Quivers for 3-manifolds: the correspondence, BPS states, and 3d $\mathcal{N}$=2 theories

We introduce and explore the relation between quivers and 3-manifolds with the topology of the knot complement. This idea can be viewed as an adaptation of the knots-quivers correspondence to Gukov-Manolescu invariants of knot complements (also known as $F_K$ or $\hat{Z}$). Apart from assigning quivers to complements of $T^{(2,2p+1)}$ torus knots, we study the physical interpretation in terms of the BPS spectrum and general structure of 3d $\mathcal{N}=2$ theories associated to both sides of the correspondence. We also make a step towards categorification by proposing a $t$-deformation of all objects mentioned above.


Introduction
In recent years many interesting relations between quantum field theory, string theory, low-dimensional topology, and representation theory have been studied. In this paper we focus on the knots-quivers correspondence [1,2] and the Gukov-Manolescu invariants of knot complements [3]. (We will abbreviate them to GM invariants and denote F K -often this symbol is also used as their name.) The core of the knots-quivers correspondence is an equality (after appropriate change of variables) between the generating series of symmetrically coloured HOMFLY-PT polynomials of a knot and a certain motivic generating series associated to the quiver. This implies that Labastida-Mariño-Ooguri-Vafa (LMOV) invariants of knots [4][5][6] can be expressed in terms of motivic Donaldson-Thomas (DT) invariants of associated quivers [7,8], which are known to be integer [9]. In consequence, by assigning a quiver to the knot, we automatically prove the integrality of LMOV invariants, which is the statement of the LMOV conjecture [4][5][6]. This was done for all knots up to 6 crossings and infinite families of T (2,2p+1) torus knots and twist knots in [2]. More systematic approach in terms of tangles provided a proof of the knots-quivers correspondence for all two-bridge knots [10] and recently for all arborescent knots [11], however finding a general proof remains to be an open problem.
Geometric and physical interpretation of the knots-quivers correspondence given in [12,13] connects it with Gromov-Witten invariants and counts of holomorphic curves, showing that the quiver encodes the construction of BPS spectrum from the basic states. This spectrum is shared by dual 3d N = 2 theories associated to the knot complement and the quiver, and respecitve counts of BPS states are given by LMOV and DT invariants. The moduli spaces of vacua of these theories are encoded in the graphs of A-polynomials [14][15][16][17][18] and their quiver versions [12,13,19,20]. Another recent results on the knots-quivers correspondence include a relation to combinatorics of counting paths [19], a consideration of more general case of topological strings on various Calabi-Yau manifolds [20], and a connection with the topological recursion [21].
The origins of GM invariants lie in attempts of categorification of the Witten-Reshetikhin-Turaev (WRT) invariants of 3-manifolds. In order to solve the problem of non-integrality of WRT invariants Gukov, Pei, Putrov, and Vafa introduced new invariants of 3-manifolds [22,23] (we will call them GPPV invariants, they are denoted byẐ and often this symbol is also used as their name). The GPPV invariant is a series in q with integer coefficients and the WRT invariant can be recovered from the q → e 2πi k limit (see [22][23][24][25][26][27] for details and generalisations). Physical origins of GPPV invariants lie in the 3d-3d corresponcendce [28][29][30]:Ẑ is a supersymmetrix index of 3d N = 2 theory with 2d N = (0, 2) boundary condition studied first in [31]. Detailed analysis of this interpretation, as well as the application of resurgence, can be found in [22][23][24], whereas [32,33] contain some recent results on GPPV invariants coming from the study of 3d-3d corresponcendce. Another interpretation of GPPV invariants as characters of 2d logarithmic vertex operator algebras was proposed in [25]. On the other hand, this work -together with [34][35][36] -initiated an exploration of the intruguing modular properties ofẐ.
GM invariants can be treated as knot complement versions of GPPV invariants, F K =Ẑ S 3 \K [3] (because of that, sometimes GM invariants are denoted and calledẐ, descending it from GPPV invariants). From the physical point of view, the GM invariant arises from the reduction of 6d N = (0, 2) theory describing M5-branes on the 3-manifold with the topology of the knot complement. Important properties analysed in [3] include the behaviour under surgeries, the agreement of the asymptotic expansion of F K with the Melvin-Morton-Rozansky expansion of the coloured Jones polynomials [37][38][39][40], relations to the Alexander polynomials, and the annihilation by the quantum A-polynomials introduced in [41,42]. These relations suggest an interesting geometric interpretation of GM invariants in terms of the annuli counting presented in [43] and based on [44]. Another recent developments include a generalisation to arbitrary gauge group [45], an adaptation of the large colour R-matrix approach to GM invariants [46], and a relation to Akutsu-Deguchi-Ohtsuki invariant for q = e 2πi k [47]. From our perspective, the most important new results are closed form expressions for a-deformed GM invariants provided in [43]. They reduce to the initial F K for a = q 2 and turn out to be closely related to HOMFLY-PT polynomials. This allows us to propose a correspondence between knot complements and quivers via GM invariants and quiver motivic generating series. Moreover, since HOMFLY-PT polynomials admit t-deformation in terms of superpolynomials [48], we can follow [43] and conjecture results for a, t-deformed GM invariants. We also study the physical intepretation of the new correspondence (for clarity we will refer to the knots-quivers correspondence of [1,2] as the standard one), generalising the results of [12,13]. It includes an exploration of the duality between 3d N = 2 theories associated to the knot complement and the quiver, as well as a study of their BPS states, which allows us to introduce knot complement analogs of LMOV invariants via DT invariants.
The rest of the paper is organised as follows. Section 2 provides a short introduction to the knotsquivers correspondence and GM invariants of knot complements. In section 3 we state our main conjecture, assigning quivers to knot complements, and study consequences for 3d N = 2 theories and their BPS spectra. In section 4 we show how these ideas can be applied and checked on concrete examples of unknot, trefoil, cinquefoil, and general T (2,2p+1) torus knots. Section 5 contains a tdeformation of previous results inspired by the categorification of HOMFLY-PT polynomials. In section 6 we conclude with a discussion of interesting open problems.

Prerequisites
In this section we recall relevant aspects of the knots-quivers correspondence, its physical interpretation, and GM invariants of knot complements.

Knot invariants
If K ⊂ S 3 is a knot, then its HOMFLY-PT polynomial P K (a, q) [49,50] is a topological invariant which can be calculated via the skein relation. More generally, coloured HOMFLY-PT polynomials P K,R (a, q) are similar polynomial knot invariants depending also on a representation R of the Lie algebra u(N ). In this setting, the original HOMFLY-PT corresponds to the fundamental representation. From the physical point of view, P K,R (a, q) is the expectation value of the knot viewed as a Wilson line in U(N ) Chern-Simons gauge theory [51].
In the context of the knots-quivers correspondence, we are interested in the HOMFLY-PT generating series: where P K,r (a, q) are HOMFLY-PT polynomials coloured by the totally symmetric representations S r (with r boxes in one row of the Young diagram), which for brevity we will call simply HOMFLY-PT polynomials. The unusual expansion variable with the negative power comes from the necessity of resolving the clash of four different conventions present in the literature (and avoiding the confusion with the quiver variables): • KRSS convention from [1,2,52,53], • EKL convention from [12,13], • FGS convention from [3,14,16,18,45], • EGGKPS convention from [43].
For simplicity, in the remaining part of the paper we will drop "a-deformed" and callÂ(μ,λ, a, q) the quantum A-polynomials.
The LMOV invariants [4][5][6] are numbers assembled into the LMOV generating function N (λ, a, q) = r,i,j N r,i,j λ −r a i q j that gives the following expression for the HOMFLY-PT generating series: Exp is the plethystic exponential: if f (t) = n a n t n and a 0 = 0, then LMOV invariants can be extracted also from the A-polynomials, see [52,53].

Quivers and their representations
A quiver Q is an oriented graph, i.e. a pair (Q 0 , Q 1 ) where Q 0 is a finite set of vertices and Q 1 is a finite set of arrows between them. We number the vertices by 1, 2, ..., m = |Q 0 |. An adjacency matrix of Q is the m × m integer matrix with entries C ij equal to the number of arrows from i to j. If C ij = C ji , we call the quiver symmetric. A quiver representation with dimension vector d = (d 1 , ..., d m ) is the assignment of a vector space of dimension d i to the node i ∈ Q 0 and of a linear map γ ij : C di → C dj to each arrow from vertex i to vertex j. Quiver representation theory studies moduli spaces of stable quiver representations. While explicit expressions for invariants describing those spaces are hard to find in general, they are quite well understood in the case of symmetric quivers [7-9, 54, 55]. Important information about the moduli space of representations of a symmetric quiver is encoded in the motivic generating series defined as where the denominator is the q-Pochhammer symbol: we obtain the generating series of motivic Donaldson-Thomas (DT) invariants Ω d,s [7,8]: The DT invariants have two geometric interpretations, either as the intersection homology Betti numbers of the moduli space of all semi-simple representations of Q of dimension vector d, or as the Chow-Betti numbers of the moduli space of all simple representations of Q of dimension vector d, see [54,55].
[9] provides a proof of integrality of DT invariants for the symmetric quivers.

Knots-quivers correspondence
The knots-quivers correspondence [1,2] is a conjecture that for each knot K there exist a quiver Q and integers n i , a i , l i , i ∈ Q 0 , such that If we substitute (2.5) and (2.9), we obtain the knots-quivers correspondence at the level of LMOV and DT invariants: N (λ, a, q) = Ω(x, q)| xi=λ n i a a i q l i . (2.12) Since DT invariants are integer, this equation implies integrality of N r,i,j , which is known as the LMOV conjecture.
From the physical point of view, DT and LMOV invariants count BPS states in 3d N = 2 theories denoted by T [L K ] and T [Q L K ] [12,13]. T [L K ] is the effective 3d N = 2 theory on the world-volume of M5-brane wrapped on the knot conormal inside the resolved conifold: The structure of T [L K ] can be read from the semiclassical limit (q = e → 1) of the HOMFLY-PT generating series: where the integral dµ µ i dzi zi corresponds to the gauge group U(1) M ×U(1) Z1 ×. . .×U(1) Z k (we single out U(1) M and its fugacity µ because for the knot complement theory it becomes a global symmetry). W T [L K ] (µ, λ, a, z i ) is the twisted superpotential with two typical kinds of contributions: Each dilogarithm is interpreted as the one-loop contribution of a chiral field with charges (n Q , n M , n Zi ) under the global symmetry U(1) Q (arising from the internal 2-cycle in the resolved conifold geometry) and the gauge group Quadratic-logarithmic terms are identified with Chern-Simons couplings among the various U(1) gauge and global symmetries, with ζ i denoting the respective fugacities. For more details see [12,16,18,28,29,56,57].
Integrating over the gauge fugacities z i using a saddle-point approximation gives the effective twisted superpotential of the theory: The moduli space of vacua of T [L K ], given by the extremal points of the effective twisted superpotential, coincides with the graph of the classical A-polynomial: (2. 16) In analogy to T [L K ], the structure of T [Q L K ] is encoded in the semiclassical limit of the motivic generating series [12]: (2.17) Using the dictionary (2.13-2.14), we can interpret the elements of (2.17) in the following way: • The integral i dyi yi corresponds to having the gauge group U(1) (1) × · · · × U(1) (m) , • Li 2 (y i ) represents the chiral field with charge 1 under U(1) (i) , • Cij 2 log y i log y j corresponds to the gauge Chern-Simons couplings, κ eff ij = C ij , • log (−1) Cii x i log y i represents the Chern-Simons coupling between a gauge symmetry and its dual topological symmetry (the Fayet-Iliopoulos coupling).
The saddle point of the twisted superpotential encodes the moduli space of vacua of T [Q L K ] and defines the quiver A-polynomials [12,13,19,20]: is a classical limit of the quantum quiver A-polynomial, which annihilates the motivic generating series:Â The general formula for the quantum quiver A-polynomial corresponding to the quiver with adjacency matrix C is given by [13] and we can see that

GM invariants
GM invariants F K =Ẑ S 3 \K implicitly depend on the gauge group and results of [3] correspond to the simplest nontrivial case of SU (2). General case is analysed in [45], but it is often very involved from computational point of view. One of the goals of [43] was overcoming these difficulties and studying the large-N behaviour of F SU(N ) K (µ, q) -the GM invariant corresponding to the symmetric representations of SU(N ). It turns out that we can introduce a-deformed GM invariants F K (µ, a, q) which capture all N by the following relation: is annihilated by the quantum a-deformed A-polynomials: Since we adapt the convention that the series expansion of F K starts from 1, as in [43], sometimes we have to rescaleλ in order to compare to the literature. From our point of view, the crucial result of [43] is the connection between a-deformed GM invariants and HOMFLY-PT polynomials. In general we expect it to be some version of the Fourier transform, but it is difficult to define it properly in general case. However, in some cases (including all T (2,2p+1) torus knots) it reduces to a simple substitution: This equality enabled obtaining closed form formulas for F K , which are essential for finding quivers. Moreover, the t-deformation of HOMFLY-PT polynomials provided by the superpolynomials [48] suggests the following t-deformation of GM invariants [43]: which we will study in section 5. For simplicity, in the remaining part of the paper we will refer to F (µ, q), F K (µ, a, q), and F K (µ, a, q, t) broadly as GM invariants, specyfying the case explicitly only when it is not obvious from the context.

Quivers for GM invariants
In this section we introduce quivers for GM invariants and study their physical interpretation in terms of BPS state counts and 3d N = 2 effective theories.

Main conjecture
The connection between GM invariants and HOMFLY-PT polynomials, together with closed form expressions for F T (2,2p+1) (µ, a, q) found in [43], suggest that the idea of knots-quivers correspondence can be reformulated for GM invariants. In other words, we expect the following: Conjecture 1 For a given knot complement M K = S 3 \K, the GM invariant F K (µ, a, q) can be written in the form where C is a symmetric m × m matrix and n i , a i , l i are fixed integers. In consequence, there exist a quiver Q which adjacency matrix is equal to C and the motivic generating series reduces to the GM invariant after the change of variables x i = µ ni a ai q li : Let us stress that we may encounter negative entries of the quiver adjacency matrix: C ij < 0, so we can either accept having two types of arrows (ordinary arrows and "antiarrows" which annihilate each other) or use the change of framing to shift all entries (for details see [2]). Moreover, there is a possiblity of the sign difference in the knot complement and quiver variables which would lead to more complicated change of variables x i = (−1) ji µ ni a ai q li , but for all analysed examples it was not the case. We can also obtain quivers for SU(N ) (and more specifically initial SU (2)) GM invariants by substituting a = q N in (3.3), which leads to (3.4)

Corollary -BPS states and 3d N = 2 effective theories
If there exist a quiver Q corresponding to the GM invariant of M K , we can compute the DT invariants using Since Q is symmetric, we immediately know that DT invariants (which are coefficients of Ω(x, q)) are integer numbers [9]. We can also make a step back and apply the change of variables x i = µ ni a ai q li to (3.5). This leads us to the definition of the knot complement analogs of LMOV invariants: which implies Since DT invariants are integer numbers, so are the knot complement analogs of LMOV invariants. This is consistent with the physical interpretation given in [12,13]: we expect that Ω d,s and N r,i,j invariants count the number of BPS states in a dual 3d We identify T [M K ] with the effective 3d N = 2 theory on R 2 × S 1 which can be engineered in two equivalent ways. One is the compactification of N M5-branes on the knot complement: When N → ∞, many protected quantities -such as the twisted superpotential -depend on the number of M5-branes only via the combination q N which can be treated as a separate variable a. The second way comes from the large-N transition from the deformed to the resolved conifold [4,58]: Then log a is a complexified Kähler parameter of X. More details, together with a description of a third way of engineering T [M K ], are available in [43].
The structure of T [M K ] can be read from the semiclassical limit of the GM invariant: Recalling the relation (2.25), we can see that W T [M K ] is the same as the effective twisted superpotential of the 3d N = 2 theory analysed in [16,18]. On the other hand, the perspective of the large-N transition explains why W T [M K ] is a Legendre transform of the superpotential of the theory T [L K ] discussed in section 2.3: We can also consider the effective twisted superpotential and introduce λ back as the variable dual to µ, which is equivalent to the saddle point equation for W eff T [L K ] and the vanishing of the A-polynomial:

12)
A i (x, y) is a classical limit of the quantum quiver A-polynomial, which annihilates the motivic generating series F Q (x, q): Applying the change of variables x i = µ ni a ai (q → 1) together with i y i = λ, we can transform A i (x, y) into the classical A-polynomial A(µ, λ, a) given by (3.10).

Examples
This section is devoted to the explicit results: computations of quivers, BPS state counts, and 3d N = 2 theories for different GM invariants of knot complements. The choice of examples on which we check ideas from section 3 is determined by the fact that the closed form formulas for F K (µ, a, q) are completely new results, so far available only for the T (2,2p+1) torus knot complements [43]. We start from the unknot complement in the unreduced normalisation. For all other cases we use the reduced one, which corresponds to the division by the unknot factor. For more details see [43]. 1

Unknot complement
The simplest example is the unknot complement, for which the GM invariant is given by [43]: Using formulas for quantum dilogarithms we can write We can see that the quiver for the unknot complement is given by which is the same as in the standard knots-quivers correspondence for the unknot. However, now the variable a appears in the change of variables for the node without the loop. For the SU(N ) GM invariants we obtain: In case of original SU(2) GM invariants, we still have same quiver (4.3) and motivic generating series (4.2), but the change of variables reduces to we have only two nonzero DT invariants: Ω (1,0),1/2 = −1, Ω (0,1),0 = 1. By definition, they lead to two nonzero knot complement analogs of LMOV invariants, exactly as in the case of the standard LMOV invariants for the unknot [4]: Alternatively, we could have obtained this result by a direct comparison between (4.1) and (3.7). As discussed in section 3, we can use the semiclassical limit of the motivic generating series and the GM invariant to obtain effective twisted superpotentials of theories T [Q M0 1 ] and T [M 01 ], whose BPS states are counted by DT invariants and knot complement analogs of LMOV invariants respectively. The limit (3.11) for the unknot complement quiver is given by ] (x, y) = Li 2 (y 1 ) + Li 2 (y 2 ) + log(−x 1 ) log y 1 + log x 2 log y 2 + 1 2 log y 1 log y 1 .

(4.8)
We can see that T [Q M0 1 ] is a U(1) (1) × U(1) (2) gauge theory with one chiral field for each group and effective Chern-Simons level one for U(1) (1) , which is the same as T [Q L0 1 ] [12]. According to (3.12), the critical point of quiver twisted superpotential given by defines the quiver A-polynomials (4.10) A 1 and A 2 are decoupled, which originates from the lack of arrows between vertices 1 and 2 in Q.
The quantum quiver A-polynomial, which annihilates F Q (x, q) and reduces to (4.10) for q → 1, is given byÂ Applying the change of variables x 1 = µ, x 2 = µa, y 1 y 2 = λ to (4.10), we recover the classical A-polynomial of the unknot 2 [16]: which is in line with section 3. We also expect that the A-polynomial of the unknot encodes the moduli space of vacua of T [M 01 ] -the theory read from the semiclassical limit of F 01 (µ, a, q): Solving we indeed find an agreement with A 01 (µ, λ, a) = 0.

Trefoil knot complement
Let us move to the trefoil knot complement and start from the closed form expression found in [43]: Using the formula for (µq −1 ; q −1 ) k and the identity we can write Then we use (ξ; q) k (q; q) l (q; q) k−l = α+β=l γ+δ=k−l ξ α+γ q − 1 2 (α+γ) −q 1/2 α 2 +γ 2 +2γl (4. 19) to get After the change of variables we have with x i = µ ni a ai q li given by The quiver adjacency matrix reads:

(4.24)
This form will be useful for obtaining quivers for all T (2,2p+1) torus knots, but we can shift the framing by 1 to have only the non-negative entries: (4.25) For the trefoil the DT spectrum is infinite, but we can find the invariants order by order in x i . Let us focus on the linear one:  Applying the change of variables x 1 = µ 2 a, x 2 = µ 2 q 3/2 , x 3 = µaq −1/2 , x 4 = µq, we obtain the (first four) knot complement analogs of LMOV invariants for the trefoil knot complement: Li 2 (y i ) + log x 1 log y 1 + log(−x 2 ) log y 2 + log(−x 3 ) log y 3 + log x 4 log y 4 − log y 1 log y 2 − 1 2 log y 2 log y 2 + 1 2 log y 3 log y 3 − log y 1 log y 4 − log y 2 log y 4 .
Using the dictionary (2.13-2.14), we can see that the gauge group of (4) and we have four chiral fields φ i with charges δ ij under U (1) (j) . We also have gauge Chern-Simons couplings determined by the quiver adjacency matrix: κ eff ij = C ij , and Fayet-Iliopoulos couplings between each gauge symmetry and its dual topological symmetry. Comparing with [12], we can see that the structure of is given by the zero locus of the following quiver A-polynomials: which are the classical limits of the annihilators of F Q (x, q): The structure of the theory T [M 31 ] is encoded in the semiclassical limit of F 31 (µ, a, q): (4.32) Extremalisation with respect to z and the introduction of the variable λ dual to µ leads to (4.33) Eliminating z * we obtain the zero locus of the A-polynomial which agrees with [16] after taking into account the rescaling of λ by a −1 .

Cinquefoil knot complement
For more complicated knot complements all expressions become more involved, so we focus on finding the corresponding quivers, having in mind that the analysis of BPS states and 3d N = 2 theories can be done analogously to the unknot and trefoil complement case. In this section we concentrate on the 5 1 = T (2,5) knot complement with a plan of generalisation to all T (2,2p+1) torus knot complements. We start from the formula for GM invariant given in [43]: where we use the q-binomial: k l = (q; q) k (q; q) l (q; q) k−l . (4.36) Following the steps from section 4.2, we obtain Now we use the formula in two iterations: to get The change of variables with x i = µ ni a ai q li given by x 5 =µ 4 aq, x 6 =µ 4 q 5/2 , x 7 =µ 3 aq 1/2 , This is the adjacency matrix corresponding to the cinquefoil knot complement. Performed computations and the matrix itself may seem invloved, but they can be considered as a simple generalisation of the trefoil knot complement case. The most important is the fact that the change of variables (4.41) can be obtained from (4.21). For indices k 1 ,l 1 ,α 1 ,β 1 ,γ 1 ,δ 1 , we just substitute In case of k 2 ,l 2 ,α 2 ,β 2 ,γ 2 ,δ 2 , we start from k 1 ,l 1 ,α 1 ,β 1 ,γ 1 ,δ 1 and shift d i → d i+4 . From the point of view of the quiver adjacency matrix, this corresponds to copying the 4 × 4 matrix C 31 to fill 8 × 8 entries: (4.45) We still need to include q −k1k2 and the q-binomial k 1 k 2 which, according to (4.39) and (4.41), con- In consequence we have to modify (4.45) to  We can see that this is consistent with (4.44). The change of variables (4.43) is a direct consequence of (4.23) and (4.41), taking into account that in F 51 (µ, a, q) we have an extra factor of µ 2k2 q k2 .

General T (2,2p+1) torus knot complements
Now we are ready to move to the general case using recursion. We assume that we know the quiver for T (2,2p+1) torus knot complement and look for the quiver for T (2,2(p+1)+1) = T (2,2p+3) . The general formula is given by [43] F T (2,2p+1) (µ, a, q) = 0≤kp≤...≤k1 so when we go from p to p + 1, the summand in F T (2,2p+1) is multiplied by This means that we copy the rightmost (also downmost and diagonal) 4 × 4 matrix blocks right (respectively down and further on the diagonal), exactly like we did for C 31 → C 51 , with the same modification as in (4.47-4.48). Therefore the quiver for T (2,2p+1) torus knot is given by We can see that, similarly to the standard knots-quivers correspondence [1,2], increasing p does not change any previously determined entries of the matrix, so it makes sense to consider the limit p → ∞ leading to an infinite quiver.
The change of variables x i = µ ni a ai q li for T (2,2p+1) torus knot complement comes directly from the generalisation of (4.23, 4.43) according to (4.50) and is given by . . .

t-deformation
In this section we propose a t-deformation of the results from sections 3-4 using F K (µ, a, q, t) given in [43]. These results are based on the t-deformation of GM invariants following the generalisation of HOMFLY-PT polynomials to superpolynomials, which is reflected in equations (2.25) and (2.26).

Quivers
The connection between GM invariants and superpolynomials [43] suggest the following t-deformed correspondence: Conjecture 2 For a given knot complement M K = S 3 \K, the GM invariant F K (µ, a, q, t) can be written in the form F K (µ, a, q, t) = d1,...,dm≥0 where C is a symmetric m × m matrix and n i , a i , l i , t i are fixed integers. In consequence, there exist a quiver Q which adjacency matrix is equal to C and motivic generating series reduces to the GM invariant after the change of variables x i = µ ni a ai q li (−t) ti : Let us look at examples, starting from the unknot complement (in unreduced normalisation). Basing on [43], we have We immediately see that the quiver is the same as in (4.3), but the change of variables is given by For all T (2,2p+1) torus knots (including trefoil and cinquefoil) the situation is similar. The general formula (in reduced normalisation) [43] F T (2,2p+1) (µ, a, q, t) = 0≤kp≤...≤k1 leads to the quiver (4.51-4.52), the only difference with respect to section 4.4 lies in the change of variables. Now . . .
For the trefoil it leads to the change of variables Comparing with the quiver adjacency matrix (4.24), we can see that in contrary to the standard knots-quivers correspondence [1,2].

BPS states and 3d N = 2 effective theories
In section 3.2 we constructed knot complement analogs of LMOV invariants basing on the DT invariants. Knowing the form of the t-deformed change of variables x i = µ ni a ai q li (−t) ti , we can define the t-deformed knot complement analogs of LMOV invariants: N (µ, a, q, t) = r,i,j,k N r,i,j,k µ r a i q j (−t) k = Ω(x, q)| xi=µ n i a a i q l i (−t) t i , (5.10) which implies F K (µ, a, q, t) =Exp N (µ, a, q, t) 1 − q = exp   n,r,i,j N r,i,j,k µ rn a in q jn (−t) kn 1 − q   = r,i,j,k,l 1 − µ r a i q j+l (−t) k −N r,i,j,k = r,i,j,k (µ r a i q j (−t) k ; q) −N r,i,j,k ∞ . (5.11) Since DT invariants are integer numbers, so are N r,i,j,k .

Future directions
We conclude with a brief description of interesting directions for future research: • It would be desirable to understand the relation between quivers found in this paper and in [1,2]. We could see that for the unknot and the unknot complement quivers are the same, but the quiver corresponding to the trefoil has 3 nodes, whereas the one associated to the trefoil complement has 4 nodes. The core of this issue probably lies in the relation between HOMFLY-PT polynomials and GM invariants. It seems that in general it should be some transformation analogous to the Fourier transform, which for some cases (including T (2,2p+1) torus knots) reduces to the substitution µ = q r . Investigating the relation between the standard and new correspondence could help in finding quivers for other knot complements. Probably it would also provide a new insight to the duality web of associated 3d N = 2 theories studied in [12,13].
• Since GPPV invariants exhibit peculiar modularity properties and are related to logarithmic conformal field theories, it would be interesting to perform a study of these aspects for GM invariants in the context of the correspondence with quivers.
• GM invariants still lack a proper mathematical definition and a proof that they are ineed topological invariants of knot complements. These goals seem ambitious, but achieving them would potentially allow to properly state and prove the conjectures proposed in this work.
• One can look for suggestions and constraints on GM invariants coming from the existence of a corresponding quiver. Such approach applied to the case of the standard knots-quivers correspondence enabled finding the formulas for HOMFLY-PT polynomials P K,r (a, q) for 6 2 and 6 3 knots [2].
• The form and meaning of the t-deformation proposed in [43] and applied here are still uncertain. Better understanding of this aspect is crucial for the categorification of 3-manifold invariants, which was the main motivation of introducing GPPV and GM invariants.