Knot Complement, ADO-Invariants and their Deformations for Torus Knots

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 formulas and/or an algorithm for certain ADO-invariants of torus knots obtained from the series invariant of complement of a knot. Furthermore, one parameter deformation of ADO_3-polynomial of torus knots is provided.


Introduction
Categorification of link invariants has been a source of fruitful interactions between physics and low dimensional topology over the past decades (see [31,32,33] for reviews). Since the advent of the Khovanov homology [17], which categorifies the Jones polynomials of links, there has been constructions 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 polynomials, Alexander, sl(N )-invariants and HOMFLY polynomial, respectively. Not only has the categorification deepened the conceptual aspects of links, but it has also provided a more powerful machinery to compute higher structural invariants beyond polynomial invariants. Furthermore, these advancements have inspired new directions in physics, which resulted in physical realizations of the link homologies. Beginning from knot Floer homology, its physical interpretation was found in [26]. A physical realization of Khovanov homology and Khovanov-Rozansky homology was first provided using topological string theory in [27]; additionally, through the conifold transition, existence of the HOMFLY homology was predicted as well. In the case of Khovanov homology, a different physical system involving D-branes was achieved in [25]. For Kauffman homology, its physical construction exemplified the role of orientifolds [28]. Even knot homology based on an exceptional Lie algebra admits a physical description [29] (see Table 1 for summary).

Polynomial
Homology Physical Realization  Table 1: A summary of link invariants and their physical realizations. Choice of an orientifold type determines so(n) or sp(n) Lie algebra. Applications of S and T-dualities are necessary to the brane system 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 blocksẐ(q) [2,3] inspired a new kind of invariant for a complement of a knot [1]. This knot invariant denoted as F K is a two-variable series that emerges fromẐ(q): complement, which arises from the integrality of the coefficients of F K -series. This in turn originates from the appearance of dimension of BPS Hilbert space of T [Y ] in the q-seriesẐ(Y, q) for a generic 3-manifold Y. Furthermore, this Hilbert space is identified with a conjectured triply graded three manifold homology H i,j BP S (Y ; b) whose (graded) Euler characteristic iŝ The WRT-invariant of Y is recovered fromẐ 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 and the ADO link invariant [9] have been discovered in [8]. This relation is conjectured to hold for all knots and for any roots of unity: 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 a generalization of the above conjecture. The rest of the paper is organized as follows. In Section 2, we briefly review the series invariant for a knot complement and the ADO-invariants. In Section 3, we present the explicit formulas and/or an algorithm for the ADO 2 , ADO 3 , ADO 4 -invariants for torus knots. Furthermore, one parameter deformation of ADO 3 -polynomial for torus knots is discussed. takes the form 2 : where f m (q) are Laurent series with integer coefficients, c ∈ Z + and ∆ ∈ Q. Moreover, x-variable is associated to Spin c -structures of M 3 K , which can be identified with H 1 (M 3 K ; Z); it has an infinite order, which is reflected as a series in F K . For applications, some classes of knots have been analyzed [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 > 1 with gcd(s, t) = 1 [1].
Prior to F K 's potential relation to the (original) ADO-invariant, it was proposed that F K possess similar characteristics of sl(2)-colored Jones polynomial through the Melvin-Morton-Rozansky conjecture [12,13](proven in [14]), and the quantum volume conjecture [15,16]: For a knot K ⊂ S 3 , the asymptotic expansion of the knot invariant F K (x, q = e ) about = 0 coincides with the Melvin-Morton-Rozansky expansion of the colored Jones polynomial in the large color limit: where x = q n is fixed, n is the color of K, P r (x) ∈ Q[x ±1 ], P 0 (x) = 1 and ∆ K (x) is the Alexander polynomial of K.
Conjecture 4. ([1] Conjecture 1.6). For any knot K ⊂ S 3 , the normalized series f K (x, q) satisfies a linear recursion relation generated by the quantum A-polynomial of K:

ADO-Invariants of Knots
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, 1)-colored tangle diagram obtained by cutting the knot (or a component of a link). From this colored and oriented tangle diagram, the ADO-invariant is constructed from a non-semisimple category of module over the unrolled quantum group U H ζ 2r (sl 2 (C)) together with the modified quantum dimension. Although we will not employ the quantum algebra construction of the ADO-invariants, for conceptual background, we give a concise review of the ingredients of this construction [9,11,10].
The first ingredient is the unrolled quantum group U H ζ 2r (sl 2 (C)), which is a C-algebra specialized at q = ζ 2r ; its generators and relations are Generators: E, F, K, K −1 , H Relations: This algebra possess a Hopf algebra structure: The second element of the construction of the ADO-invariant is a functor RT between a category of colored oriented tangle diagrams COD and a category Rep of representations of U H ζ 2r (sl 2 (C)).
The objects of COD are framed colored oriented (1, 1)-tangle diagrams and morphisms are equivalence classes of the tangle diagrams whose equivalence relations are generated by the tangle moves(see [9] Section 2). For the target category, its objects are vector spaces V and morphisms are linear maps between them. The image of the RT functor is where V α is a vector space assigned to K (or to an open component of a link 3 ) and d(V α ; r) is the modified quantum dimension, This modified dimension replaces the usual quantum trace, which vanishes in this context. Moreover, it makes ADO(K) an isotopy invariant.

ADO-Invariants of Torus Knots
Recently, evidences for a relation between F K at specific values of roots of unity and the ADO invariants were discovered for (positive) trefoil, the figure-8 and 5 2 knots [8]. Furthermore, this relation is conjectured to hold for any roots of unity and for all knots (Conjecture 1). We strengthened this connection between the two invariants by presenting an explicit formula or an algorithm for ADO 3 and ADO 4 invariants of T (2, 2s + 1), s ∈ Z + and ADO 2 of T (3, w), w > 3.

ADO 3 Invariants of T (2, 2s + 1)
The ADO 3 invariants of T (2, 2s + 1) are divided in three types depending on their coefficient pattern. The general formula are as follows.

Algorithm for ADO 4 Invariants of T (2, 2s + 1)
Explicit formulas for ADO 4 invariants of T (2, 2s + 1) for s ∈ Z ≥7 are constructed inductively. Torus knots are divided into four sets and each set has its own seed ADO 4 [T (2, 2s + 1)] together with a pattern of coefficients that generates the invariant for higher values of 2s + 1. We present an algorithm for obtaining explicit expressions.
where the semicolon means that the next term has a power of x lowered by three. The coefficients of the first and the third sets are differ by signs as well as the second and the fourth sets. ADO 4invariant of the first knot in each set is a seed for next knot in the set. This pattern continues for all the subsequent knots in each set. The fundamental seed invariants can be easily computed using the torus knot formula F T (s,t) in Section 2.1.

Examples
Let us demonstrate the algorithm through examples. For T (2,15) in the first set, the first step of the algorithm yields Next step is to use the coefficients from the seed ADO 4 [T (2, 7)] but its powers of x adjusted appropriately. Step Since the above expression ends in (1 − i2)x 4 , we need to reflect the coefficients about this term until a constant term is reached. This results in The application of the last step leads to One can check that F T (2,15) (x, ζ 4 ) obtained from the ADO 4 [T (2, 15)] using Conjecture 1 agrees with the direct computation of F T (2,15) (x, ζ 4 ) from Section 2.1.
For T (2,17) in the second set, the seed invariant is ADO 4 [T (2,9)] and application of the first and second steps produce Step After the refection about x 4 -term

The last step results in
One can again verify that F T (2,17) obtained from ADO 4 [T (2,17)] matches with the direct method.
Formulas for ADO 4 invariants become lengthy as the winding number along the longitude of a torus increases so their expressions are recorded in the Appendix.

ADO 2 Invariants of T (3, r)
For this knot, ADO 2 divides into two types: 1. K = T (3, r = odd > 3) : 2. K = T (3, r = even > 2) : It is obvious that exponent of x decreases by two between every two consecutive terms, which is where change of sign occurs. Sign of a constant term is fixed such that the sign follows the pattern of coefficients. We move onto the deformation of the ADO-polynomial.

Deformed ADO 3 -invariants of T (2, 2s + 1)
A link between superpolynomial defined in [26] and F K was discovered in [7]. Specifically, two parameter deformations F K (x, q, a, t) was introduced, which motivated to define t-deformed ADOpolynomial. In this Subsection, we present t-deformed version of ADO 3 -invariants for T (2, 2s + 1) knots.
Reduced superpolynomial for negative torus knots carrying symmetric representation S r of SU (N ) is stated in [30]: P S r [T (2, −(2s + 1)); q, a, t] = a q pr r where s ∈ Z + , r is the dimension of S r and k 0 ≡ r. Note that the convention for negative torus knot in [7] is T (2, 2s + 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 : Fixing a = q N and t = −1, F K (x, q, a, t) becomes the original F K (x, q) for torus knots. 4 Different specialization of a, namely, a = −t −1 yields a refined Alexander polynomial [7], Using Conjecture 2, a refined ADO 3 -polynomial for T (2, 2s + 1), s ∈ Z + is where O(1/tx)-terms are determined by the t-deformed Weyl symmetry of the ADO p -invariant, The suppressed polynomial terms follow the same power and coefficient patterns of the previous terms. The three formulas for the original ADO 3 [T (2, 2s + 1); x] coalesce into one formula through the t-deformation. We next present a few examples.
• K = T (2, 5) We start from F K (x, q, a, t) for T (2, −5), We next apply the mirror map to reverse the orientation of K, Setting a = −1/t, we get a refined Alexander polynomial of K (upon multiplication by a monomial), Further fixing t = −1, it reduces to the Alexander polynomial of K. Moreover, this refined polynomial possess the t-deformed Weyl symmetry for the refined Alexander polynomial, A refined ADO 3 -polynomial of K is computed via Conjecture 2 as A refined Alexander polynomial of K having the refined Weyl symmetry is

Appendix
We record ADO 4 invariants of torus knots obtained from the algorithm together with the results in Section 3.3.