Link polynomial calculus and the AENV conjecture

Using the recently proposed differential hierarchy (Z-expansion) technique, we obtain a general expression for the HOMFLY polynomials in two arbitrary symmetric representations of link families, including Whitehead and Borromean links. Among other things, this allows us to check and confirm the recent conjecture of arXiv:1304.5778 that the large representation limit (the same as considered in the knot volume conjecture) of this quantity matches the prediction from mirror symmetry consideration. We also provide, using the evolution method, the HOMFLY polynomial in two arbitrary symmetric representations for an arbitrary member of the one-parametric family of 2-component 3-strand links, which includes the Hopf and Whitehead links.

{q r−j }{q s−j } D j (1) to obtain formulas in the case of the Whitehead link, and of the Borromean rings: Here {x} = x − x −1 and D k = {Aq k }. These expressions are manifestly symmetric under the permutation of r and s or r, s and t. These results did not appear in literature so far 2 and were tested at particular values of r, s and t indicated above (the Whitehead HOMFLY polynomials were calculated up to r + s ≤ 12 and the Borromean ones up to r + s + t ≤ 12). These answers are the normalized HOMFLY, obtained by division over a product of two or three unknots S * r S * s or S * r S * s S * t , where the quantum dimension Note that in the case of links (not knots), the normalized HOMFLY is not a polynomial. The first term is unity, as conjectured in [29], this is a generic feature of normalized knot polynomials in the topological framing.
In the paper, we also apply the evolution method of [14,29] to the family of 2-component 3-strand links (see Figure 2), which includes the Hopf link, the Whitehead link, L7a6, L9a36, L11a360 etc in accordance with the classification of [33]. In this way, we obtain the HOMFLY polynomials for this whole family in the case of the both symmetric representations, the result reads: [j]! · {q} j · · p a,b=0 This formula has no form of a q-hypergeometric polynomial. Therefore, the existing software implementing Zeilberger's algorithm for the hypergeometric sums [34,35] can not be immediately used to obtain the quantum A-polynomials. For this purpose, the formula still has to be reshuffled like it is done in the case of the Hopf (1) and the Whitehead (2) links, i.e. for k = 0 and k = 2 respectively. In fact, revealing the differential hierarchy structure in this formula for generic k is a challenging problem of its own. We will return to this issue elsewhere.

Recursion relations for link polynomials
Knot polynomials depend on various variables and satisfy various interesting relations [36]. Of relevance for the AENV conjecture is dependence on the "spins" r, s, t. For links there are two kinds of such relations: those relating evolution in r and s and involving the single spin r only. The former ones are very easy to observe. The latter ones are usually complicated, but instead they can be looked for in the same way as in the case of knots: with the help of computer programs implementing Zeilberger's algorithm for the hypergeometric sums [34,35]. Recursion relations in r are also sometime called "quantum A-polynomials".
In the remaining part of this section we just list these formulas for the Hopf and Whitehead links, equations for the Borromean rings is too huge to be included into the text. Equations look a little simpler for nonnormalized polynomials H r,s,t = S * r S * s S * t H r,s,t , also it is their large representation asymptotic that is to be compared with [1].

Unknot
The normalized HOMFLY polynomial for the unknot is just unity, i.e. it does not depend on the representation, However, for the non-normalized polynomial H U r = S * r the same equation looks already a little non-trivial: For the purposes of the present paper it is useful to consider the restricted Ooguri-Vafa (OV) partition function which includes a sum over symmetric representations only: we denote it by a bar. This restriction means that in the usual OV partition function where p k = i z k i are auxiliary time variables (sources) and S R are the Schur functions, one leaves only one Miwa variable z i = z, i.e. p k = z k . Then,Z For the unknot all the four types of Ooguri-Vafa functions are straightforwardly evaluated: Here we used the by-now-standard notation for the time variables at the topological locus, where the Schur functions turn into quantum dimensions, The capital R and small r are used to denote all and only symmetric representations respectively, the calligraphic Z, H and ordinary letters Z, H denote non-reduced and reduced polynomials and partition functions. The first two lines of (9) are avatars of the Cauchy formula for the Schur functions, while the last two are just direct corollaries of definition of the Schur polynomials (note that sometime one use differently normalized time variables t k = 1 k p k , where this definition looks even simpler). The partition functionZ |U (z) denoted through Ψ U (A|z) in [1] is just a further restriction of (11) to the topological locus (10): As a corollary of (6), it satisfies a difference equation in z. Indeed, rewriting (6) as one gets for the generating function or where the multiplicative-shift operators are defined byT ± z f (z) = f (q ±1 z). An important additional observation is that the action of the dilatation operator T A on the unknot function Ψ U (A|z) is closely related to the action of T z : while

Hopf link
There are numerous different representations for the colored HOMFLY polynomial of the Hopf link besides (1).
To begin with, one could use the celebrated Rosso-Jones formula [10,11,12,18,13]: where the cut-and-join operatorŴ =Ŵ [2] a,b (a + b)p a p b ∂ ∂p a+b + abp a+b ∂ 2 ∂pa∂p b . According to [1], it is most convenient to begin from the alternative representation for the Hopf link used in ref. [17]: where the r-shifted topological locus is Given these formulas, one can write [1] the restricted Ooguri-Vafa partition function for the Hopf link as In other words,Z what implies for the non-normalized HOMFLY polynomials: For the normalized polynomials one gets, after using (6): Additional equations are obtained by the substitution x ↔ y, i.e. r ↔ s.
Complementary to these recursions in s and r there is a simple relation, which involves shifts in the both directions and can be checked from the manifest expression for the HOMFLY polynomial: Note that it looks more concise when written in terms of the non-normalized HOMFLY. It turns out that (25) remains almost the same in the case of more complicated links, hence we call it "simple" relation.

Hopf link recursions from eq.(1)
Our main task is, however, to deduce recurrence relations from still another representation of the HOMFLY polynomial for the Hopf knot, that is, from (1) What follows from (26) just immediately by changing summation variable from k to k − 1, is which can also be rewritten in two other ways: and For each given r ≥ s these are finite recursions to H H r−s,0 = 1 (we remind that for a link of unknots this quantity is symmetric under the permutation of r and s). As a corollary, the Ooguri-Vafa generating function satisfiesZ Eq.(31) is a recursion in a more tricky sense than (27): the Ooguri-Vafa functions are power series, i.e. the series with only non-negative powers of x and y. Then (31) allows one to express the coefficients in front of a given power through those at lower powers, and thus reconstruct the entire series. The derivation of (31) makes use of the identity which is an important complement of (6). Note that (27) is a similar complement of (24). However, it is (27) which is a straightforward implication of (26): derivation of (24) from this starting point is somewhat more transcendental. It uses the fact that the hypergeometric polynomials often satisfy difference relations w.r.t. its parameters, not only arguments, and (26) is exactly of this type. Indeed, it can be rewritten in terms of the q-factorials (n)! = and acquires a form of the q-hypergeometric polynomials (for the q-hypergeometric functions of type 3 F 1 , note that the (−) k in this case is absorbed into the q-factorials). In this case, the recursion in the parameter s can, of course, be found "by hands", but this is almost impossible for more general series of this type like (2) and (3). Therefore, it makes sense to apply the standard software [37], one should only divide the quadratic form in the exponent of q by two, because the program uses q instead of q 2 . The program gives the equations in the case of reduced and non-reduced polynomials in the form and where When we sum (34) and (35) over k from 0 to infinity, then only the lower summation limit at k = 0 contributes at the r.h.s., because both h(k) and h(k) vanish at large k > min(r, s). Since and this immediately leads to (23) and (24). It deserves noting that instead of the first order difference equations with a non-vanishing free term, one can write down a homogeneous equation of the second order in the shift operator: This follows directly from (23) in the form Equation (40) can be obtained by applying the operator annihilating the unknot S * r , (6) to (23): since the r.h.s. of (23) depends on r only though S * r , one immediately obtains a homogeneous equation (of the second order).

Whitehead link
Now we switch to the Whitehead link, with the HOMFLY polynomial in symmetric representations given by (2): We are again interested in recurrence relations in r and s. First of all, one can check that it satisfies the "simple" relation, which is practically the same as (25) for the Hopf link: The only difference is that in (25) there is a factor q 2 in front of the second item at the l.h.s. We remind that the HOMFLY polynomial in this formula is non-normalized, H W r,s = H W r,s S * r S * s . Second, the "natural" recursion for (2) is somewhat less trivial than (27): r,s (A|q) is just the first term in the additional hierarchy With this definition, H r,s = 1 only for r, s > m and this relation is indeed a recursion with a finite number of steps needed to find any particular term in these polynomials. In accordance with (45), the Ooguri-Vafa partition functionZ is just the first member of the hierarchy, and a direct counterpart of (31) is obtained when one multiplies (45) by Eq.(44) can also be rewritten in terms of the OV partition function, this time without any decomposition: Third, the most non-trivial recursion in s only (with r and A fixed) can be again obtained with the help of the program [37], this time it should be applied to the q-hypergeometric polynomial (43): Similar to the Hopf case, one obtains for the non-normalized quantities the equation and this leads to the equation As in the Hopf case, the r.h.s. of this equation depends on r only though the unknot S * r , i.e. one can again apply the operatorÔ r (42) in order to get a (fifth order) homogeneous equation.

Borromean rings
For the Borromean 3-component link, the normalized HOMFLY polynomial is given by (3): Note that the corrections to 1 are of order {q} 4 . Also for r or s or t = [0] the answer is just unity: because when one component of the Borromean link is removed, the other two are two independent unknots. The "simple" relation for non-normalized H B is literally the same as (44): Of course, this time there are two more relations for the pairs r, t and s, t.
Rewriting (55) as a hypergeometric polynomial: and in non-normalized case as we can apply the program of [37] to get the recursion relation in s (or r or t), which is now an order six difference equation. It is, however, too huge to be presented here -but can be easily generated by MAPLE or Mathematica. For illustrative purposes we present just the two simplest items in the equation for non-normalized HOMFLY polynomial:

More three-component links
In paper [1] there are three more three-component links considered.

Spectral curves
The linear recurrence relations in the previous section are actually written in terms of two operatorsQ i andP i , which act on the representation index of HOMFLY polynomials as follows: These operators satisfy the commutation relation and they commute in the limit q = e −→ 1. Therefore, in this limit in an appropriate basis, which is in fact provided by the (restricted) Ooguri-Vafa functions, one can substitute the difference equations by a vanishing condition for a system of polynomials. All together they define an algebraic variety, which is called the spectral variety, associated with the given link or knot. Moreover, this spectral variety is known to coincide with the classical A-polynomial. The AENV conjecture [1] is about these classical A-polynomials, which are independently calculated by topological methods, and our purpose in this section is to demonstrate that they are indeed obtained as the spectral curves for our knot polynomials. This check proves the AENV conjecture for the Whitehead and the Borromean links.

Unknot
At small (q = e ) the Ooguri-Vafa function (12) behaves as follows and the genus zero free energy of the unknot is The spectral curve is the algebraic relation between µ = exp z ∂W (A|z) ∂z and z. From (64) and one obtains the spectral curve for the unknot: with the Seiberg-Witten differential log µ dz z . Changing the variables, one can rewrite it as [1]: Equivalently one can obtain the same spectral curve (66) as the q = 1 limit of (15), provided the action of the dilatation operatorT is substituted by the quasi-momentum µ: Useful in applications is also the expansion Still for our purposes it is desirable to derive Σ U directly from (6). This is, of course, straightforward: Since for the generating OV function Ψ(z) = r H r z r P Ψ(z|A) = zΨ(z|A), this equation is just the same as (69).

Hopf link
In the small limit of (31), there are two possibilities: one is that Z Hopf is not sensitive to the small (by ) shift of its parameters, i.e. is regular in the limit q → 1; then with where N is an arbitrary constant. This "homogeneous part" of solution is singular at the point q = 1 and, hence, predominates over (75). This solution is just the one suggested in [1] to describe the non-trivial phase "2" of the Hopf link OV partition function: the double Fourier transform with the weight exp ipξ+ip ′ η iŝ This function before and after the Fourier transform satisfies the peculiarly simple differential equations like Our next task is to derive them directly from the recurrence relation (23). This is straightforward. In the phase 2 of [1], when inhomogeneous terms at the r.h.s. are suppressed like in (76) (we denote this approximation by ∼ =), With operatorP s substituted by its eigenvalue z s , we get: After the change of variables (67) the first equation turns into 1 x r − 1 y s = 0 (84) and the variety becomes simply Of course, instead from (23) one could start from relation (24) for the normalized polynomials, The procedure of getting the varieties can be described by a more formal sequence of steps: take the difference equation, rewrite them as an operator polynomial ofQ i andP i , make substitutionŝ and then put q = 1. In particular, in the Hopf case one can start with the homogeneous equation of the second order (40) and a similar equation for the shift w.r.t. r and immediately obtain the spectral curve as the intersection of products of the unknot curves and (85).

Whitehead link
Again in the limit of small there is a "trivial" solution it is the same for all links made from the same number of unknots.
What distinguishes different linkings of the two unknots is another W (2) in another phase, solving the homogeneous equations at the l.h.s. of (48) and (54). The corresponding variety Σ WH (2) lies in the intersection of two varieties, which arise in the double scaling limit of (44) and (54) respectively. The first one is simple: This formula can be also obtained (44) by the formal procedure described in the previous subsection: make substitutions r = log µ r log q 2 , s = log µ s log q 2 (i.e. µ r = q 2r , µ s = q 2s ),P r −→ z r ,P s −→ z s , then put q = 1 and obtain (89). This equation coincides with the first formula in V K (2) for the Whitehead link from [1, s.7.3] after the change of variables Similarly, the more complicated second formula (54) by the same procedure is converted into a product of A 2 µ 2 s −1 A 5 µr µ 2 s and the irreducible equation: This factorization is immediately seen in (54), because in the limit (90) all the factors D 2s+... become the same, and the only term without such a factor (underlined in (54)) is proportional to {q 2 } and vanishes in the limit. Eq.(92) coincides with the second formula in V K (2) for the Whitehead link from section 7.3. of [1] after the same change of variables (91). Of course, along with (92) there is another equation, obtained by the substitution r ↔ s.
We discussed here only the l.h.s. of (54). As was explained in s.2.4, one can obtain the fifth order homogeneous equation by acting on (54) with the operatorÔ r canceling the unknot S * r . However, in the q = 1 limit it reduces just to the unknot factor, as in the Hopf case, i.e. comes from Σ WH (1×1)

Borromean rings
With the same procedure one can immediately generate the spectral variety for the Borromean rings from the recurrent relations of s.2.5. In particular, the "simple" relations (56) lead to exactly the same, as (89). Two more equations of this type exist for the two other pairs of variables (r, t) and (s, t).
The variety Σ B (3) is provided by the intersection of (93) with that described by the limit (90) of the "complicated" equation: Here (λ, µ, ν) is any permutation of the triple (µ r , µ s , µ t ), and z = z µ . The coefficients ξ B ij are Making the change of variables (91): one obtains from (94)

Spectral curves
Equations, derived in the previous subsections describe various hypersurfaces in the space of (µ, z)-parameters.
In this short subsection we collect all that one can learn about the spectral varieties V of the sequence unknot-Hopf-Whitehead-Borromean rings from the double scaling limit −→ 0, r −→ ∞ of the recurrence relations for the HOMFLY polynomials obtained in section 2. These varieties can be described either as intersections of Σ's at different phases, or just as intersections of all varieties obtained in the limit q = 1 from all recurrent relations for the given link.
An alternative interesting question is how much the knot polynomials for other simple links deviate from this simple structure. In this section we provide results about the link family, to which the Hopf and Whitehead links naturally belong and study the evolution a la [18,29] along this family (see Figure 2).
Thus, we consider the family of the L 2k+1 = [k, 1; 2] links (it is also easy to extend to [k, l; 2], however increasing the length of braid with counter-oriented strands from 2 to m is more problematic). We present the results in the form of the differential expansion of [24], generalizing the dream-like formulas of [14] for the figure eight knot.
The series L 2k+1 in representation [r] ⊗ [1] . These are all links of the two unknots, thus the answer is symmetric under the permutation of r and s. In the following list we refer also to notation from the link classification in [33]. Note that k in this family is always even, k = 2m. Clearly, Looking at these formulas one can get an impression that the powers of A increase with r, however this is not true: they cancel between the two structures in this formula. This is getting clear from an alternative expression (107) below. In fact, one can consider these formulas from the point of view of the evolution method of [18] and [29]. The product of representations [r] ⊗ [1] = [r + 1] + [r, 1], and thus one expects that with some parameters a and b, which can depend on A and q, but not on k. This is indeed true, with i.e.
This time it is clear that the powers of A are limited, however, instead the differential structure of (103) is obscure. An alternative formulation, where the both properties are transparent is as follows: For m ≤ 3 the standard summation rule is implied in the expression for f 0 : e.g.
One way to do this is to return to the evolution method. In generic symmetric representations [r] and [s] Clearly, with the obvious notation D r ! ≡ r j=0 D j (note that the product starts from j = 0 and includes r + 1 factors, also note that according to this definition D −1 ! = 1 and D −2 ! = 1/D −1 ), The underbraced ratio respects r ↔ s symmetry, because it is equal to q r+s−j − q j (q r−s + q s−r ) + (2q j − q −j )q −r−s {q} (117) In general σ (p,j) r,s = (−) p+j · q (p−1)(p−2j) · q j(j+1) 2 −j(r+s) [j]! · {q} j · · p a,b=0 To apply Zeilberger's programs and obtain recursion relations one should begin with the differential-hierarchy analysis a la [24] to convert the answers to the hypergeometric form, a generalization of (1)-(3). This will be done elsewhere.

Evolution of three-component Borromean rings
This time we consider the evolution in parameter k = 2m defined in picture 5. The k = 0 member of the family is the Borromean link, while L8n5 also considered in [1], arises at k = −4.
The evolution formula this time is and it should be analyzed just in the same way as (111) was in the previous section.
Here we just mention the first non-trivial result: sufficient to define H k 1,s,t for the entire family,when one of the three representations is fundamental, while the other two are arbitrary. Note that for k = 0 there is a symmetry only between two of the three representations (between r and s), thus there is no symmetry between s and t in (120).