Knot Complement, ADO-Invariants and their Deformations for Torus Knots
aa r X i v : . [ m a t h . G T ] J u l Knot Complement, ADO-Invariants and their Deformations forTorus Knots
John ChaeDepartment of Physics and QMAP, UC Davis, 1 Shields Ave, Davis, CA, 95616, [email protected]
Abstract
A relation between the two-variable series knot invariant and the Akutus-Deguchi-Ohtsuki(ADO)-invariant was conjectured recently. We reinforce the conjecture by presenting explicit formulasand/or an algorithm for certain
ADO -invariants of torus knots obtained from the series invari-ant of complement of a knot. Furthermore, one parameter deformation of
ADO -polynomial oftorus knots is provided. CONTENTS ADO -Invariants of Torus Knots 6
ADO Invariants of T (2 , s + 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Algorithm for ADO Invariants of T (2 , s + 1) . . . . . . . . . . . . . . . . . . . . . 63.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.4 ADO Invariants of T (3 , r ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.5 Deformed ADO -invariants of T (2 , s + 1) . . . . . . . . . . . . . . . . . . . . . . . . 9 Categorification of link invariants has been a source of fruitful interactions between physics andlow dimensional topology over the past decades (see [31, 32, 33] for reviews). Since the adventof the Khovanov homology [17], which categorifies the Jones polynomials of links, there has beenconstructions of other homological theories, for example, knot Floer homology [18, 19], Khovanov-Rozansky homology [20] and HOMFLY homology [21] that categorify the well-known link poly-nomials, Alexander, sl ( N )-invariants and HOMFLY polynomial, respectively. Not only has the1ategorification deepened the conceptual aspects of links, but it has also provided a more powerfulmachinery to compute higher structural invariants beyond polynomial invariants. Furthermore,these advancements have inspired new directions in physics, which resulted in physical realizationsof the link homologies. Beginning from knot Floer homology, its physical interpretation was foundin [26]. A physical realization of Khovanov homology and Khovanov-Rozansky homology was firstprovided using topological string theory in [27]; additionally, through the conifold transition, ex-istence of the HOMFLY homology was predicted as well. In the case of Khovanov homology, adifferent physical system involving D-branes was achieved in [25]. For Kauffman homology, itsphysical construction exemplified the role of orientifolds [28]. Even knot homology based on anexceptional Lie algebra admits a physical description [29] (see Table 1 for summary). Polynomial Homology Physical Realization
Alexander sl (1 |
1) Knot Floer homology M5-M2 branes on the de-formed conifoldJones sl (2) Khovanov homology M5-M2 branes on the de-formed conifold or D3-NS5brane system sl ( N )-invariants sl ( N ) Khovanov-Rozansky ho-mology M5-M2 branes on the de-formed conifoldHOMFLY HOMFLY homology M5-M2 branes on the resolvedconifold so ( n ) /sp ( n )-invariants& Kauffman Kauffman homology D4-brane & Orientifold sys-tem on the resolved conifoldHyperpolynomial e homology M5-M2 branes on the resolvedconifoldTable 1: A summary of link invariants and their physical realizations. Choice of an orientifold typedetermines so ( n ) or sp ( n ) Lie algebra. Applications of S and T-dualities are necessary to the branesystem in the case of Khovanov homology(see [25] for details).In recent years, a physical approach to categorification of the Witten-Reshetikhin-Tureav (WRT)-invariant of 3-manifold [24, 22, 23], namely, homological blocks ˆ Z ( q ) [2, 3] inspired a new kind ofinvariant for a complement of a knot [1]. This knot invariant denoted as F K is a two-variable seriesthat emerges from ˆ Z ( q ): F K ( x, q ) := ˆ Z ( M K ; x / , n, q ) | q | < , where M K is a complement of a knot K in S , n ∈ Z , c ∈ Z + and ∆ ∈ Q . Physical interpretationof F K is that it counts BPS states of a 3d N = 2 supersymmetric theory T [ M K ] on the knot The RHS of the definition of F K is a two-variable version of ˆ Z b ( q ). integrality of the coefficients of F K -series. This in turn originatesfrom the appearance of dimension of BPS Hilbert space of T [ Y ] in the q-series ˆ Z ( Y, q ) for a generic3-manifold Y. Furthermore, this Hilbert space is identified with a conjectured triply graded threemanifold homology H i,jBP S ( Y ; b ) whose (graded) Euler characteristic isˆ Z b [ Y ; q ] = X i,j ( − i q j dim H i,jBP S ( Y ; b ) ∈ − c q ∆ Z [[ q ]] | q | < . The WRT-invariant of Y is recovered from ˆ Z b [ Y ; q ] as q goes to a root of unity (see for details [2]Section 2).Among mathematical developments of F K [4, 5, 6], evidences for a relationship between F K andthe ADO link invariant [9] have been discovered in [8]. This relation is conjectured to hold for allknots and for any roots of unity: Conjecture 1. ([8] Conjecture 3) For any knot K in S , F K ( x, q ) | q = ζ p = (cid:16) x / − x − / (cid:17) ADO p ( K ; x, ζ p )∆ K ( x p ) ζ p = e i π/p , p ∈ Z + . This conjecture was verified for specific values of p for positive trefoil and the figure-8 knots [8].Another advancement was an introduction of a refinement of F K ( x, q ) [7]. It was shown that F K ( x, q ) admits two parameter deformation through the superpolynomial [26, 30]. This led to ageneralization of the above conjecture. Conjecture 2. ([7] Conjecture 4) For any knot K in S , there exists a t-deformation of thesymmetric ADO p -polynomial of K for SU ( N ), ADO SU ( N ) p [ K ; x, t ] := (∆ K ( x p , − ( − t ) p )) N − lim q → e i π/p F K ( x, q, a = − q N /t, t ) , p ∈ Z + and t = − ADO p [ K ; x ] (up to rescaling of x).The rest of the paper is organized as follows. In Section 2, we briefly review the series invariantfor a knot complement and the ADO-invariants. In Section 3, we present the explicit formulasand/or an algorithm for the ADO , ADO , ADO -invariants for torus knots. Furthermore, oneparameter deformation of ADO -polynomial for torus knots is discussed. Acknowledgments.
I am grateful to Sergei Gukov for his valuable suggestions on a draft ofthis paper. I would like thank Angus Gruen for helpful conversations.
A series invariant F K for a complement of a knot M K was introduced [1]. It has variousproperties such as the gluing formula and the (Dehn) surgery formula. This knot invariant F K : F K ( x, q ) = 12 ∞ X m ≥ m odd ( x m/ − x − m/ ) f m ( q ) ∈ c q ∆ Z [ x ± / ][[ q ± ]] , where f m ( q ) are Laurent series with integer coefficients, c ∈ Z + and ∆ ∈ Q . Moreover, x-variableis associated to Spin c -structures of M K , which can be identified with H ( M K ; Z ); it has an infiniteorder, which is reflected as a series in F K . For applications, some classes of knots have beenanalyzed [1, 5]. One of them is torus knot, which is relevant for our purpose. Hence we display F K for positive torus knots T ( s, t ), s, t > gcd ( s, t ) = 1 [1]. F T ( s,t ) ( x, q ) = 12 q ( s − t − / ∞ X m ≥ m odd ǫ ( s, t ) m ( x m/ − x − m/ ) q m − ( st − s − t )24 st ,ǫ ( s, t ) m = − , m ≡ st + s + t or st − s − t (mod 2 st )1 , m ≡ st + s − t or st − s + t (mod 2 st )0 , otherwise . Prior to F K ’s potential relation to the (original) ADO-invariant, it was proposed that F K pos-sess similar characteristics of sl (2)-colored Jones polynomial through the Melvin-Morton-Rozanskyconjecture [12, 13](proven in [14]), and the quantum volume conjecture [15, 16]: Conjecture 3. ([1] Conjecture 1.5). For a knot K ⊂ S , the asymptotic expansion of the knotinvariant F K ( x, q = e ~ ) about ~ = 0 coincides with the Melvin-Morton-Rozansky expansion of thecolored Jones polynomial in the large color limit: F K ( x, q = e ~ ) x / − x − / = ∞ X r =0 P r ( x )∆ K ( x ) r +1 ~ r , where x = q n ~ is fixed, n is the color of K, P r ( x ) ∈ Q [ x ± ], P ( x ) = 1 and ∆ K ( x ) is the Alexanderpolynomial of K. Conjecture 4. ([1] Conjecture 1.6). For any knot K ⊂ S , the normalized series f K ( x, q )satisfies a linear recursion relation generated by the quantum A-polynomial of K:ˆ A K (ˆ x, ˆ y, q ) f K ( x, q ) = 0 , where f K := F K ( x, q ) / ( x / − x − / ). Colored generalization of the Alexander polynomial for framed colored and oriented knot (link)was introduced in [9]. This knot invariant(ADO invariant) is based on (1 , Implicitly, there is a choice of group; originally, the group used is SU (2). U Hζ r ( sl ( C )) together with the modified quantum dimension. Althoughwe will not employ the quantum algebra construction of the ADO-invariants, for conceptual back-ground, we give a concise review of the ingredients of this construction [9, 11, 10].The first ingredient is the unrolled quantum group U Hζ r ( sl ( C )), which is a C -algebra specializedat q = ζ r ; its generators and relations areGenerators: E, F, K, K − , H Relations: KK − = K − K = 1 KE = ζ r EK KF = ζ − r F K [ E, F ] = K − K − ζ r − ζ − r KH = HK [ H, E ] = 2 E [ H, F ] = − F E r = F r = 0 . This algebra possess a Hopf algebra structure:∆( E ) = 1 ⊗ E + E ⊗ K ǫ ( E ) = 0 S ( E ) = − EK − ∆( F ) = K − ⊗ F + F ⊗ ǫ ( F ) = 0 S ( F ) = − KF ∆( H ) = 1 ⊗ H + H ⊗ ǫ ( H ) = 0 S ( H ) = − H ∆( K ) = K ⊗ K ǫ ( K ) = 1 S ( K ) = K − ∆( K − ) = K − ⊗ K − ǫ ( K − ) = 1 S ( K − ) = K The second element of the construction of the ADO-invariant is a functor RT between a categoryof colored oriented tangle diagrams COD and a category Rep of representations of U Hζ r ( sl ( C )). RT : COD −→ Rep.
The objects of COD are framed colored oriented (1 , RT ( T ) = < T > Id V ∈ End C ( V ),which enables to define ADO ( K ) r := d ( V α ; r ) < T >, where V α is a vector space assigned to K (or to an open component of a link ) and d ( V α ; r ) is themodified quantum dimension, d ( V α ; r ) = − ζ r (1 − r )2 r ζ α +12 r − ζ − α − r ζ rα r − ζ − rα r , α ∈ ( C \ Z ) ∪ ( r Z − . This modified dimension replaces the usual quantum trace, which vanishes in this context. More-over, it makes
ADO ( K ) an isotopy invariant. ADO-invariant is independent of choice of a component of a link that is cut (see for details Section 5 in [9]). ADO -Invariants of Torus Knots
Recently, evidences for a relation between F K at specific values of roots of unity and the ADOinvariants were discovered for (positive) trefoil, the figure-8 and 5 knots [8]. Furthermore, thisrelation is conjectured to hold for any roots of unity and for all knots (Conjecture 1). We strength-ened this connection between the two invariants by presenting an explicit formula or an algorithmfor ADO and ADO invariants of T (2 , s + 1) , s ∈ Z + and ADO of T (3 , w ) , w > . ADO Invariants of T (2 , s + 1) The
ADO invariants of T (2 , s + 1) are divided in three types depending on their coefficientpattern. The general formula are as follows.1. For K = T (2 , s + 1) = T (2 , , T (2 , , T (2 , , T (2 , , · · · ADO ( x ) = ζ x s + ζ x s − + ( ζ − ζ − ) x s − − ζ − x s − − ζ − x s − + ζ x s − + ζ x s − + ( ζ − ζ − ) x s − − ζ − x s − − ζ − x s − + · · · + ( ζ − ζ − ) + ( x → /x ) .
2. For K = T (2 , s + 1) = T (2 , , T (2 , , T (2 , , T (2 , , · · · ADO ( x ) = ζ − x s + ζ − x s − + ( ζ − − x s − − x s − − x s − + ζ − x s − + ζ − x s − + ( ζ − − x s − − x s − − x s − + · · · − x → /x ) .
3. For K = T (2 , s + 1) = T (2 , , T (2 , , T (2 , , T (2 , , · · · ADO ( x ) = x s + x s − + (1 − ζ ) x s − − ζ x s − − ζ x s − + x s − + x s − + (1 − ζ ) x s − − ζ x s − − ζ x s − + · · · + 1 + ( x → /x ) . All the explicit x terms are polynomials and power of x decreases by two after one cycle of a coef-ficient combination. We next move onto
ADO -invariants, whose explicit formula can be obtainedalgorithmically. ADO Invariants of T (2 , s + 1) Explicit formulas for
ADO invariants of T (2 , s + 1) for s ∈ Z ≥ are constructed inductively.Torus knots are divided into four sets and each set has its own seed ADO [ T (2 , s + 1)] togetherwith a pattern of coefficients that generates the invariant for higher values of 2 s + 1. We presentan algorithm for obtaining explicit expressions. 6 lgorithm
1. Beginning with x s , write a polynomial with coefficients c i following one of the four patterns(shown below) that T (2 , s + 1) belong to: c s x s + c s − x s − + c s − x s − + c s − x s − + c s − x s − + c s − x s − , c n ∈ C .
2. Add a polynomial starting with x s − with exponent pattern and coefficients given by ADO [ T (2 , s − /x terms.As a consequence of the normalization factor ( x / − x − / ) in Conjecture 1, we obtain the symmetricversion of ADO-invariants. Their coefficients c n are divided into four types:1. − i, − i, − i − , − i − , − , −
1; for { T (2 , , T (2 , , T (2 , , · · ·}
2. 1 , , − i, − i, − i, − i ; for { T (2 , , T (2 , , T (2 , , · · ·} i, i, i + 1 , i + 1 , ,
1; for { T (2 , , T (2 , , · · ·} − , − , − i, − i, i, i ; for { T (2 , , T (2 , , · · ·} ,where the semicolon means that the next term has a power of x lowered by three. The coefficientsof the first and the third sets are differ by signs as well as the second and the fourth sets. ADO -invariant of the first knot in each set is a seed for next knot in the set. This pattern continuesfor all the subsequent knots in each set. The fundamental seed invariants can be easily computedusing the torus knot formula F T ( s,t ) in Section 2.1. ADO [ T (2 , − ix − ix + ( − − i ) x + ( − − i ) x − x − x − ix − i x + 1 − i x → /x ) .ADO [ T (2 , x + x + (1 − i ) x + (1 − i ) x − ix − ix + x − ix − i x − − i
2+ ( x → /x ) .ADO [ T (2 , ix + ix + (1 + i ) x + (1 + i ) x + x + x + ix + ix + (1 + i ) x + (1 + i x + (1 + i ) x + ix + ( i − x − x → /x ) .ADO [ T (2 , − x − x + ( − i ) x + ( − i ) x + ix + ix − x − x + ( − i ) x + ( − i ) x + ix + (1 + i ) x + x + (1 + i ) x + ix + ( i − x − i x → /x ) . For completeness, we display the
ADO -invariants of T (2 ,
3) [8] and T (2 , ADO [ T (2 , ix + ix + (1 + i ) x + 1 + i x → /x ) .ADO [ T (2 , − x − x + ( − i ) x + ( − i ) x + ix + (1 + i ) x + 1 + ( x → /x ) . .3 Examples Let us demonstrate the algorithm through examples. For T (2 ,
15) in the first set, the first stepof the algorithm yieldsStep 1 = − ix − ix + ( − − i ) x + ( − − i ) x − x − x . Next step is to use the coefficients from the seed
ADO [ T (2 , − ix − ix + ( − − i ) x + ( − − i ) x − x − x − ix − ix + ( − − i ) x + ( − − i ) x − x − x − ix − i x + (1 − i x . Since the above expression ends in (1 − i x , we need to reflect the coefficients about this termuntil a constant term is reached. This results inStep 3 = − ix − ix + ( − − i ) x + ( − − i ) x − x − x − ix − ix + ( − − i ) x + ( − − i ) x − x − x − ix − i x + (1 − i x − i x − ix − . The application of the last step leads to
ADO [ T (2 , − ix − ix + ( − − i ) x + ( − − i ) x − x − x − ix − ix + ( − − i ) x + ( − − i ) x − x − x − ix − i x + (1 − i x − i x − ix −
1+ ( x → /x ) . One can check that F T (2 , ( x, ζ ) obtained from the ADO [ T (2 , F T (2 , ( x, ζ ) from Section 2.1.For T (2 ,
17) in the second set, the seed invariant is
ADO [ T (2 , x + x + (1 − i ) x + (1 − i ) x − ix − ix . Step 2 = x + x + (1 − i ) x + (1 − i ) x − ix − ix + x + x + (1 − i ) x + (1 − i ) x − ix − ix + x − ix − i x + ( − − i x . After the refection about x -termStep 3 = x + x + (1 − i ) x + (1 − i ) x − ix − ix + x + x + (1 − i ) x + (1 − i ) x − ix − ix + x − ix − i x + ( − − i x − i x − ix + 1 . The last step results in
ADO [ T (2 , x + x + (1 − i ) x + (1 − i ) x − ix − ix + x + x + (1 − i ) x + (1 − i ) x − ix − ix + x − ix − i x + ( − − i x − i x − ix + 1 + ( x → /x ) . F T (2 , obtained from ADO [ T (2 , T (2 ,
19) is
ADO [ T (2 , ix + ix + (1 + i ) x + (1 + i ) x + x + x + ix + ix + (1 + i ) x + (1 + i ) x + x + x ix + ix + (1 + i ) x + (1 + i x + (1 + i ) x + ix + ( − i ) x − x . The last two steps produce
ADO [ T (2 , ix + ix +(1+ i ) x +(1+ i ) x + x + x + ix + ix +(1+ i ) x +(1+ i ) x + x + x + ix + ix + (1 + i ) x + (1 + i x + (1 + i ) x + ix + ( − i ) x − x + ( − i ) x + ix + (1 + i ) x + 1 + i x → /x ) . Similarly,
ADO [ T (2 , ADO [ T (2 , ADO [ T (2 , − x − x + ( − i ) x + ( − i ) x + ix + ix − x − x + ( − i ) x + ( − i ) x + ix + ix − x − x + ( − i ) x + ( − i ) x + ix + (1 + i ) x + x + (1 + i ) x + ix + ( − i ) x + ( − i x + ( − i ) x + ix + (1 + i ) x + 1 + ( x → /x ) . Formulas for
ADO invariants become lengthy as the winding number along the longitude of atorus increases so their expressions are recorded in the Appendix. ADO Invariants of T (3 , r ) For this knot,
ADO divides into two types:1. K = T (3 , r = odd >
3) :
ADO = x r − + x r − − x r − − x r − + x r − + x r − + · · · ± x → /x ) . K = T (3 , r = even >
2) :
ADO = − x r − − x r − + x r − + x r − − x r − − x r − + · · · ± x → /x ) . It is obvious that exponent of x decreases by two between every two consecutive terms, which iswhere change of sign occurs. Sign of a constant term is fixed such that the sign follows the patternof coefficients. We move onto the deformation of the ADO-polynomial.
ADO -invariants of T (2 , s + 1) A link between superpolynomial defined in [26] and F K was discovered in [7]. Specifically, twoparameter deformations F K ( x, q, a, t ) was introduced, which motivated to define t-deformed ADO-polynomial. In this Subsection, we present t-deformed version of ADO -invariants for T (2 , s + 1)knots. 9educed superpolynomial for negative torus knots carrying symmetric representation S r of SU ( N ) is stated in [30]: P S r [ T (2 , − (2 s + 1)); q, a, t ] = (cid:18) aq (cid:19) pr r X k =0 k X k =0 · · · k s − X k s =0 q (2 r +1)( k + ··· + k s ) − P si =1 k i − k i t k + ··· + k s ) × ( q r ; q − ) k ( − at/q ; q ) k ( q ; q ) k (cid:20) k k (cid:21) q · · · (cid:20) k s − k s (cid:21) q , ( w ; q ) m := m Y i =1 (1 − wq i − ) (cid:20) wn (cid:21) q := ( q ; q ) w ( q ; q ) n ( q ; q ) w − n , where s ∈ Z + , r is the dimension of S r and k ≡ r . Note that the convention for negative torusknot in [7] is T (2 , s + 1) for s ∈ Z + , which is opposite of the convention used in this article. In[7], it was shown that P S r can be converted into a two parameter deformation of F K by replacing q r by x and dropping the overall factor ( a/q ) pr : F T (2 , − (2 s +1)) ( x, q, a, t ) = ∞ X k =0 k X k =0 · · · k s − X k s =0 x k + ··· + k s ) − k q ( k + ··· + k s ) − P si =2 k i − k i t k + ··· + k s ) × ( x ; q − ) k ( − at/q ; q ) k ( q ; q ) k (cid:20) k k (cid:21) q · · · (cid:20) k s − k s (cid:21) q . Fixing a = q N and t = − F K ( x, q, a, t ) becomes the original F K ( x, q ) for torus knots. Differentspecialization of a, namely, a = − t − yields a refined Alexander polynomial [7], F K ( x, q, − t − , t ) = ∆ K ( x, t ) . Using Conjecture 2, a refined
ADO -polynomial for T (2 , s + 1), s ∈ Z + is ADO [ T (2 , s + 1); x, t ] = ( tx ) s + ζ − t ( tx ) s − + (cid:18) ζ t − ζ − (cid:19) ( tx ) s − − ζ t ( tx ) s − − t ( tx ) s − +( tx ) s − + ζ − t ( tx ) s − + (cid:18) ζ t − ζ − (cid:19) ( tx ) s − − ζ t ( tx ) s − − t ( tx ) s − + · · · + O (cid:18) tx (cid:19) , where O (1 /tx )-terms are determined by the t-deformed Weyl symmetry of the ADO p -invariant, ADO SU (2) p (1 /x, t ) = ADO SU (2) p ( ζ − p t − x, t ) . The suppressed polynomial terms follow the same power and coefficient patterns of the previousterms. The three formulas for the original
ADO [ T (2 , s + 1); x ] coalesce into one formula throughthe t-deformation. We next present a few examples. Specifically, additional manipulations are needed to arrive at F K ( x, q ) for torus knots ([7] Subsection 5.2). K = T (2 , F K ( x, q, a, t ) for T (2 , − F T (2 , − ( x, q, a, t ) = ∞ X k =0 k X k =0 x k + k ) − k q k + k − k k t k + k ) (cid:0) x ; q − (cid:1) k (cid:16) − atq ; q (cid:17) k ( q ; q ) k (cid:20) k k (cid:21) q . We next apply the mirror map to reverse the orientation of K, x /x q /q a /a t /t. Setting a = − /t , we get a refined Alexander polynomial of K (upon multiplication by a monomial),∆ K ( x, t ) = t x + 1 t x − t x − x + 1 . Further fixing t = −
1, it reduces to the Alexander polynomial of K. Moreover, this refined polyno-mial possess the t-deformed Weyl symmetry for the refined Alexander polynomial,∆ K (1 /x, t ) = ∆ K ( x/t , t ) . A refined
ADO -polynomial of K is computed via Conjecture 2 as ADO [ T (2 , x, t ] = ( tx ) + ζ − t ( tx ) + (cid:18) ζ t − ζ − (cid:19) ( tx ) − ζ t ( tx ) − t − ζ − t tx )+ (cid:18) t − ζ (cid:19) tx ) + ζ − t tx ) + ζ tx ) . This formula carries the t-deformed Weyl symmetry of the
ADO -invariant. Moreover, fixing t = − x ζ x , the refined polynomial becomes the original ADO -polynomial, ζ − x + ζ − x + ( ζ − − x − x − x → /x ) . • K = T (2 , F K for T (2 , −
7) is F T (2 , − ( x, q, a, t ) = ∞ X k =0 k X k =0 k X k =0 x k + k + k ) − k q k + k + k − k k − k k t k + k + k ) (cid:0) x ; q − (cid:1) k (cid:16) − atq ; q (cid:17) k ( q ; q ) k × (cid:20) k k (cid:21) q (cid:20) k k (cid:21) q .
11 refined Alexander polynomial of K having the refined Weyl symmetry is∆ T (2 , ( x, t ) = − t x − t x + 1 t x + tx − tx − tx + 1 t . A refined
ADO -polynomial of K is ADO [ T (2 , x, t ] = ( tx ) + ζ − t ( tx ) + (cid:18) ζ t − ζ − (cid:19) ( tx ) − ζ t ( tx ) − t ( tx ) + 1 − ζ − t tx ) − ζt tx ) + (cid:18) ζ − t − (cid:19) tx ) + ζt tx ) + 1( tx ) . This polynomial possess the t-deformed Weyl symmetry of the
ADO -invariant and after special-izing t = − x ζ x , it becomes x + x + (1 − ζ ) x − ζ x − ζ x + 1 + ( x → /x ) , which is the original ADO -polynomial for K. • K = T (2 , T (2 , ( x, t ) = t x + 1 t x − t x − t x + t x + 1 t x − t x − x + 1 . A refined
ADO -polynomial of K is ADO [ T (2 , x, t ] = ( tx ) + ζ − t ( tx ) + (cid:18) ζ t − ζ − (cid:19) ( tx ) − ζ t ( tx ) − t ( tx ) +( tx ) + ζ − t ( tx ) + (cid:18) ζ t − ζ − (cid:19) + 1 t tx + ζ tx ) − ζ t tx ) − t tx ) + (cid:18) ζ t − ζ − (cid:19) tx ) + 1 t tx ) + ζ tx ) . This polynomial is invariant under the refined Weyl symmetry of the
ADO -invariant and becomesthe original ADO -polynomial after setting t = − x ζ x , ζ x + ζ x + ( ζ − ζ − ) x − ζ − x − ζ − x + ζ x + ζ x + (cid:0) ζ − ζ − (cid:1) + ( x → /x ) . Appendix
We record
ADO invariants of torus knots obtained from the algorithm together with the resultsin Section 3.3. ADO [ T (2 , − ix − ix − (1+ i ) x − (1+ i ) x − x − x − ix − ix − (1+ i ) x − (1+ i ) x − x − x − ix − ix − (1 + i ) x − (1 + i ) x − x − x − ix − ix + (1 − i ) x − ix − ix − x − ix − ix + (1 − i ) + ( x → /x ) .ADO [ T (2 , x + x + (1 − i ) x + (1 − i ) x − ix − ix + x + x + (1 − i ) x + (1 − i ) x − ix − ix + x + x + (1 − i ) x + (1 − i ) x − ix − ix + x − ix − ix − (1 + 2 i ) x − ix − ix + x − ix − ix − (1 + 2 i ) + ( x → /x ) .ADO [ T (2 , ix + ix + (1 + i ) x + (1 + i ) x + x + x + ix + ix + (1 + i ) x + (1 + i ) x + x + x + ix + ix + (1 + i ) x + (1 + i ) x + x + x + ix + ix + (1 + i ) x + (1 + i x + (1 + i ) x + ix + ( − i ) x − x + ( − i ) x + ix + (1 + i ) x + (1 + i x + (1 + i ) x + ix + ( − i ) x − x → /x ) .ADO [ T (2 , − x − x + ( − i ) x + ( − i ) x + ix + ix − x − x + ( − i ) x + ( − i ) x + ix + ix − x − x + ( − i ) x + ( − i ) x + ix + ix − x − x + ( − i ) x + ( − i ) x + ix + (1 + i ) x + x + (1 + i ) x + ix + ( − i ) x + ( − i x + ( − i ) x + ix + (1 + i ) x + x + (1 + i ) x + ix + ( − i ) x − i x → /x ) .ADO [ T (2 , − ix − ix − (1+ i ) x − (1+ i ) x − x − x − ix − ix − (1+ i ) x − (1+ i ) x − x − x − ix − ix − (1 + i ) x − (1 + i ) x − x − x − ix − ix − (1 + i ) x − (1 + i ) x − x − x − ix − ix +(1 − i ) x − ix − ix − x − ix − ix +(1 − i ) x − ix − ix − x → /x ) .ADO [ T (2 , x + x + (1 − i ) x + (1 − i ) x − ix − ix + x + x + (1 − i ) x + (1 − i ) x − ix − ix + x + x + (1 − i ) x + (1 − i ) x − ix − ix + x + x + (1 − i ) x + (1 − i ) x − ix − ix + x − ix − i x + ( − − i x − i x − ix + x − ix − i x + ( − − i x − i x − ix + 1 + ( x → /x ) . DO [ T (2 , ix + ix +(1+ i ) x +(1+ i ) x + x + x + ix + ix +(1+ i ) x +(1+ i ) x + x + x + ix + ix + (1 + i ) x + (1 + i ) x + x + x + ix + ix + (1 + i ) x + (1 + i ) x + x + x + ix + ix + (1+ i ) x + (1+ 2 i ) x + (1+ i ) x + ix − (1 − i ) x − x − (1 − i ) x + ix + (1+ i ) x + (1 + 2 i ) x + (1 + i ) x + ix − (1 − i ) x − x − (1 − i ) x + ix + (1 + i ) x + (1 + 2 i ) + ( x → /x ) .ADO [ T (2 , − x − x − (1 − i ) x − (1 − i ) x + ix + ix − x − x − (1 − i ) x − (1 − i ) x + ix + ix − x − x − (1 − i ) x − (1 − i ) x + ix + ix − x − x − (1 − i ) x − (1 − i ) x + ix + ix − x − x − (1 − i ) x − (1 − i ) x + ix + (1 + i ) x + x + (1 + i ) x + ix − (1 − i ) x − (1 − i ) x − (1 − i ) x + ix + (1+ i ) x + x + (1+ i ) x + ix − (1 − i ) x − (1 − i ) x − (1 − i ) x + ix + (1 + i ) x + 1 + ( x → /x ) .ADO [ T (2 , − ix − ix − (1+ i ) x − (1+ i ) x − x − x − ix − ix − (1+ i ) x − (1+ i ) x − x − x − ix − ix − (1 + i ) x − (1 + i ) x − x − x − ix − ix − (1 + i ) x − (1 + i ) x − x − x − ix − ix − (1 + i ) x − (1 + i ) x − x − x − ix − ix + (1 − i ) x − ix − ix − x − ix − ix + (1 − i ) x − ix − ix − x − ix − ix + (1 − i ) + ( x → /x ) . References [1] S. Gukov and C. Manolescu, A two-variable series for knot complements, arXiv:1904.06057.[2] S. Gukov, D. Pei, P. Putrov and C. Vafa, BPS spectra and 3-manifold invariants,arXiv:1701.06567.[3] S. Gukov, P. Putrov and C. Vafa, Fivebranes and 3-manifold homology,
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