On knots, complements, and 6j-symbols
Hao Ellery Wang, Yuanzhe Jack Yang, Hao Derrick Zhang, Satoshi Nawata
PPrepared for submission to JHEP
On knots, complements, and j -symbols Hao Ellery Wang, Yuanzhe Jack Yang, Hao Derrick Zhang, and Satoshi Nawata
Department of Physics and Center for Field Theory and Particle Physics, Fudan University, 220,Handan Road, 200433 Shanghai, China
E-mail: [email protected], [email protected],[email protected], [email protected]
Abstract:
This paper investigates the relation between colored HOMFLY-PT and Kauff-man homology,
SO( N ) quantum j -symbols and ( a, t ) -deformed F K . First, we present asimple rule of grading change which allows us to obtain the [ r ] -colored quadruply-gradedKauffman homology from the [ r ] -colored quadruply-graded HOMFLY-PT homology forthin knots. This rule stems from the isomorphism of the representations ( so , [ r ]) ∼ =( sl , [ r ]) . Also, we find the relationship among A -polynomials of SO and SU -type comingfrom a differential on Kauffman homology. Second, we put forward a closed-form expressionof SO( N )( N ≥ quantum j -symbols for symmetric representations, and calculate thecorresponding 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 doubletwist knots with positive braids. Using the conjectural expressions, we derive t -deformedADO polynomials. Keywords:
Knot homology, Super- A -polynomial, j -symbols, F K , ADO polynomial a r X i v : . [ h e p - t h ] D ec ontents A -polynomial 4 A -polynomials of SO-type 6 SO( N ) quantum j -symbols 9 SO( N ) quantum j -symbols for symmetric representations 93.2 SO( N ) fusion matrices for symmetric representations 11 F K and ADO polynomials 14 F K of double twist knots 144.2 t -deformed ADO polynomials 16 A.1 ADO polynomials of double twist knot K m,n ( m, n ∈ Z + ) K m + , − n ( m, n ∈ Z + ) Witten’s celebrated paper [1] has shown that the Chern-Simons theory provides a natu-ral framework for quantum invariants of 3-manifolds and knots. The Chern-Simons pathintegral on a 3-manifold M with the action S = k π (cid:90) M Tr (cid:18) A ∧ dA + 23 A ∧ A ∧ A (cid:19) , is a quantum invariant of M , known as the Witten-Reshetikhin-Turaev (WRT) invariant[1, 2]. If a Wilson loop is included along a knot in S , the expectation value of the Wilsonloop is a quantum invariant of the knot: J GR ( K ; q ) := (cid:104) W R ( K ) (cid:105) = (cid:82) [ D A ] e iS W R ( K ) (cid:82) [ D A ] e iS , where the parameter q is expressed by the level k and the dual Coxeter number h ∨ of thegauge group G as q = exp (cid:18) πik + h ∨ (cid:19) . (1.1)– 1 –ince the Chern-Simons functional integral on a three-manifold with boundary is an ele-ment of the Hilbert space on the boundary which is isomorphic to the space of WZNWconformal blocks, quantum knot invariants could be constructed by using braiding and fu-sion operations on WZNW conformal blocks [3–7]. When G = SU( N ) , it corresponds to atwo-variable colored HOMFLY-PT polynomial P R ( K ; a, q ) , where R is a representation of SU( N ) and a = q N . When G = SO( N ) , it corresponds to colored Kauffman polynomial F R ( K ; a, q ) , where R is a representation of SO( N ) and a = q N − . The quantum knotinvariants 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 [8],i.e. the categorifications of quantum knot invariants. Khovanov constructed a bi-gradedhomology which itself is a knot invariant, and its q -graded Euler characteristic is the Jonespolynomial J ( K ; q ) of a knot K . In [9], M. Khovanov and L. Rozansky constructed a triply-graded homology of a knotwhose graded Euler characteristic is the HOMFLY-PT polynomial. And in [10], it wasconjectured that the HOMFLY-PT homology is endowed with structural properties. Sim-ilarly, the existence and structural properties of Kauffman homology were conjectured in[11] although the rigorous definition of a triply-graded homology theory categorifying theKauffman polynomials has not been given yet.This line of studying knot homology has been pursued in the colored cases. In [12, 13],quadruply-graded colored HOMFLY-PT homology was proposed and various structuresand symmetries were uncovered. In [14], it was further conjectured that there exist thequadruply-graded colored Kauffman homologies with rich structural properties. The firstpart of this paper investigates the relationship between the colored HOMFLY-PT homologyand Kauffman homology. In §2.1, we show a rule of changing variables that transformsthe Poincaré polynomial of [ r ] -colored HOMFLY-PT homology into that of [ r ] -coloredKauffman homology for thin knots.As another important development, the volume conjecture [15, 16] gives a remarkablerelationship between quantum invariants of a knot K and classical geometry of the knotcomplement S \ K . It states that the large color asymptotic behavior of colored Jonespolynomial J [ r ] ( K ; q ) of a hyperbolic knot provides the hyperbolic volume of its knot com-plement. This was further generalized in [17, 18], which gives the relation between the largecolor 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 varietyof SL(2 , C ) -representation of the knot group [19] so that the generalized volume conjec-ture paves a way to complex Chern-Simons theory. There are further generalizations ofthe conjecture to incorporate the case of [ r ] -colored HOMFLY-PT polynomials and theircategorifications [14, 20–22], leading to a notion of super- A -polynomials. Since there is a We define the normalized quantum invariant J GR ( K ; q ) = J GR ( K ; q ) J GR ( ; 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. – 2 –ifferential that relates colored Kauffman homology to HOMFLY-PT homology, we inves-tigate the influence of the differential on super- A -polynomials in §2.2. This brings aboutConjecture 2.1 that provides an intriguing relationship among A -polynomials of SU and SO -type.As briefly mentioned, quantum invariants can be evaluated with braiding and fusionmatrices on the WZNW models. When q is a root of unity (1.1), the fusion matrices ofthe WZNW models are equivalent to quantum j -symbols of the corresponding quantumgroups (up to normalizations (3.11)) [3, 23–27]. The j -symbols were introduced by Wigner[28, 29] and Racah [30] in the study of angular momenta, and they are beautifully expressedby a hypergeometric series F . In the case of sl , the explicit expression for quantum j -symbols was obtained in [31], which is a natural quantization with a q -hypergeometric series ϕ . Nonetheless, it is notoriously involved to obtain explicit expressions for j -symbolsfrom the definition with the Clebsch-Gordan coefficients ( j -symbols), in particular, forhigher ranks [32]. However, the information of quantum j -symbols with all symmetricrepresentations was extracted from quantum knot invariants for U q ( sl N ) [33], and it wasformulated in terms of the q -hypergeometric series ϕ in [34]. In §3, we study the SO( N ) quantum j -symbols with symmetric representations along this line. The formulae for theclassical SO( N ) 6 j -symbols with all representations symmetric were obtained by Ališauskas[35, 36]. We quantize these formulae and compute SO( N ) fusion matrices of the WZNWmodels for the case of R = , . We check that the fusion matrices reproduce the coloredKauffman polynomials of non-torus knots. This gives a piece of evidence that the naturalquantization of the Ališauskas’ formulae will be valid for the SO( N ) quantum j -symbols.The developments of knot invariants and volume conjectures are inseparably boundup 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 [2]when q is a root of unity (1.1). In the recent study of the 3d/3d correspondence, an analyticcontinuation (cid:98) Z of the WRT invariant has been proposed as a BPS q -series [37, 38]. Thisinvariant can be thought of as a generating series of BPS states in a 3d N = 2 theory T [ M ] . Also it can be regarded as a partition function of complex Chern-Simons theoryon M defined on a unit disk | q | < whose radial limit q (cid:38) e πi/ ( k + h ∨ ) becomes the WRTinvariant. Its generalization to a knot complement, denoted as F K ( x, q ) = (cid:98) Z ( S \ K ) , isgiven in [39] to make surgeries and TQFT method work. It is observed that the asymptoticexpansion of F K agrees with the Melvin-Morton-Rozansky expansion [40–43] of the coloredJones polynomials. Moreover, F K is conjectured to be annihilated by the quantum A -polynomial [44, 45] of the corresponding knot K , ˆ A ( K ; ˆ x, ˆ y, q ) F K ( x, q ) = 0 . (1.2)Moreover, based on [46], it is proposed in [47] that the limit q → ζ p = e πi/p of F K leads tothe p -th ADO polynomials [48].The development of F K is currently following that of quantum knot invariants althoughits definition is not given yet. In [49], an extension of (cid:98) 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 [50]. – 3 –ollowing 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 deriveclosed-form formulae of t -deformed ADO polynomials. A -polynomial The categorifications of quantum knot invariants shed new light on knot theory not onlybecause knot homologies are more powerful than quantum invariants but also becausethey are functorial [51]. Moreover, they are endowed with rich structural properties [52],and various differentials give remarkable relationships among themselves [10, 12, 13, 53].In particular, some relations between HOMFLY-PT and Kauffman homology have beenuncovered in [11, 14]. The rich structural properties also help us find closed-form expressionsof Poincaré polynomials of cyclotomic type [54] for various knots and links.In this section, we will show a new relationship between colored HOMFLY-PT andKauffman homology for thin knots. Also, we investigate the effect of differentials on super- A -polynomials of SU and SO -type. It has been apparent that the rich structural properties become manifest if we intro-duce quadruple-gradings to HOMFLY-PT and Kauffman homology. The quadruply-gradedHOMFLY-PT homology ( H HOMFLY-PT R ( K )) i,j,k,l with ( a, q, t r , t c ) -gradings was introducedin [13]. With this grading, we can associate a δ -grading to every generator x of the [ r ] -colored HOMFLY-PT homology defined as, δ ( x ) := a ( x ) + q ( x )2 − t r ( x ) + t c ( x )2 . A knot K is called homologically-thin if all generators of H HOMFLY-PT [ r ] ( K ) have the same δ -grading equal to r S ( K ) , where S ( K ) is the Rasmussen s -invariant of K [51]. Otherwise,it is called homologically-thick. In general, thick knots possess more intricate structures[10, 12].In a similar manner, the structral properties of quadruply-graded Kauffman homology ( H Kauffman R ( K )) i,j,k,l were studied in [14]. In the following discussion in this section, we alsouse the tilde versions, (cid:102) H HOMFLY-PT [ r s ] ( K ) [13], (cid:102) H Kauffman [ r s ] ( K ) [14], and their correspondingPoincaré polynomials (cid:102) P [ r s ] ( K ; a, Q, t r , t c ) := (cid:88) i,j,k,l a i Q j t kr t lc dim (cid:16) (cid:102) H HOMFLY-PT [ r s ] ( K ) (cid:17) i,j,k,l , (2.1) (cid:102) F [ r s ] ( K ; a, Q, t r , t c ) := (cid:88) i,j,k,l a i Q j t kr t lc dim (cid:16) (cid:102) H Kauffman [ r s ] ( K ) (cid:17) i,j,k,l , (2.2)where the gradings of H and (cid:102) H are related by F [ r s ] ( K ; a, q, t r , t c ) := (cid:102) F [ r s ] ( K ; a, q s , t r q − , t c q ) , (2.3)– 4 – [ r s ] ( K ; a, q, t r , t c ) := (cid:102) P [ r s ] ( K ; a, q s , t r q − , t c q ) . (2.4)It turns out that there are differentials, called universal and diagonal in [14], thatrelate [ r ] -colored Kauffman and HOMFLY-PT homology. Furthermore, the well-knownisomorphism of the representations [14] ( so , [ r ]) (cid:39) (cid:0) sl , [ r ] (cid:1) , leads to an isomorphism between bi-graded homologies H so , [ r ] ( K ) ∼ = H sl , [ r ] ( K ) . (2.5)For a thin knot K thin , this can be stated in terms of the Poincaré polynomials [14, (4.57)] (cid:102) F [ r ] ( K thin ; a = q , Q = q, t r = q − , t c = qt ) = (cid:102) P [ r ] ( K thin ; a = q , Q = q , t r = q − , t c = qt ) . (2.6) − − Q a
00 8462 4253 73 95 115126 t r t c − − Q a
03 22 34 55 6
Figure 1 : An isomorphism between [1 ] -colored HOMFLY-PT homology (left) and un-colored Kauffman homology (right) for the trefoil. Generators with the same color areidentified under the grading change (2.7).Moreover, we find an isomorphism between [1 ] -colored HOMFLY-PT homology andthe uncolored Kauffman homology of the trefoil and the figure-eight. For example, there is aone-to-one correspondence between their generators with the same color, shown in Figure 1.This motivates us to uplift (2.5) to an isomorphism between HOMFLY-PT and Kauffmanhomologies in the case of a thin knot K thin with grading change (cid:16) (cid:102) H Kauffman [ r ] ( K thin ) (cid:17) i,j,k,l ∼ = (cid:16) (cid:102) H HOMFLY-PT [ r ] ( K thin ) (cid:17) i + j − k, − i − j + k,i + j,l , (2.7)up to some overall grading shifts proportional to rS ( K )2 in ( a, Q, t r ) degrees. This can beexpressed in terms of the Poincaré polynomials (cid:102) F [ r ] ( K thin ; a, Q, t r , t c ) = (cid:18) Q at r (cid:19) rS ( K )2 (cid:102) P [ r ] ( K thin ; a Q − t r , a Q − t r , a − Q , t c ) . (2.8)We would like to emphasize that the isomorphism holds only for thin knots, but it isnot true for thick knots. For thick knots, the dimensions of [ r ] -colored quadruply-graded– 5 –OMFLY-PT homology and [ r ] -colored quadruply-graded Kauffman homology are different[12, Appendix B].Remark that the relation (2 . only provides the method to obtain the Poincaré poly-nomial of [ r ] -colored Kauffman homology from that of [ r ] -colored HOMFLY-PT homologyfor a thin knot, but it does not allow us to use this relation in the opposite direction sincethe grading shifts of variables in (2.8) are not linearly independent.In [55, 56], the Poincaré polynomial of quadruply-graded HOMFLY-PT homology ofthe double twist knot K m,n colored by rectangular Young tableau [ r s ] has been given. (SeeFigure 2 for the notation.) Since the double twist knots are 2-bridge knots which are knownto 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 (2.8). Toverify the relations (2.7) and (2.8), we have checked that the colored Kauffman homologyobtained by this substitution satisfies all the differentials and properties [14] of the coloredKauffman homology for double twist knots. We also attached the Mathematica files thatshow these properties on the arXiv page. A -polynomials of SO-type The study of the large color behaviors of colored Jones polynomials led us to the volumeconjecture [16, 17]. The generalization of the volume conjecture to the large color behaviorof the Poincaré polynomial led to the so-called super- A -polynomial [14, 22]. As brieflymentioned above, there are the differentials that relate [ r ] -colored Kauffman homology andHOMFLY-PT homology. In this subsection, we will investigate the effect of the differentialon super- A -polynomials.To this end, let us first recall the generalized volume conjecture [17, 18] and thesuper- A -polynomials [22]. Here we use Poincaré polynomials of triply-graded homology Q [ r ] ( K ; a, q, t ) only with t r -grading, also called the superpolynomial, defined as Q [ r ] ( K ; a, q, t ) := Q [ r ] ( K ; a, q, t r = t, t c = 1) , (2.9)where Q = P or F represents Poincaré polynomials of either HOMFLY-PT or Kauffmanhomology. If a knot K satisfies the exponential growth property [12, 52] (such as thin knotsand torus knots) Q [ r ] ( K ; a, q, t ) = (cid:104) Q [1] ( K ; a, q = 1 , t ) (cid:105) r , it is conjectured [57, Conjecture 4.4] that the superpolynomial colored by a symmetricrepresentation can be expressed as Q [ r ] ( K ; a, q, t ) = (cid:88) l + l + ··· + l n = r (cid:34) rl , l , . . . , l n (cid:35) q a (cid:80) ni =1 a i l i t (cid:80) ni =1 t i l i q (cid:80) ni,j =1 Q i,j l i l j + (cid:80) ni =1 q i l i , (2.10)where ( a i , q i , t i ) are constants and Q i,j is a quadratic form. Here the q -multibinomial isdefined by (cid:34) rl , l , . . . , l n (cid:35) q := ( q ; q ) r ( q ; q ) l ( q ; q ) l · · · ( q ; q ) l n . – 6 –n the large color limit q = e (cid:126) → , r → ∞ , a = fixed , t = fixed , x = q r = fixed , (2.11)a superpolynomial asymptotes to the form Q [ r ] ( K ; a, q, t ) r →∞ , (cid:126) → ∼ x = q r exp (cid:18) (cid:126) (cid:90) log y dxx + · · · (cid:19) , (2.12)where the integral is carried out on the zero locus of classical super-A-polynomial A ( K ; x, y, a, t ) = 0 . The ellipsis in (2.12) represents the subleading terms that possess regular behaviors underthe limit (cid:126) → .If Q [ r ] ( K ; a, q, t ) is written as a summation only over n variables l , · · · , l n as in (2.10),then its behavior under the large color limit could be approximated as Q [ r ] ( K ; a, q, t ) ∼ (cid:90) e (cid:126) (cid:16) (cid:102) W ( K ; z , ··· ,z n ; x,a,t )+ O ( (cid:126) ) (cid:17) dx n (cid:89) i =1 dz i , (2.13)where z i = q l i , x = q r . The leading asymptotic behavior (2.12) with respect to (cid:126) comesfrom the saddle points exp (cid:32) z i ∂ (cid:102) W ( K ; z , · · · , z n ; x, a, t ) ∂z i (cid:33) (cid:12)(cid:12)(cid:12)(cid:12) z i = z ∗ i = 1 . (2.14)and the zero locus of the classical super-A-polynomial is determined by exp (cid:32) x ∂ (cid:102) W ( K ; z ∗ , · · · , z ∗ n ; x, a, t ) ∂x (cid:33) = y . (2.15)Now let us consider the relationship between super- A -polynomials of SU -type and SO -type that comes from the universal differential. According to [14, §4.5], the universaldifferential d univ → has ( a, q, t ) -degree (0,2,1), which relates the [ r ] -colored Kauffman homologyand the [ r ] -colored HOMFLY-PT homology as F [ r ] ( K ; a, q, t ) = q rS ( K ) P [ r ] ( K ; aq − , q, t ) + (1 + q t ) f ( a, q, t ) . (2.16)This implies F [ r ] ( K ; a, q, − q − ) = q rS ( K ) P [ r ] ( K ; aq − , q, − q − ) . (2.17)For a knot satisfying the exponential growth property, the superpolynomial admits anexpression (2.10). Therefore, it is straightforward to verify that the leading order of thelarge color asymptotic behavior of the right hand side of (2.17) is the same as that of thecolored HOMFLY-PT polynomial although the change of variables a → aq − , t = − q − isnon-trivial. As a result, we have the large color asymptotic behavior F [ r ] ( K ; a, q, − q − ) ∼ (cid:90) e (cid:126) ( (cid:102) W SU ( K ; z i ; x,a )+ O ( (cid:126) ) ) dxdz i . (2.18)– 7 –hus, it encodes the information about the a -deformed A -polynomial A SU ( K ; x, y, a ) ofSU-type. On the other hand, the decategorification limit t = − is just the colored Kauff-man polynomial F [ r ] ( K ; a, q, − and its large color limit yields a -deformed A -polynomial A SO ( K ; x, y, a ) of SO-type. Therefore, the super- A -polynomial A SO ( K ; x, y, a, t ) of SO-type contains both A SO ( K ; x, y, a ) and A SU ( K ; x, y, a ) at the t = − specialization. Conjecture 2.1.
For a knot satisfying the exponential growth property, the zero locus ofthe super- A -polynomial of SO-type at t = − A SO ( K ; x, y, a, t = −
1) = 0 includes two branches A SO ( K ; x, y, a ) = 0 and A SU ( K ; x, y, a ) = 0 . Let us take the trefoil as an example in which the super- A -polynomial of SO -type canbe calculated. The colored superpolynomial of the trefoil [14, (5.8)] and the correspondingtwisted superpotential [14, (6.15)] can be written as F [ r ] ( ; a, q, t )= r (cid:88) k =0 k (cid:88) j =0 r − k (cid:88) i =0 a i − k +3 r q k − j (1+ r )+ r (2 r − i (2 j +2 r − t i − j + r ) (cid:34) rk (cid:35) q (cid:34) kj (cid:35) q (cid:34) r − ki (cid:35) q × ( − a t q r − ; q ) j ( − q t ; q ) r − k ( − aq − t ; q ) i . (2.19)and (cid:102) W SO ( ; x, w, v, z, a, t ) = log (cid:16) wx z (cid:17) log a − log v log x + (log x ) + log w log( vx )+ 2 log (cid:0) wxv (cid:1) log t − π − Li ( x ) + Li ( v ) + Li ( zv − )+ Li ( w ) + Li ( xz − w − ) + Li ( − a t x ) − Li ( − a t xv )+ Li ( − t ) − Li ( − txz − ) + Li ( − at ) − Li ( − atw ) , (2.20)where we introduce the variables as w = q i , v = q j and z = q k . Obviously, the t = − specialization leads to the large color limit of colored Kauffman polynomial (cid:102) W SO ( ; x, w, v, z, a, t = −
1) = (cid:102) W SO ( ; x, w, v, z, a ) . (2.21)On the other hand, at the t = − q − specialization, the expression (2.19) collapses to k = r and i = 0 due to the term ( − q t ; q ) r − k in the summand so that it is written bythe summation only over j . Correspondingly, the twisted superpotential encodes the largecolor limit of colored HOMFLY-PT polynomial (cid:102) W SO ( ; x, w = 1 , v, z = x, a, t = −
1) = (cid:102) W SU ( ; x, v, a ) . (2.22)The saddle point equations (2.13) and (2.14) yield the classical super- A -polynomial of SO -type of the trefoil [14, (6.19)], and its t = 1 specialization can be consequently writtenas A SO ( ; x, y, a, t = −
1) = (1 − ax ) A SO ( ; x, y, a ) A SU ( ; x, y, a ) , (2.23)– 8 –here A SO ( ; x, y, a ) = a x − a x − a xy + y , (2.24) A SU ( ; x, y, a ) = y − ya + xya − x ya − xy a − x a + x a + 2 x ya + x ya − x ya . (For A SU ( ; x, y, a ) , see [22, (2.25)] with a → a .) It is evident that it satisfies Conjecture2.1. In principle, we can compute super- A -polynomials of SO-type for the double twist knotsusing the result in §2.1. However, it is beyond the computational capabilities of currentdesktops to solve the saddle point equations (2.13) and (2.14) for them. However, since thecolored Kauffman homology of a double twist knot is endowed with the universal differential d univ → , we can easily check the relations corresponding to (2.21) and (2.22) at the level of thetwisted superpotentials for the double twist knots. In this way, we can confirm Conjecture2.1 for the double twist knots.Note that the diagonal differentials [14, §4.6] also relate colored Kauffman and HOMFLY-PT homology. However, they do not bring us any interesting results once we take largecolor limits. SO( N ) quantum j -symbols j -symbols are ubiquitous in physics and mathematics, and they reveal very rich symme-tries. In particular, quantum knot invariants can be evaluated by using quantum j -symbols[3, 23, 31]. Despite their importance and universality, it is difficult to evaluate them inhigher ranks. Obtaining a closed-form expression is moreover indispensable to study theirsymmetries as in [28, 34].Kauffman polynomials (homology) colored by symmetric representations for non-torusknots obtained in §2.1 in principle encodes the information about SO( N ) quantum j -symbols. Hence, we attempt to extract the information in this section. First, we quan-tize SO( N ) 6 j -symbols with all symmetric representations obtained by Ališauskas [35, 36].Next, we give some evidence to its validity by using representation theory and computingKauffman polynomials. SO( N ) quantum j -symbols for symmetric representations The j -symbols of a Lie group G take the form (cid:40) R R R R R R (cid:41) G , where the arguments are representations of G , and they obey the fusion rule, R ∈ ( R ⊗ R ) ∩ ( R ⊗ R ) , R ∈ ( R ⊗ R ) ∩ ( R ⊗ R ) .Ališauskas has obtained the closed-form expression for classical SO( N ) 6 j -symbolswith all representations symmetric [35, 36]. Kirillov and Reshetikhin naturally quantizedthe classical j -symbols of SU(2) [31] by bringing the hypergeometric series F to ϕ .– 9 –herefore, we conjecture that we can naturally quantize Ališauskas’ results to a closed-form expression of quantum SO( N ) ( N ≥ ) j -symbols for all symmetric representations, (cid:40) a b ed c f (cid:41) SO( N ) = (cid:18) [2 c + N − q [2 d + N − q [2 e + N − q [2] q dim q [ c ] dim q [ d ] dim q [ e ] (cid:19) / × (cid:32) c d e (cid:33) − N (cid:88) l (cid:48) ( − ( c + d − e ) / f + N + l (cid:48) (cid:2) l (cid:48) + N − (cid:3) q × (cid:40) b f + N − d + N − f + N − ( b + N ) − l (cid:48) + N − (cid:41) SU(2) × (cid:40) a f + N − c + N − f + N − ( a + N ) − l (cid:48) + N − (cid:41) SU(2) × (cid:40) a b + N − e + N − b + N − ( a + N ) − l (cid:48) + N − (cid:41) SU(2) × (cid:32) [ l (cid:48) ] q ![ N − q ![ l (cid:48) + N − q ! (cid:33) / , (3.1)where the curly braces on the right hand side are the quantum j -symbols of SU(2) [31].Since we are only dealing with all symmetric representations, a, b, c, d, e, f are the numberof boxes in the corresponding Young tableau of one row. Here a quantum number is definedas [ x ] q = q x − q − x q − q − . (3.2)Also, a q -factorial is defined as the product of quantum numbers [ x ] q ! = [ x ] q [ x − q · · · [ x − (cid:98) x (cid:99) ] q , (3.3)where (cid:98) x (cid:99) is the floor of x so that the definition of q -factorial works for both integer andhalf-integer arguments. The quantum dimension of the SO( N ) symmetric representation [ l ] is given by dim q [ l ] = [ l + N − q ![ l ] q ![ N − q ! ([ l + N − q + [ l ] q ) , (3.4)and the special quantum j -symbols are defined as (cid:32) l l l (cid:33) N = 1[ N/ − q ! (cid:18) [ J + N − q ![ N − q ![ J + N/ − q ! × (cid:89) i =1 [2 l i + N − q [ J − l i + N/ − q ![2] q dim q [ l i ] [ J − l i ] q ! (cid:33) / , (3.5)where J = ( l + l + l ) .Note that R , R can be non-symmetric representations even when R = R = R = R are symmetric representations. (Some fusion rules can be found in (3.10).) Furthermore,– 10 –here are several relations from representation theory, which would lead to identities amongthe quantum j -symbols. In the previous part of this paper, we used the notation [ r s ] todenote the rectangular representations. From now on, the Young tableau ( r ≥ r ≥ · · · )is denoted by [ r , r , · · · ] .• The isomorphism of representations, ( so , [ r ]) (cid:39) ( sl , [ r, r ]) , would lead to equations, (cid:40) [ r ] [ r ] [2 l − j, j ][ r ] [ r ] [2 l (cid:48) − j (cid:48) , j (cid:48) ] (cid:41) SO(6) = (cid:40) [ r, r ] [ r, r ] [2 l, l − j, j ][ r, r ] [ r, r ] [2 l (cid:48) , l (cid:48) − j (cid:48) , j (cid:48) ] (cid:41) SU(4) , (3.6)where ≤ l, l (cid:48) ≤ r , ≤ j ≤ l , ≤ j (cid:48) ≤ l (cid:48) . We have checked that this propertyholds using SU(4) quantum j -symbols in [58], and the result in the next subsection.Since now we only have the closed form expression of SO( N ) quantum j -symbols forsymmetric representations, by setting j = j (cid:48) = 0 , the above equation reduces to (cid:40) [ r ] [ r ] [2 l ][ r ] [ r ] [2 l (cid:48) ] (cid:41) SO(6) = (cid:40) [ r, r ] [ r, r ] [2 l, l ][ r, r ] [ r, r ] [2 l (cid:48) , l (cid:48) ] (cid:41) SU(4) , (3.7)where ≤ l ≤ r . Thus we can easily obtain SU(4) quantum j -symbols of such type,using our closed formula (3.1).• Also, there is an isomorphism between Lie algebras, so (cid:39) sl ⊕ sl . Restrictingourselves to the case where all representations are symmetric, we have (cid:40) [ r ] [ r ] [2 l ][ r ] [ r ] [2 l (cid:48) ] (cid:41) SO(4) = (cid:40) [ r ] [ r ] [2 l ][ r ] [ r ] [2 l (cid:48) ] (cid:41) SU(2) , (3.8)where ≤ l, l (cid:48) ≤ r .• Although our formula only works for SO( N ) ( N ≥ ), it’s still worthy to note thatthere is another isomorphism of representations, ( so , [ r ]) (cid:39) ( sl , [2 r ]) . This leads to (cid:40) [ r ] [ r ] [2 l ][ r ] [ r ] [2 l (cid:48) ] (cid:41) SO(3) (cid:12)(cid:12)(cid:12)(cid:12) q → q = (cid:40) [2 r ] [2 r ] [4 l ][2 r ] [2 r ] [4 l (cid:48) ] (cid:41) SU(2) , (3.9)where ≤ l, l (cid:48) ≤ r . SO( N ) fusion matrices for symmetric representations Since the fusion rule of
SO( N ) symmetric representations is given by [ r ] ⊗ [ r ] = r (cid:77) l =0 l (cid:77) j =0 [2 l − j, j ] , (3.10)we need full SO( N ) fusion matrices including the case that R and R are non-symmetricto evaluate Kauffman polynomials colored by symmetric representations,. At a root of unity q , the WZNW fusion matrix and the corresponding quantum j -symbol are related by a R R (cid:34) R R R R (cid:35) = (cid:15) { R i } (cid:112) dim q R dim q R (cid:40) R R R R R R (cid:41) , (3.11)– 11 –here (cid:15) { R i } = ± and dim q R is the quantum dimension of the representation R . Since SO( N ) representations are real, for colored knot invariants, we only need to consider fusionmatrix of the type a ts (cid:104) R RR R (cid:105) , where t, s ∈ R ⊗ R .To obtain full fusion matrices, we can make use of their properties and symmetries [28]• The fusion matrix is symmetric, a ts (cid:34) R RR R (cid:35) = a st (cid:34) R RR R (cid:35) . (3.12)• The fusion matrix is orthonormal, (cid:88) s a ts (cid:34) R RR R (cid:35) a sm (cid:34) R RR R (cid:35) = δ tm . (3.13)• The entries in the row and column, which corresponds to the trivial representation,are known, a ∅ s (cid:34) R RR R (cid:35) = (cid:15) Rs (cid:112) dim q s dim q R , (3.14)where ∅ is the trivial representation.Note that j -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 theentries when t and s are symmetric representations (3.1), and the entries corresponding tothe trivial representation (3.14). Then we can solve the rest entries by using the fact thatthe fusion matrix is symmetric and orthonormal. In this paper, we will present our resultsof the SO( N ) fusion matrices a ts (cid:104) R RR R (cid:105) for R = , .1. R = (cid:40) R R tR R s (cid:41) = q R s = ∅ t = ∅ [ N ] q [ N − q ([ N − q +[2] q ) − N − q − N − q [ N − q [ N − q ([ N ] q +[2] q ) ,where the quantum dimension dim q R = [ N − q + 1 .2. R = – 12 – R R tR R s (cid:41) = q R s = ∅ t = ∅ b b b b b b b b b b b b b b b b b b b b b b b b b ,where the quantum dimension dim q R = [ N − q + [ N − q [ N ] q / [2] q , and b = − q − q N − q N + q N − q N + q N + q N + q N (1 + q )( − q N )( q + q N + q N + q N ) ,b = b = q N +2 + 1 q N +2 + q N + q + 1 ,b = b = q N (cid:0) q − (cid:1) (cid:0) q N +2 + 1 (cid:1) q N + q N +4 − q N − q ,b = b = (cid:0) q − (cid:1) q N q N + q N +4 + q N + q ,b = b = (cid:0) q − (cid:1) q N − − q N ,b = − q N + q N +2 +3 q N +4 − q N +6 − q N +8 − q N +2 +3 q N +4 + q N +6 − q N +8 + q N +6 − q N + q ( q + 1) ( q N + 1) ( q N − q ) ( q N +4 − ,b = b = − (cid:0) q − (cid:1) q N ( q N + 1) ( q N − q ) ,b = b = q N (cid:0) q − (cid:1) (cid:0) q N +2 + 1 (cid:1) ( q − q N ) ( q N +4 + q N +4 − q N − ,b = b = (cid:0) q − (cid:1) q N q N +4 + q N +4 − q N − ,b = q N (cid:0) q − (cid:1) (cid:0) q +1 (cid:1) (cid:0) q N − q N +2 + q N +6 − q N +2 − q N +4 − q N +6 + q N +8 + q N +2 − q N +6 + q N +4 + q (cid:1) ( q N −
1) ( q N +1) ( q N − q ) ( q N + q ) ( q N − q ) ( q N +2 − ,b = b = − q N (cid:0) q − (cid:1) (cid:0) q + 1 (cid:1) (cid:0) q N +2 + 1 (cid:1) ( q N + 1) ( q N − q N +2 + q N +4 − q ) ( q N +2 − ,b = b = (cid:0) q − (cid:1) (cid:0) q + 1 (cid:1) q N − ( q N −
1) ( q N +2 − ,b = q N (cid:0) q − (cid:1) (cid:0) q + 1 (cid:1) (cid:0) q N +2 − q N +4 + q N +6 + q N +4 − q N − q (cid:1) ( q N − q N +2 + q N +4 − q ) ( q N +2 −
1) ( q N +4 + q N +4 − q N − ,b = b = − (cid:0) q − (cid:1) (cid:0) q + 1 (cid:1) q N ( q N +2 −
1) ( q N +4 + q N +4 − q N − ,b = q N +2 (cid:0) q − (cid:1) (cid:0) q + 1 (cid:1) (cid:0) q N − q (cid:1) ( q N +6 −
1) ( q N +2 −
1) ( q N +4 + q N +4 − q N − . Using these two
SO( N ) fusion matrices and braiding operator eigenvalues, we havecomputed colored Kauffman polynomials for double twist knots when R = , . The– 13 –esults coincide with F [ r ] ( K ; a = q N − , q, t = − obtained in §2.1, which comes from thePoincaré polynomial of [ r ] -colored HOMFLY-PT homology [56, (3.4)] by using the relation(2.8). F K and ADO polynomials In the study of the 3d/3d correspondence, a new invariant (cid:98) Z of a closed 3-manifold wasproposed as a BPS q -series [37, 38]. To incorporate TQFT methods and surgery formula,an analogous series F K for a knot complement S \ K [39] was introduced. Although amathematical definition of ˆ Z or F K has not been fleshed out yet, there are many approachesto understand them. A recipe to compute (cid:98) Z for a plumbed 3-manifold with a negativedefinite linking matrix is given in [38]. We can also use resurgence to calculate (cid:98) Z or F K [39, 59]. Also, for a positive braid knot K , F K ( x, q ) can be computed using the R -matrix forVerma modules [60]. In particular, closed-form expressions of F K for the double twist knotswith positive braids are given in [60], based on the formulae of colored Jones polynomials[61, 62]. The expressions are compatible with Habiro’s cyclotomic expansion [54]. In thissection, we will generalize the results of [60] to the ( a, t ) -deformed case [50], and we willsee by explicit computations that the ADO invariants [46–48] are compatible with thecyclotomic expansions. F K of double twist knots We study the double twist knots, K s,t ( s, t ∈ Z ) , whose notation and the correspondingknots in Rolfsen table are shown in figure 2. Among them, two classes possess only positivebraids. 2s2t Knot Name K , K , K , K , − K , K , K , − K , − K , Figure 2 : Double twist knot K s,t and corresponding knots in Rolfsen table. Double twist knot K m,n ( m, n ∈ Z + ) The cyclotomic expansions of the [ r ] -colored superpolynomials for the double twist knots K m,n ( m, n ∈ Z + ) are given in [63, 64]. In fact, by substitution q r = x and changing the– 14 –pper limit from r to ∞ , we can obtain the corresponding ( a, t ) -deformed F K : F K m,n ( x, a, q, t ) = ( − tx ) log q ( − at ) − ∞ (cid:88) k =0 g ( k ) K m,n ( x, a, q, t ) , (4.1)where g ( k ) K m,n ( x, a, q, t ) = k (cid:88) j =0 (cid:0) x ; q − (cid:1) k ( − at/q ; q ) k (cid:0) − axt ; q (cid:1) j ( q ; q ) k × ( − j x k − j a − j q k t − j +2 k (cid:34) kj (cid:35) q Tw ( j ) K m,n ( a, q, t ) . (4.2)The twist factor is defined as Tw ( j ) K m,n ( a, q, t ) = j (cid:88) l =0 ( − l q l ( l +1) − jl tw ( l ) m ( a, q, t ) tw ( l ) n ( a, q, t ) (cid:34) jl (cid:35) q , (4.3)with tw ( l ) m ( a, q, t ) = (cid:88) ≤ b ≤···≤ b m − ≤ b m = l m − (cid:89) i =1 (cid:16) a b i q b i ( b i − t b i (cid:17) (cid:20) b i +1 b i (cid:21) q . (4.4)Note that, to connect the standard notation of F K and the ADO invariants, we rescale thevariables a → a / and q → q / from the superpolynomial (2.9) F K m,n ( x = q r , a, q, t ) = P [ r ] ( K m,n ; a / , q / , t ) . When we write the ( a, t ) -deformed F K as a four-variable series explicitly, we wouldusually omit the prefactor ( − tx ) log q ( − at ) − . However, we have to keep it to verify theproperties (4.10), (4.11), (4.12) of ( a, t ) -deformed F K . If we take a = − t − q N , the prefactorsimply reduces to ( − tx ) N − . Double twist knot K m + , − n ( m, n ∈ Z + ) Similarly, we can find the cyclotomic expansions of the [ r ] -colored superpolynomial fordouble twist knot K m + , − n ( m, n ∈ Z + ) by using the structural properties of HOMFLY-PT homology. Then, we obtain the ( a, t ) -deformed F K for it: F K m + 12 , − n ( x, a, q, t ) = ( − tx n ) log q ( − at ) − ∞ (cid:88) k =0 g ( k ) K m + 12 , − n ( x, a, q, t ) , (4.5)where g ( k ) K m + 12 , − n ( x, a, q, t ) = k (cid:88) j =0 (cid:0) x ; q − (cid:1) k ( − at/q ; q ) k (cid:0) − axt ; q (cid:1) j ( q ; q ) k × x k − j (cid:34) kj (cid:35) q q j ( j − k t − j +2 k tw ( j ) m ( a, q, t ) tw ( k ) n ( x, q, t ) . (4.6)– 15 – new twist factor is defined as tw ( k ) n ( x, q, t ) = (cid:88) ≤ b ≤···≤ b n − ≤ b n = k n − (cid:89) i =1 (cid:16) x b i q − b i ( b i +1 − t b i (cid:17) (cid:20) b i +1 b i (cid:21) q . (4.7)The prefactor becomes simply ( − tx n ) N − when taking a = − t − 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 [60] upon the specialization a = q , t = − . If we further take a = q N , t = − , it reducesto F SU( N ) ,symK ( x, q ) [49], which is the F K with gauge group G = SU( N ) associated withsymmetric representations.It is conjectured in [50] that the ( a, t ) -deformed F K has the following property, ˆ A ( K ; ˆ x, ˆ y, a, q, t ) F K ( x, a, q, t ) = 0 . (4.8)We have studied the classical super- A -polynomial in §2.2, and ˆ A ( K ; ˆ x, ˆ y, a, q, t ) is its quan-tum version [63], which annihilates the colored superpolynomial of K . The operators in thequantum super- A -polynomial act on ( a, t ) -deformed F K as ˆ xF K ( x, a, q, t ) = xF K ( x, a, q, t ) , ˆ yF K ( x, a, q, t ) = F K ( xq, a, q, t ) . (4.9)In the case of , , , the conjectural ( a, t ) -deformed F K can be annihilated by the cor-responding super- A -polynomial in [63], the detail can be seen in the Mathematica file.It is proposed in [50] that ( a, t ) -deformed F K satisfy the following properties: F K ( x, − t − , q, t ) = ∆ K ( x, t ) , (4.10) F K ( x, − t − q, q, t ) = 1 , (4.11) lim q → F K ( x, − t − q N , q, t ) = 1∆ K ( x, t ) N − , (4.12)where ∆ K ( x, t ) is the t -deformed Alexander polynomial introduced in [50], which can beobtained from the superpolynomial, ∆ K ( x, t ) = P ( K ; a = − t − , q = x, t ) . (4.13)We have checked that our closed formulae of the ( a, t ) -deformed F K for the double twistknots satisfy (4.10). Also, we give an analytical proof of (4.11) and (4.12) in Appendix A. t -deformed ADO polynomials As studied in [47], at the limit q → ζ p = e πip , F K ( x, q ) is equal to the p -th ADO polynomial[48] up to the Alexander polynomial ∆ K ( x p ) . For ( a, t ) -deformed F K at radial limit, it isnatural to consider a t -deformation of the p -th ADO polynomial. The formal definition isgiven in [50] as, ADO K ( p ; x, t ) = ∆ K ( x p , − ( − t ) p ) lim q → ζ p F K ( x, − t − q , q, t ) . (4.14)– 16 –n this subsection, we derive closed-form expressions of t -deformed ADO polynomials ofthe double twist knots. As in Appendix A, it turns out that the closed formulae (4.1)and (4.5) of cyclotomic type are suitable to compute the ADO polynomial up on the limit q → ζ p = e πip in which an infinite sum truncates to a finite one [65]. (See also [66] for asimilar topic.)First, we will obtain a closed-form expression of t -deformed Alexander polynomials.Although ∆ K ( x, t ) can be easily calculated from the superpolynomial by using (4.13), wecompute it from (4.1) and (4.5) via the substitution (4.10). Notice that both the formulaeshare a common term ( − at/q ; q ) k , where k runs from to ∞ . Upon specialization a = − t − ,it becomes ( q − ; q ) k and vanishes for k > . Therefore, the infinite summation becomes arather simple finite one. Subsequently, they are given by ∆ K m,n ( x, t ) = − t + (cid:18) t + 1 t − tx − tx (cid:19) S m ( − t ) S n ( − t ) , ∆ K m + 12 , − n ( x, t ) = t n [ n ] tx − t n − [ n + 1] tx − t n [ n ] tx (cid:18) t + 1 t − tx − tx (cid:19) S m ( − t ) , (4.15)where S l ( x ) = l − (cid:80) i =0 x i , and the definition of quantum number [ n ] tx is given in (3.2). Wepresent our formulae this way, so that the Weyl symmetry [50] of the t -deformed Alexanderpolynomial becomes manifest, ∆ K ( x − , t ) = ∆ K ( t − x, t ) . (4.16)After detailed derivation shown in Appendix A, we obtain closed-from expressions ofthe t -deformed ADO polynomials for double twist knots K m,n and K m + , − n , ADO K m,n ( p ; x, t ) = ( − tx ) − p p − (cid:88) k =0 lim q → ζ p g ( k ) K m,n ( x, − t − q , q, t ) , ADO K m + 12 , − n ( p ; x, t ) = ( − tx n ) − p p − (cid:88) k =0 lim q → ζ p g ( k ) K m + 12 , − n ( x, − t − q , q, t ) . (4.17)Note that there is no dependence on t -deformed Alexander polynomial because the radiallimit of ( a, t ) -deformed F K is proportional to the inverse of the factor ∆ K ( x p , − ( − t ) p ) ,which cancels the same factor in (4.14). Especially, we can analytically derive that theseformulae give rise to ADO K ( p = 1; x, t ) = 1 , ADO K ( p = 2; x, t ) = ∆ K ( x, t ) , (4.18)which implies that the t -deformed ADO polynomial is a generalization of t -deformed Alexan-der polynomial.In Table 1, 2, 3, we summarize the results of t -deformed ADO polynomials for thetrefoil, knot and knot. – 17 – ADO ( p ; x, t ) ( − tx ) + t + ( − tx ) − ζ (cid:104) ( tζ x ) + t ( tζ x ) + (cid:0) t − ζ − (cid:1) + t ( tζ x ) − + ( tζ x ) − (cid:105) Table 1 : t -deformed ADO polynomials for the left-handed trefoil p ADO ( p ; x, t ) (1 − t )( − tx ) − t − t + (1 − t )( − tx ) − ζ (cid:2) (cid:0) tζ + t (cid:1) ( tζ x ) + (cid:0) − t + t ζ + t (cid:1) ( tζ x ) + 2 + ζ + ( − ζ ) t +2 t + t ζ + t + (cid:0) − t + t ζ + t (cid:1) ( tζ x ) − + (cid:0) tζ + t (cid:1) ( tζ x ) − (cid:3) Table 2 : t -deformed ADO polynomials for the knot p ADO ( p ; x, t ) (1 − t + t )( − tx ) − t − t + t + (1 − t + t )( − tx ) − ζ (cid:2) (cid:0) tζ + t ζ + t (cid:1) ( tζ x ) + (cid:0) − − ζ ) t + 2 ζ t + t + ζ t + t (cid:1) ( tζ x )+2 + ζ + ( − ζ ) t + (1 − ζ ) t + 2 ζ t + t + ζ t + t + (cid:0) − − ζ ) t + 2 ζ t + t + ζ t + t (cid:1) ( tζ x ) − + (cid:0) tζ + t ζ + t (cid:1) ( tζ x ) − (cid:3) Table 3 : t -deformed ADO polynomials for the knotMore generally, the higher rank t -deformed ADO is defined as [50] ADO
SU( N ) K ( p ; x, t ) = ∆ K ( x p , − ( − t ) p ) N − lim q → ζ p F K ( x, − t − q N , q, t ) . (4.19)It is also feasible to derive the closed-form expression for the t -deformed ADO SU( N ) K ( p ; x, t ) from (4.1) and (4.5) ADO
SU( N ) K m,n ( p ; x, t ) = 1 p p − (cid:88) l =0 ( − tx ) l +1 − p ∆ K m,n ( x p , − ( − t ) p ) ( N − p − lp S p ( ζ N − l − p ) × p − (cid:88) k =0 lim q → ζ p g ( k ) K m,n ( x, − t − q N , q, t ) , – 18 – DO SU( N ) K m + 12 , − n ( p ; x, t ) = 1 p p − (cid:88) l =0 ( − tx n ) l +1 − p ∆ K m + 12 , − n ( x p , − ( − t ) p ) ( N − p − lp × S p ( ζ N − l − p ) p − (cid:88) k =0 lim q → ζ p g ( k ) K m + 12 , − n ( x, − t − q N , q, t ) . (4.20)The detailed derivation is also shown in the Appendix A. Especially, we can analyticallyverify that they become ADO
SU( N ) K (1; x, t ) = 1 , ADO
SU( N ) K (2; x, t ) = 12 (cid:88) l,j =0 ( − j ( N − l ) (cid:2) ∆ K ( x , − ( − t ) ) (cid:3) l + N − [∆ K ( x, t )] − l , (4.21)and they obey the recursion relation [50] ADO
SU( N + p ) ( p ; x, t ) = ∆ K ( x p , − ( − t ) p ) p − ADO
SU( N ) ( p ; x, t ) . (4.22)Note that colored Jones polynomials of cyclotomic type [54, 61, 62] at a root of unity q = ζ p provide the corresponding ADO invariants [65] even if a closed-form expressionof F K is not known. Therefore, one can evaluate the ADO invariants from colored Jonespolynomials in a similar fashion for arbitrary double twist knots such as K m,n with m, − n ∈ Z + [65]. In this paper, we presented a simple rule of grading change that would allow us to ob-tain [ r ] -colored quadruply-graded Kauffman homology from [ r ] -colored quadruply-gradedHOMFLY-PT homology for thin knots. We check the grading change by consistency withdifferentials and symmetries for colored Kauffman homology of double twist knots. We alsofind 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 = − . With the natural quantization of SO( N )( N ≥
4) 6 j -symbols for symmetric representations given by Ališauskas, we calculate SO( N ) fusion matrices for R = , and compute Kauffman polynomials for the corre-sponding colors. We check the validity of the quantization from representation theory andKauffman 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 + , − 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 (3.1) is rather involved and there must be a similar formula for it. It is alsodesirable to find a closed-form expression of j -symbols when fusions R and R are non-symmetric. Moreover, it has been found that [67] the j -symbol in AdS is the Lorentzianinversion of a crossing-symmetric tree-level exchange amplitude, and the one-loop vertexcorrection in φ -theory in AdS d +1 is given by a spectral integral over the j -symbol for– 19 – O( d + 1 , . It would be interesting that our research results would bring some inspirationto the calculations of these j -symbols in the AdS background.So far the study of the volume conjecture involves only the large color limit of symmetricrepresentations. It is very important to formulate volume conjectures with arbitrary colors,relating to the moduli space of flat SL( N, C ) connections over the knot complement. Sincecolored knot homology is endowed with many colored differentials, we expect a behaviorsimilar 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 fromthe corresponding superpolynomials. However, It is desirable to develop a general methodto obtain a closed-form expression of F K for arbitrary knots. Acknowledgments
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 j -symbols, and he alsothanks Ryo Suzuki for identifying the reference [36] about SO( N ) 6 j -symbol. We also wouldlike to thank Sunghyuk Park for identifying the relationship between the t -deformed ADOpolynomials of and ∗ . This work was supported by the National Science Foundationunder Grant No. NSF PHY-1748958. A Derivation of the ADO polynomials
In this Appendix, we derive the t -deformed p -th ADO polynomial (4.14) and its higher rankgeneralization (4.19) from F K ( x, a, q, t ) (4.1) and (4.5). We use F K as a short for, lim q → ζ p F K ( x, a = − t − q N , q, t ) , and similarly for other functions discussed in this section. We use q and ζ p interchangeably.When an integer k is displayed as k = k p + k , it is implied that k , k ∈ N , and ≤ k ≤ p − .Our closed formulae are built from q -binomials and q -Pochhammer symbols and wewill first discuss their behavior when q goes to roots of unity. The q -Lucas theorem [68]states that, (cid:20) lk (cid:21) q = ζ p = (cid:18) l k (cid:19)(cid:20) l k (cid:21) q = ζ p , (A.1)where l = l p + l , k = k p + k . For q -Pochhammers, we have ( a ; q ) k = (1 − a p ) k ( a ; q ) k , (A.2)where q = ζ p , k = k p + k . In the following discussion, We will break F K ( x, a, q, t ) intoparts. 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 and a finite one over k ,– 20 –here k = k p + k and q = ζ p , and the infinite summation over k can be repackaged bythe generalized binomial theorem, − z ) n = ∞ (cid:88) i =0 (cid:18) n + i − i (cid:19) z i . (A.3)As a result, an ADO polynomial can be expressed as a finite summation of over k . A.1 ADO polynomials of double twist knot K m,n ( m, n ∈ Z + ) For double twist knots K m,n ( m, n ∈ Z + ) , we first analyze ( a, t ) -deformed F K at radiallimit q = ζ p . Each component of (4.1) behaves as follows.• The twist factor tw ( l ) m ( a, q, t ) (4.4) now becomes tw ( l ) m = (cid:88) ≤ b ≤···≤ b m − ≤ b m = l m − (cid:89) i =1 ( − q N ) b i q b i ( b i − t b i (cid:20) b i +1 b i (cid:21) q . (A.4)We decompose the summation variables as, b i = α i p + β i , and l = l p + l . Using the q -Lucas theorem, (A.4) can be written as a product of two summations, one over α i ’sand another one over β i ’s. The summation over β i ’s would give tw ( l ) m . Performingthe summation over α i ’s, we obtain tw ( l ) m = S l m (( − t ) p ) tw ( l ) m , (A.5)where S l m ( x ) := ( S m ( x )) l := (cid:18) m − (cid:80) i =0 x i (cid:19) l .• The twist factor Tw ( j ) K m,n ( a, q, t ) (4.3) becomes Tw ( j ) K m,n = j (cid:88) l =0 ( − l q l ( l +1) − jl tw ( l ) m tw ( l ) n (cid:20) jl (cid:21) q . (A.6)We write the summation variables as j = j p + j , and l = l p + l . Because of thefact that q p = 1 , the q -Lucas theorem and (A.5), we obtain, Tw ( j ) K m,n = (1 − S m (( − t ) p ) S n (( − t ) p )) j Tw ( j ) K m,n . (A.7)• Now g ( k ) K m,n ( x, a, q, t ) in (4.2) becomes g ( k ) K m,n = k (cid:88) j =0 ( x ; q − ) k ( xt q N ; q ) j ( xt ) k − j q k − Nj (cid:20) N − kk (cid:21) q (cid:20) kj (cid:21) q Tw ( j ) K m,n , (A.8)where we have used the fact that ( q N − ; q ) k ( q ; q ) k = (cid:20) N − kk (cid:21) q . (A.9)– 21 –e decompose the variables as j = j p + j , k = k p + k and N − A p + A .Plugging (A.1), (A.2) and (A.7) into (A.8), we have g ( k ) K m,n = (cid:18) A + k k (cid:19) (1 − x p ) k (cid:2) − (cid:0) − x p t p (cid:1) S m (( − t ) p ) S n (( − t ) p ) (cid:3) k g ( k ) K m,n . (A.10)• F K m,n ( x, a, q, t ) becomes F K m,n = ( − tx ) N − ∞ (cid:88) k =0 g ( k ) K m,n . (A.11)Given k = k p + k , N − A p + A , we have F K m,n = ( − tx ) N − p − (cid:88) k =0 g ( k ) K m,n × ∞ (cid:88) k =0 (cid:18) A + k k (cid:19) (1 − x p ) k (cid:2) − (cid:0) − x p t p (cid:1) S m (( − t ) p ) S n (( − t ) p ) (cid:3) k . (A.12)Using the generalized binomial theorem, we obtain F K m,n = ( − tx ) N − p − (cid:80) k =0 g ( k ) K m,n [ x p + (1 − x p ) (1 − x p t p ) S m (( − t ) p ) S n (( − t ) p )] A +1 . (A.13)Recall the closed formulae of t -deformed Alexander polynomials (4.15), we can write F K m,n = ( − tx ) A +1 − p ∆ K m,n ( x p , − ( − t ) p ) A +1 p − (cid:88) k =0 g ( k ) K m,n . (A.14)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 = N − , A = 0 , we have lim q → F K m,n ( x, − t − q N , q, t ) = g (0) K m,n ∆ K m,n ( x, t ) N − = 1∆ K m,n ( x, t ) N − , (A.15)in accordance with (4.12). Finally, for the t -deformed ADO polynomials, we have
ADO
SU( N ) K m,n ( p ; x, t ) = ∆ K m,n ( x p , − ( − t ) p ) N − F K m,n = ( − tx ) A +1 − p ∆ K m,n ( x p , − ( − t ) p ) A ( p − A p − (cid:88) k =0 g ( k ) K m,n , (A.16)where N − A p + A .Now let us consider some simple cases. If p = 1 , then A = 0 , we have ADO
SU( N ) K m,n ( p = 1; x, t ) = g (0) K m,n = 1 . (A.17)For N = 2 and p = 2 , then A = A = 0 , we have ADO
SU(2) K m,n ( p = 2; x, t ) = ( − tx ) − (cid:16) g (0) K m,n + g (1) K m,n (cid:17) = ∆ K m,n ( x, t ) . (A.18)– 22 – .2 ADO polynomials of double twist knots K m + , − n ( m, n ∈ Z + ) Following the same procedure in the last subsection, we decompose the closed-form expres-sion of F K m + 12 , − n ( x, a, q, t ) (4.5) into parts:• The twist factor tw ( k ) n ( x, q, t ) (4.7) now becomes tw ( k ) n ( x, q, t ) = (cid:88) b ≤ b ≤···≤ b n − ≤ b n = k n − (cid:89) i =1 ( x b i q − b i ( b i +1 − t b i ) (cid:20) b i +1 b i (cid:21) q . (A.19)We write b i = α i p + β i , k = k p + k , and we can obtain tw ( k ) n = S k n ( x p t p ) tw ( k ) n . (A.20)• Now g ( k ) K m + 12 , − n in (4.6) becomes g ( k ) K m + 12 , − n = k (cid:88) j =0 ( x ; q − ) k ( xt q N ; q ) j x k − j (cid:20) N − kk (cid:21) q (cid:20) kj (cid:21) q q j ( j − k t − j +2 k tw ( j ) m tw ( k ) n . (A.21)We write k = k p + k , j = j p + j , and N − A p + A , and obtain g ( k ) K m + 12 , − n = (cid:8) (1 − x p ) ( − t ) p (cid:2) ( − xt ) p + (cid:0) x p t p − (cid:1) S m (( − t ) p ) (cid:3) S n (cid:0) x p t p (cid:1)(cid:9) k × (cid:18) A + k k (cid:19) g ( k ) K m + 12 , − n . (A.22)• F K m + 12 , − n ( x, a, q, t ) now becomes F K m + 12 , − n = ( − tx n ) N − ∞ (cid:88) k =0 g ( k ) K m + 12 , − n . (A.23)We write k = k p + k , N − A p + A . Again, with the help of the generalizedbinomial theorem, we can get rid of the infinite summation over k and obtain F K m + 12 , − n = ( − tx n ) A +1 − p ∆ K m + 12 , − n ( x p , − ( − t ) p ) A +1 p − (cid:88) k =0 g ( k ) K m + 12 , − n , (A.24)where the t -deformed Alexander polynomial is given in (4.15).When p = 1 , then A = N − , A = 0 , we have lim q → F K m + 12 , − n ( x, − t − q N , q, t ) = g (0) K m + 12 , − n ∆ K m + 12 , − n ( x, t ) N − = 1∆ K m + 12 , − n ( x, t ) N − , (A.25)– 23 –n accordance with (4.12). Finally, for the t -deformed ADO polynomials, we have
ADO
SU( N ) K m + 12 , − n ( p ; x, t ) = ∆ K m + 12 , − n ( x p , − ( − t ) p ) N − F K m + 12 , − n = ( − tx n ) A +1 − p ∆ K m + 12 , − n ( x p , − ( − t ) p ) A ( p − A p − (cid:88) k =0 g ( k ) K m + 12 , − n , (A.26)where N − A p + A .Now we consider some simple cases. When p = 1 , A = 0 , we have ADO
SU( N ) K m + 12 , − n ( p = 1; x, t ) = g (0) K m + 12 , − n = 1 . (A.27)For N = 2 , p = 2 , we have ADO K m + 12 , − n ( p = 2; x, t ) = ( − tx n ) − ( g (0) K m + 12 , − n + g (1) K m + 12 , − n ) = ∆ K m + 12 , − n ( x, t ) . (A.28) A.3 Final formulae
In conclusion, for gauge group
SU( N ) , the t -deformed p -th ADO polynomials are given by ADO
SU( N ) K m,n ( p ; x, t ) = ( − tx ) A +1 − p ∆ K m,n ( x p , − ( − t ) p ) A ( p − A p − (cid:88) k =0 g ( k ) K m,n , ADO
SU( N ) K m + 12 , − n ( p ; x, t ) = ( − tx n ) A +1 − p ∆ K m + 12 , − n ( x p , − ( − t ) p ) A ( p − A p − (cid:88) k =0 g ( k ) K m + 12 , − n , where N − A p + A , A , A ∈ N and ≤ A ≤ p − . It is easily seen that the recursionrelation conjectured in [50] holds, ADO
SU( N + p ) K ( p ; x, t ) = ∆ K ( x p , − ( − t ) p ) p − ADO
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