c = 1 specialization To obtain the full expression, t c -gradings remain to be determined. First of all, the binomials in (5.5) are replaced by the t 2 c -binomials: e.g.
. In addition, the factors with red color at k = r accord to the homology H * ( H Kauffman (3 1 ), d univ → ) with respect to the universal differential d univ → and therefore is identical to the form Moreover, the form for the red factor is roughly of the form Q 2k P [k] (3 1 ; λQ −1 , Q, t r , t c ) while the t c -grading has to be modified in general. On the other hand, the factors colored in blue are killed by the universal differential d univ → . (See Figure 11.) This can be realized by uplifting the term (1 + Q 2 t r ) r−k in (5.5) to the t 2 c -Pochhammer symbol (−Q 2 t r t c ; t 2 c ) r−k , which is very natural, judging from the homological elements in the top λ-degree in Figure  11. In a similar fashion, the t 2 c -Pochhammer symbol (−λQ −1 t r t * c ; t 2 c ) i is substituted for the term (1 + λQ −1 t r ) i in (5.5), although the t c -degrees in the argument has to be fixed. To incorporate t c -gradings appropriately, the explicit expression (C.5) of the [2]-colored quadruply-graded Kauffman homology is inevitable. 2 By fixing t c -gradings in such a way that all the properties in §4 are satisfied, we find the Poincaré polynomial of the [r]-colored quadruply-graded Kauffman homology of the trefoil Apart from the refined exponential growth property and the universal differential, one can check that the formula has the following properties.  [(4.50)] Figure 14 and Figure 15] 4.69) and Figure 12] 2 What is written in this paragraph was explained to S.N. by Marko Stošić. S.N. would like to thank him. Figure 13] The Poincaré polynomial of the homology with respect to the colored differential d + [r]→[ ] is given by Then, one can verify (4.71): It is straightforward from (4.28) to obtain the Poincaré polynomial of the [r]-colored triplygraded Kauffman homology of the trefoil in the t r -grading We verify that the expression reduces to colored Kauffman polynomial computed by the Rosso-Jones formula [14] at t = −1 up to 4 boxes. In addition, the mirror/transposition symmetry (4.30) tells us the Poincaré polynomial of the [1 r ]-colored triply-graded Kauffman homology of the trefoil in the t c -grading For instance, the homology diagrams of the [1 2 ]-colored Kauffman homology of the trefoil are depicted in Figure 10 for the triply-gradings and Figure 16 for the quadruSple-gradings. Especially, in Figure 10, one can see the action of the differential d −2 , providing the [2]colored sp(2) homology of the trefoil P sp(2) = q 8 + q 20 t 2 + q 24 t 2 + q 24 t 3 + q 28 t 3 + q 32 t 4 + q 36 t 5 + q 40 t 5 + q 44 t 6 . Having studied the [r]-colored Kauffman homology of the trefoil, the next goal is to obtain the [r, r]-colored HOMFLY homology of the trefoil using the relation to the [r]colored Kauffman homology predicted in §4. First, it is useful to review the case of r = 1 [12]. The expression for the [1, 1]-colored HOMFLY homology follows from (5.1) via the mirror/transposition symmetry A simple calculation involving (5.3) and (5.11) confirms that F [1] (3 1 ; λ = q 5 , q, q −1 , qt) = P [1,1] (3 1 ; a = q 4 , q, q −1 , qt) . (5.12) In fact, when comparing Figure 2 with Figure 1 in the t c -grading, it is easy to see the one-to-one correspondence between the generators of Kauffman homology and those of HOMFLY homology. In addition, the differential d + (blue color). Like the colored Kauffman homology (5.5), with the great help of the refined exponential growth property (4.16), one can evaluate the t c = 1 specialization of the [r, r]-colored quadruply-graded HOMFLY homology P [r,r] (3 1 ; a, Q, t r , t c = 1) = P [1,1] On the other hand, the isomorphism (sl(4), [r, r]) ∼ = (so(6), [r]) can be seen in the identity with the t c -grading at the naive specialization λ = q 5 and a = q 4 since the trefoil is homologically thin. Thus, this relation helps us determine the t c -degrees in (5.13) by using the formula (5.7). Consequently, the Poincaré polynomial of the [r, r]colored quadruply-graded HOMFLY homology of the trefoil can be written as It is easy to confirm that the formula reproduces the This is actually expected since the colored differentials in the [r, r]-colored HOMFLY homology have their own counterparts in the [r]-colored Kauffman homology as we see in §4, and we have seen that the formula (5.7) analogous to (5.15) is endowed with the correct differential structure. As in the Kauffman homology, the Poincaré polynomial of the [r, r]-colored triplygraded HOMFLY homology of the trefoil in the t r -grading immediately follows: P [r,r] (3 1 ; a, q, t r = t, t c = 1) = P [r,r] (3 1 ; a, q 2 , q −1 t, q) = r k=0 k j=0 r−k i=0 a 4r q −2(j+k+(2+j)r−r 2 −i(2+j+r)) t 2(2i−3j+k+2r) r k The mirror/transposition symmetry yields the Poincaré polynomial of the [2 r ]-colored triply-graded HOMFLY homology of the trefoil in the t c -grading Setting r = 3, the formula reproduces the [2, 2, 2]-colored HOMFLY homology of the trefoil in §4.5 of [13].

Figure-eight
In this subsection, we obtain the [r]-colored Kauffman homology and the [r, r]-colored HOMFLY homology of the figure-eight. The strategy is the same as the case of the trefoil although the size of the homology is bigger and therefore the computations are more tedious. Hence, we will not repeat the detailed explanations for the method. (4 1 ), d univ → ). The red and blue dots correspond to the fact colored by red and blue in (5.21) respectively.
As in the case of the trefoil, let us start with writing the Poincaré polynomial of the [r]-colored quadruply-graded HOMFLY homology of the figure-eight (5.20) The expression P [r] (4 1 ; a, q, tq −1 , q) for the triply-graded homology in the t r -grading is equal to (3.3) in [25]. The uncolored HOMFLY homology of the figure-eight is given in [17] F (4 1 ; λ, Q, t r , t c ) whose homology diagram is presented in Figure 3. Unlike the case of the trefoil, the colored Kauffman polynomials of the figure-eight are not available to date. However, using the refined exponential growth property, the representation theoretic relation and the differential property, one can uniquely determine the [2]-colored Kauffman homology of the figure-eight F (41; λ, q, tr = t, tc = 1) +4t + 14q 2 t + 12q 4 t + 2q 6 t + q 2 t 2 + 4q 4 t 2 + 7q 6 t 2 + 2q 8 t 2 + q 8 t 3 + q 10 t 3 +λ 4 q 5 + 10 q 3 + 6 q + 2 q 7 t + 3 q 5 t + 1 q 3 t + 3t q 3 + 14t q + 14qt + 3q 3 t + t 2 q + 10qt 2 +16q 3 t 2 + 7q 5 t 2 + 3q 3 t 3 + 9q 5 t 3 + 7q 7 t 3 + q 9 t 3 + 2q 7 t 4 + 3q 9 t 4 + q 11 t 4 +λ 2 1 q 6 + t + 3t q 4 + 4t q 2 + 10t 2 + 4t 2 q 2 + 4q 2 t 2 + 3t 3 + 11q 2 t 3 + 9q 4 t 3 + q 6 t 3 +q 2 t 4 + 5q 4 t 4 + 9q 6 t 4 + 3q 8 t 4 + q 6 t 5 + 4q 8 t 5 + 3q 10 t 5 + q 12 t 6 +λ 3 t 2 q 3 + t 2 q + 3t 3 q + 3qt 3 + 3qt 4 + 5q 3 t 4 + 2q 5 t 4 + q 3 t 5 + 5q 5 t 5 + 4q 7 t 5 + 2q 7 t 6 + 3q 9 t 6 +q 11 t 6 + q 11 t 7 + q 13 t 7 +λ 4 t 4 + q 2 t 5 + q 4 t 5 + q 4 t 6 + 2q 6 t 6 + q 8 t 7 + q 10 t 7 + q 12 t 8 (5.22) Here, the expression is written in the triply-grading with the t r -grading, and it has 625 generators. Using the δ-grading (4.25) where the S-invariant S(4 1 ) of the figure-eight is 0, one can assign the t c -gradings in (5.22). Note that the colored Kauffman homology obeys the following identity since the figure-eight is the same as its mirror image: The method to obtain the [r]-colored Kauffman homology of the figure-eight is the same as in the case of the trefoil although it is more tedious due to its size. The refined exponential growth property determines the t c = 1 specialization of the Poincaré polynomial of the [r]-colored quadruply-graded Kauffman homology of the figure-eight. Then, the t c -gradings are fixed by the differential structure and the [2]-colored Kauffman homology, yielding the full expression By construction, the red factors in (5.7) are very close to Q 2k P [k] (4 1 ; λQ −1 , Q, t r , t c ) and the blue factors are killed by the universal differential d univ → due to the presence of the t 2 c -Pochhammer (−Q 2 t r t c ; t 2 c ) r−k . One can check that the formula satisfies all the structural properties innate in the [r]-colored Kauffman homology. Subsequently, the Poincaré polynomial of the [r]-colored triply-graded Kauffman homology of the figure-eight in the t r -grading can be expressed by The mirror/transposition symmetry provides the Poincaré polynomial of the [1 r ]-colored triply-graded Kauffman homology of the trefoil in the t c -grading The t c = 1 specialization of the Poincaré polynomial of the [r, r]-colored quadruply-graded HOMFLY homology of the figure-eight is determined by the refined exponential growth property and the t c -gradings can be eventually given by using (5.24). As a result, we can write a closed form expression The Poincaré polynomial of the [r, r]-colored triply-graded HOMFLY homology of the figure-eight in the t r -grading P [r,r] (4 1 ; a, q, t r = t, t c = 1) = P [r,r] (4 1 ; a, q 2 , q −1 t, q) Indeed, setting r = 2, this formula decategorifies at t = −1 to the HOMFLY polynomial colored by [2,2]-representation written in (E.1) of [61]. The Poincaré polynomial of the [2 r ]-colored triply-graded HOMFLY homology of the figure-eight in the t c -grading can be written as

Super-A-polynomials
In the last fifteen years, remarkable results have been obtained by looking at the large color behaviors of colored Jones invariants, i.e. the volume conjectures. (See a comprehensive review [62] and references therein.) It is apparent that the volume conjecture [63,64] is a key to understanding the relationship between quantum invariants of a knot K and classical geometry of the knot complement S 3 \K. Surprisingly, the large color behavior of colored Jones invariants is dominated by SL(2, C) flat connections rather than SU (2) [65]. Hence, it is more directly related to analytically continued SL(2, C) Chern-Simons theory [65][66][67]. Let us briefly review the conjecture below. Let g be either sl(2) or so (3). Certainly, the corresponding gauge group G in Chern-Simons theory is either SU (2) or SO(3) respectively. Since representations of g are specified by Young tableaux with a single row, the colored g quantum invariants of a knot can be expressed by J g [r] (K, q). If one takes the double scaling limit r → ∞ and q = e → 1 with q r = x fixed, the invariant J g [r] (K; q) is conjectured to take the form where the integral is carried out on the zero locus A G C (K; x, y) = 0 of the A-polynomial. It is known that the A-polynomial A G C (K; x, y) of a knot K is the character variety of G Crepresentation ρ : π 1 (S 3 \K) → G C of the fundamental group of the knot complement [68]. Note that the complexification G C of the gauge group G is either SL(2, C) or P SL(2, C), respectively. In fact, the moduli space M flat (SL(2, C), T 2 ) of SL(2, C) flat connections on the boundary torus is a hyper-Kähler manifold C × × C × /Z 2 , where C × × C × is spanned by the holonomy eigenvalues of the SL(2, C) gauge connection along the meridian x and the longitude y, and Z 2 is the Weyl group symmetry of the gauge group SL(2, C). where Ξ = Z 2 × Z 2 is generated by (x, y) → (−x, y) and (x, y) → (x, −y) [69]. In addition, it is shown in [69] that the P SL(2, C) character variety can be written in terms of the SL(2, C) character variety Consecutively, the volume conjecture has been extended to the quantum version, called the quantum volume conjecture or the AJ conjecture [65,70,71]. Namely, the quantization of the A-polynomial becomes the q-holonomic function of the knot invariants: where the operatorsx andŷ act on the set of the colored quantum invariants aŝ Therefore, the q-difference equation of the colored quantum invariants of minimal order amounts to the quantum A-polynomial A G C (K;x,ŷ; q) = k j=0 b j (x, q)ŷ j where taking q = e = 1 gives the classical A-polynomial A G C (K; x, y).
Recently, generalizations of these conjectures have been proposed by incorporating [r]-colored HOMFLY polynomials and their categorifications [20,24,52,72]. Specifically, the q-difference equation and the large color behavior of the Poincaré polynomial of a [r]-colored HOMFLY homology are called the quantum and classical super-A-polynomial [24]. In this paper, we call it the super-A-polynomial A SU (K; x, y; a, t) of SU -type. The explicit computations have been performed for the (2, 2p + 1)-torus knots and the twist knots [24][25][26].
Let us extend the notion of super-A-polynomials by including Poincaré polynomials of [r]-colored Kauffman homology. In the limit where the integral is carried out on the zero locus of the classical super-A-polynomial of SO-type We conjecture that the q-difference equation of minimal order for the Poincaré polynomials of Kauffman homology, b k (x; λ, q, t)F [r+k] (K; λ, q, t, 1) + · · · + b 0 (x; λ, q, t)F [r] (K; λ, q, t, 1) = 0 , (6.11) provides the quantum super-A-polynomial of SO-type where the operatorsx andŷ act on F [r] (K; λ, q, t, 1) as in (6.6), so that its classical limit q → 1 is equal to A SO (K;x,ŷ; λ, t) up to factors. The same procedure for the [r]-colored Kauffman polynomial F [r] (K; λ, q) leads to the quantum A SO (K;x,ŷ; λ, q) and classical λ-deformed A-polynomial A SO (K; x, y; λ) of SO-type. We emphasize that the t = −1 specialization of the super-A-polynomial is not necessarily equal to the λ-deformed A-polynomial though it always contains the λ-deformed A-polynomial. Furthermore, the λ = 1, t = −1 specialization of the super-A-polynomial embraces the P SL(2, C) character variety. The statement holds true for SU -type as well.
As conjectured in [13,20], we also predict that there is the relation between the super-A-polynomial of SO-type and the Poincaré polynomial of the uncolored Kauffman homology for any knot K: (6.13)
Since the expression (5.8) involves triple summations, it is very difficult to find the q-difference equation for it. Nevertheless, one can check whether the classical super-Apolynomial satisfies the condition for the quantizability [73]. It is argued in [65,73] that the integral of the one-form log y dx x along a one-cycle γ on the algebraic curve A SO (K; x, y; λ, t) = 0 must be subject to the Bohr-Sommerfeld condition in order for a classical super-A-polynomial A SO (K; x, y; λ, t) to be quantizable. If one writes the Newton polygon of A SO (K; x, y; λ, t), its faces correspond to punctures of the algebraic curve A SO (K; x, y; λ, t) = 0. Then, the Bohr-Sommerfeld condition around a puncture amounts to all roots of the corresponding face polynomial are roots of unity. Thus, the necessary condition for the quantizability is that the classical super-A-polynomial A SO (K; x, y; λ, t) is tempered [73]. Face Face polynomials The Newton polygon of the super-A-polynomial A SO (3 1 ; x, y; λ, t) and its face polynomials are shown in Figure 5 and Figure 6. Writing A SO (3 1 ; x, y; λ, t) = i,j c i,j (λ, t)x i y j , the Newton polygon is designed by plotting red circles for monomials c ij = 0 and yellow crosses for monomials at the special limit c ij (λ = 1, t = −1) = 0. The faces of the Newton polygons are denoted by the dotted line in Figure 5. For a given face, we rename the monomial coefficients on the face as c k . Then, the face polynomial is defined to be f (z) = k c k z k . Assuming that the variables λ, t are roots of unity, the quantizability condition requires that all roots of f (z) constructed for all faces of the Newton polygon must be roots of unity. Therefore, it is easy to see from Figure 6 that the classical super-A-polynomials A SO (3 1 ; x, y; λ, t) satisfy the necessary condition of quantizability.
At t = −1, the super-A-polynomial becomes while the analysis for the large color behavior of the Kauffman polynomials F [r] (K; λ, q) leads to the λ-deformed A-polynomial of SO-type A SO (3 1 ; x, y; λ) = y − xyλ 2 − x 6 λ 7 + x 7 λ 7 . (6.21) Therefore, we can clearly see that A SO (3 1 ; x, y; λ, t = −1) = A SO (3 1 ; x, y; λ) although A SO (3 1 ; x, y; λ, t = −1) contains A SO (3 1 ; x, y; λ). Furthermore, the λ = 1, t = −1 specialization of the super-A-polynomial can be written as whereas the P SL(2, C) character variety of the trefoil is expressed by In fact, the large color limit of the [r]-colored so(3) quantum invariant of the trefoil, i.e. F [r] (3 1 ; λ = q 2 , q, t = −1, 1), provides only the non-abelian branch y − x 6 = 0. Hence, ignoring the trivial factor 1 − x = 0, the specialization of the super-A-polynomial contains not only the abelian branch 1 − y = 0 but also the extra non-abelian branch x 3 + y = 0. We postpone further study of understanding its meaning to future work. In the case of the colored so(3) quantum invariants, we can find the q-difference equation, providing the quantum P SL(2, C) character variety of the trefoil A simple computation shows that it reduces to (6.23) up to trivial factors at q = 1.

Figure-eight
Now, let us consider the figure-eight. In the limit (6.8), the expression (5.25) behaves as F [r] (4 1 ; λ, q, t, 1) ∼ e 1 2 W SO (4 1 ;x,w,v,z) dwdvdz , (6.25) with the twisted superpotential In this case, the saddle points are given by the following system of equations: .

(6.27)
With a current desktop computer, it is difficult to solve the above set of the equations for general value of t. 3 Thus, we solve it only for the special case t = −1 which gives us the λ-deformed classical A-polynomial (6.28) In the sequel, one can confirm that the λ-deformed classical A-polynomial A SO (4 1 ; x, y; λ) obeys the quantizability condition, which can be seen in Figure 7 and Figure 8.  At λ = 1, it reduces to the A-polynomial associated to the P SL(2, C) character variety of the figure-eight

Face Face polynomials
The third factor matches with the non-abelian branch of the P SL(2, C) character variety of the figure-eight. (See Example 1 in §6.0.12. of [74].) In fact, the SL(2, C) character variety of the figure-eight is given by Therefore, A SO (4 1 ; x, y; λ = 1) is equal to A SL(2,C) (4 1 ; x 1/2 , y 1/2 )A SL(2,C) (4 1 ; x 1/2 , −y 1/2 ) up to the trivial factors as stated in (6.4). In addition, we can obtain the q-difference equation for the colored so(3) quantum invariants shown in Table 5.
7 3d/3d correspondence One of the most remarkable developments in recent years has been the program of studying a duality arising from the compactification of the 6d (2,0) superconformal field theory on a certain manifold. Particularly, the partially twisted compactification of the 6d (2,0) theory with Lie algebra g on a 3-manifold M leads to a 3d N = 2 supersymmetric gauge theory . This correspondence has been lately placed on a rigorous footing by the localization technique [79][80][81].
The explicit IR descriptions of T sl (2) [M ] for a large class of 3-manifolds M have been established by abelian N = 2 gauge theories, with possibly non-perturbative superpotentials that preserve U (1) R [77,78]. In particular, the cases in which a 3-manifold M is a knot complement S 3 \K have been intensively investigated [25,77,78,82,83]. Moreover, in this setting, the relation between "holomorphic blocks" B α (x; q) and colored Jones polynomials has been investigated in [82]. Roughly speaking, a holomorphic block B α (x; q) is the partition function on S 1 × R, labelled by a choice of vacuum α at the asymptotic boundary of spatial R 2 whose form is with fugacity x for the U (1) flavor symmetry. Note that the fugacity x is identified with the holonomy eigenvalue x of the gauge connection along the meridian of the tubular neighborhood of K in SL(2, C) Chern-Simons theory. Given an abelian N = 2 gauge theory, the holomorphic block can be schematically expressed as where we choose the appropriate integration cycle Γ α ⊂ (C × ) s , with s being the number of the U (1) gauge groups. The variables z ∈ (C * ) s are complexified scalars in the gauge multiplets, and each chiral multiplet contributes a single block to the integrand. Note that the parameter w i = w i (z, x) depends on the scalars z and the fugacity x. Here, the theta-functions θ(w; q) encode contributions of Chern-Simons and Fayet-Iliopoulos (FI) terms. Then, it was observed in [82] that the holomorphic block B(x = q r ; q) of T sl (2) [S 3 \K] at x = q r coincides with the stable limit of the colored Jones polynomial J [r] (K; q). Nevertheless, the holomorphic block B(x = q r ; q) of T sl (2) [S 3 \K] does not reproduce the colored Jones polynomial exactly. This appears to be related to the fact that the way the theory T sl (2) [S 3 \K] is constructed captures only the non-abelian branch, but not the abelian branch of the SL(2, C) character variety. Equivalently, this can be rephrased that the large color limit of colored Jones polynomials provides only the non-abelian branch. 4 On the other hand, as we have seen in §6, a super-A-polynomial encode much richer information than a ordinary A-polynomial. The most important fact is that it intrinsically encompasses the abelian branch. Thus, it seems more appropriate to consider the 3d/3d correspondence in this setting. As discussed in [24,25], the parameters a or λ, and t can be interpreted as fugacities in the index for certain global symmetries U (1) bulk and U (1) F in the context of N = 2 gauge theory. Thus, super-A-polynomials carry important information about N = 2 gauge theories with those symmetries. Taking into account these features, it would certainly be interesting to elucidate the relation between the "refined" holomorphic blocks and the Poincaré polynomials of knot homology [84].
As in the sl(2) case, the IR descriptions of T so(3) [S 3 \K] can be constructed by abelian N = 2 gauge theories. In fact, the Kaluza-Klein reduction on S 1 with q → 1 brings (7.2) to the form given in (6.14) and (6.25). In the leading contribution, a single block B ∆ yields a dilogarithm (Li 2 ) to the twisted superpotential. Hence, each dilogarithm (Li 2 ) term in the twisted superpotential W expresses the contribution from a chiral field φ i . In (6.14) and (6.25), the parameters w, v and z can be interpreted as scalars in U (1) gauge multiplets. Therefore, if the j-th dilogarithm term in W is ±Li 2 (x n 1 (−t) n 2 a n 3 z n 4 i ), then the chiral field φ j will have charges ±n 1 , ±n 2 , ±n 3 , ±n 4 respectively under the U (1) x , U (1) F , U (1) bulk global symmetries and U (1) z i gauge groups. In addition, the theta-functions reduces to the term − log x i ·log z j for the FI coupling and the term k ij 2 log z i ·log z j for the supersymmetric Chern-Simons coupling k ij 4π A i ∧ dA j . Using this dictionary, one can read off the theory T so(3) [S 3 \3 1 ] and T so(3) [S 3 \4 1 ] where the U (1) charges of matter contents are depicted in Table 3 and Table 4. Table 3. U (1) charges of the chiral fields corresponding to the twisted superpotential W SO (3 1 ; x, w, v, z). Table 4. U (1) charges of the chiral fields corresponding to the twisted superpotential W SO (4 1 ; x, w, v, z).

Future directions
The study on the structure of the [r]-colored Kauffman homology that we have implemented raises several questions. Although we focused on thin knots of the simplest class so far, it would be desirable to study whether these structural properties hold for the Kauffman homology of thick knots. However, the size of colored Kauffman homology for thick knots would be too big to study with our current technique. Nevertheless, it is worth making the following comments. Given the uncolored Kauffman homology of the (3, 4)-torus knot T 3,4 (Appendix B in [12]), its uncolored so(4) homology should be isomorphic to the homology with respect to the d 4 differential. However, if we use the differential whose (λ, q, t r , t c )degree is deg d 4 = (−1, 3, −1, −1) as in (4.43), the homology H * (H (T 3,4 ), d 4 ) would be 41-dimensional while the uncolored so(4) homology of T 3,4 should be 25-dimensional, which follows from the 5-dimensional uncolored sl(2) homology of T 3,4 . This indicates that one has to take into account the spectral sequence [57] for the d 4 differential in the case of thick knots.
Another important problem concerns the interpretation of the differentials inherent in colored Kauffman homology. In [17], the differentials in uncolored Kauffman homology are elucidated by the deformations of the Landau-Ginzburg potential in B-model. One way to relate the colored Kauffman homology to the Landau-Ginzburg B-model is the algebraic model proposed in [13]. In this model, the colored HOMFLY homology of a torus knot is identified with a space of differential forms on a certain reduced moduli space. This description is certainly suitable for the interpretation in terms of the Landau-Ginzburg model. Hence, the route via the algebraic model may connect to the explanation of the differentials in terms of the Landau-Ginzburg B-model . Furthermore, it was observed in [13] that the colored differentials present in the HOM-FLY homology can be associated with the representation of the Lie superalgebras sl(n|m). Therefore, it would be intriguing to consider the relation to Lie superalgebra and its physical meaning in the context of the colored Kauffman homology.
It is also worthwhile to find a new way to look at the λ-deformed A-polynomials of SO-type. It is conjectured in [52] that the a-deformed A-polynomials of SU -type can be identified with the augmentation polynomial of knot contact homology [85,86]. This conjecture has been verified in the cases of the (2,2p+1)-torus knots [24] and the twist knots [25,26]. It is natural to ask whether or not one can construct augmentation polynomials in knot contact homology analogous to λ-deformed A-polynomials of SO-type.
Finally, one may be interested in extracting information about quantum 6j-symbols for U q (so(N )) from the expressions for the invariants of the trefoil and the figure-eight obtained in §5 as done in [87] for U q (sl(N )). For this purpose, the relations from representation theory will be also useful. In fact, quantum 6j-symbols for U q (so(N )) would enable us to evaluate colored so(N ) quantum invariants for any knot. The feasibility of this approach is currently under investigation.

A Conventions and notations
Polynomial invariants The reduced g quantum invariant of a knot K colored by a representation R of g J g R (K; q) The unreduced g quantum invariant of a knot K colored by a representation R of g P R (K; a, q) The reduced HOMFLY polynomial of a knot K colored by a representation R P R (K; a, q) The unreduced HOMFLY polynomial of a knot K colored by a representation R P (R,S) (K; a, q) The unreduced HOMFLY polynomial of a knot K colored by a composite representation (R, S) F R (K; λ, q) The reduced Kauffman polynomial of a knot K colored by a representation R F R (K; λ, q) The unreduced Kauffman polynomial of a knot K colored by a representation R

Knot homology
The reduced quadruply-graded HOMFLY homology colored by a representation R with (a, q, t r , t c )-gradings ( H HOMFLY The reduced quadruply-graded HOMFLY homology colored by a representation R with (a, Q, t r , t c )-gradings H * (H HOMFLY The reduced quadruply-graded Kauffman homology colored by a representation R with (λ, q, t r , t c )-gradings ( H Kauffman The reduced quadruply-graded Kauffman homology colored by a representation R with (λ, Q, t r , t c )-gradings The homology of H Kauffman R (K) with respect to a differential d acting on (H Kauffman The homology of H Kauffman R (K) with respect to a differential d acting on ( H Kauffman R (K)) i,j,k,. with (λ, Q, t r , t c )-gradings Poincaré polynomials P R (K; a, q, t r , t c ) := i,j,k, The SL(2, C) character variety of the complement of a knot K in S 3 A P SL(2,C) (K; x, y) The P SL(2, C) character variety of the complement of a knot K in S 3 A P SL(2,C) (K;x,ŷ; q) The quantum P SL(2, C) character variety of the complement of a knot K in S 3 A SU (K; x, y; a) The a-deformed A-polynomial of SU -type for a knot K A SO (K; x, y; λ) The λ-deformed A-polynomial of SO-type for a knot K A SO (K;x,ŷ; λ, q) The quantum λ-deformed A-polynomial of SO-type for a knot K A SU (K; x, y; a, t) The super-A-polynomial of SU -type or a knot K A SO (K; x, y; λ, t) The super-A-polynomial of SO-type for a knot K