Are Khovanov-Rozansky polynomials consistent with evolution in the space of knots?

R-coloured knot polynomials for m-strand torus knots Torus[m,n] are described by the Rosso-Jones formula, which is an example of evolution in n with Lyapunov exponents, labelled by Young diagrams from R⊗m. This means that they satisfy a finite-difference equation (recursion) of finite degree. For the gauge group SL(N ) only diagrams with no more than N lines can contribute and the recursion degree is reduced. We claim that these properties (evolution/recursion and reduction) persist for Khovanov-Rozansky (KR) polynomials, obtained by additional factorization modulo 1 + t, which is not yet adequately described in quantum field theory. Also preserved is some weakened version of differential expansion, which is responsible at least for a simple relation between reduced and unreduced Khovanov polynomials. However, in the KR case evolution is incompatible with the mirror symmetry under the change n −→ −n, what can signal about an ambiguity in the KR factorization even for torus knots.

The first step in the story is of course the choice of parametrization of the space itself. Knot diagrams are often considered as glued from braids, and each m-strand braid has The first task in the theory of knot polynomials is to describe their dependence on any of the integer-valued parameters n. This approach is called evolution method [56][57][58][59][60], because as a function of n the polynomial, obtained by the modified RT method [26][27][28][29], is a sum of Lyapunov-like exponents: over the Young diagrams Q in the product R ⊗m with n-independent coefficients C. In other words, the polynomial satisfies a finite difference equation (recursion) in n, which can be integrated to a system Here we use the finite difference operator (1.5) and do not divide by 1 − x to simplify most formulas below. Also, we define the shift factor in ∇ (m) x as x rather than x m , because this simplifies formulas for superpolynomials. If some p of the expansion coefficients C are vanishing, the degree of the equation is actually reduced by p. In particular, this happens for N < m|R|, because all the diagrams Q with more than N lines do not contribute. There are also some "accidental" omissions for HOMFLY polynomials (like C [1], [22] in the case of torus H [4,n] ), but they are usually lifted after t-deformation to superpolynomials.
While evolution recursion is a direct consequence of RT approach, it is not yet so evident for KR method (see the first paper in [37][38][39] for a tedious derivation in the simplest example). Thus it is a meaningful question, if KR polynomials satisfy a recursion (1.4), and if its degree is further reduced at small N ?
In the present paper we analyse the available list of KR polynomials for the simplest torus knots with m = 2, 3, 4, 5 in the fundamental representation R = [1] = and observe that the answer is "yes": not only recursion is true, but its degree drops down at N < m, exactly as it does for the corresponding superpolynomials -where this followed from the super-Rosso-Jones formulas of [56].
If one assumes the recursion, then it provides an answer for infinitely many KR polynomials with all positive integer n once the first few are known -what is a major achievement by itself, given the complexity of direct calculations in KR formalism.

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However, as already observed in [61], for some knot families the evolution formulas for superpolynomials provide a unified description only for positive values of the evolution parameter n, while fail to provide pure positive or pure negative polynomials for negative n. In this paper we find that the same happens to KR polynomials even for the torus family with more than two strands. In result, these functions of n do not respect the mirror symmetries n −→ −n, at least in the usual way: while for torus superpolynomials from [56], for their KR counterparts, i.e. the left hand sides are understood as analytic continuations from positive n to negative, provided by the evolution formulas. Despite mirror symmetry is not a true symmetry of knot theory, associated link diagrams are not Reidemeisterequivalent, this is still an unpleasant feature, signalling about some kind of non-analyticity of the KR reductions in the space of knots (or, perhaps, about non-adequate coordinatisation of this space by parameters like n). Of course, the problem (1.7) disappears in cases when KR reduction is trivial -exhausted by just a substitution a = q N in the superpolynomial, like it happens for reduced 2-strand (m = 2) torus polynomials at N = 2. For the same reason mirror symmetry survives in the 3-strand (m = 3) case for reduced torus polynomials at N = 2, 3 and for the unreduced ones at N = 2.
As to another symmetry, acting on the family of knot polynomials, [m, n] ↔ [n, m], which is topological and relates Reidemeister-equivalent link diagrams, it is respected by our evolution formulas: both are true, are quantum dimensions of representations Q, given by the standard hook formula, the eigenvalues of cut-and-join (actually, Casimir) operator are κ Q = (i,j)∈Q (i − j) and the integer-valued coefficients C Q are defined by the Adams rule Since the only dependence on n is in the power of q, this HOMFLY satisfies a homogeneous finite-difference equation where and the degree of the difference equation is naively the number of Young diagrams Q ∈ R 2m . This fact is independent on the actual value of the coefficients C Q , i.e. from the point of view of the equation, they are the free parameters, parametrising its solution.
In this sense the equation does not provide too much information about the HOMFLY polynomials, still it reflects an important property -evolution rule -of the torus family with particular number m of strands. However, when some additional information is available, the degree of the equation can be decreased. For example, if the spectrum of Casimir eigenvalues κ Q is degenerate, i.e. if two or more of them coincide, the corresponding Q should appear only once in the product in (2.4). Such degeneracy is often preserved by the t-deformation.
If instead some of the coefficients C are vanishing, the corresponding ∇ can be omitted and the degree of the difference equation gets lower. The lift from HOMFLY to superpolynomials, however, usually lifts such degenerations: even if C was vanishing for HOMFLY, its t-deformation does not, see examples below.
Another story is when vanishing are some dimensions D Q . For A = q N contributing are only Q with no more than N columns, because otherwise D Q (q, q N ) = 0. The order of the difference equation is reduced accordingly. In particular, for N = 2 R and Q are symmetric representations (two-column representations of sl 2 are equivalent to the singlecolumn ones). The main claim of the present paper will be that this reduction is preserved by the t-deformation -even after additional reduction from super-to KR polynomial. Still, in this section we continue with HOMFLY.
If one adjusts the framing "phase" F R so that it cancels the contribution from one of the κ Q , F R = q 2 m κ Q 0 , then (2.4) can be promoted (integrated) to a more restrictive non-homogeneous equation with degree lower by one (i.e. one of the free parameters in its solution can be fixed): These framings are usually different from topological and The second equality in (2.7) follows from transformation law for HOMFLY polynomial from R = to transposed R tr = = R and thus is an accidental feature of the fundamental representation. Relation (2.7) persists in the case of superpolynomials, but generically it does not survive in the case of their KR reductions. At m = 2 the fundamental HOMFLY in this framing is and satisfies∇ Here and below we use the standard notation: satisfies the second-order difference equation we get the first degeneracy: C [2,2] = 0. Thus the equation is of the order three, This time the degree of equation is further decreased at N = 3 and N = 2: satisfies the second-order while Jones polynomial satisfies the first-order∇ With increasing strand number m we get the following pattern:

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Arrows mark the changes which take place at the level of super-and Khovanov polynomials.
Reversing the logic, evolution recursion describes generic solutions of the homogeneous difference equation: from∇ it follows that with p arbitrary integration constants C i . Each of these parameters appears in nonhomogeneous equation of degree p − 1, obtained by integrating (2.24) w.r.t. one of the commuting difference operators∆ x k : where the r.h.s. is independent of n.
Alternatively parameters C k can be defined from "initial conditions", i.e. the values of H n at some p different values of the evolution parameter ("time") n. Used for this purpose below are the directly calculated KR polynomials for relatively small knots. Things would be greatly simplified if also their twins at negative values of n could be used. However, the transformation n ↔ −n acts badly on the evolution formulas for KR polynomials, and this way to extend the set of available initial conditions does not seem to work.

Recursion for torus superpolynomials
Superpolynomials depending on one extra deformation parameter t = −1 could be defined from an analogue of the rule (2.3) with MacDonald instead of the Schur polynomials -and already this would make them depending on q and t, -but actually there are additional correcting c-factors, discovered in [56] and better described in two different ways in [67] and [68][69][70].
Explicit expressions for the simplest reduced superpolynomials in topological framing are [56]: where n = ±1 mod m. Note that this parameter takes the opposite values for the knots [m, −n] and [m, n] in (1.6). Here we used our usual notation {x} = x − x −1 and the standard MacDonald variables A, q, t, used in [56], are related to DGR variables of [35] by A 2 = −a 2 t, q = −qt, t = q. Torus superpolynomials can be converted from Laurent to true superpolynomials by extracting a power of appears here instead of the writhe number n(m − 1) in the exponents of the framing factor in (3.1). This explains the appearance of normalization factors in the following sections of this paper.
Superpolynomials are supposed to be positive Laurent polynomials of the DGR variables a, q, t, i.e., all the coefficients should be non-negative integers. However, this is not fully true for above expressions: with the change of n they switch from pure positive to pure negative polynomials. Moreover, this is a typical anomaly: by insertion of additional overall factor of (−) n−1 m one could cure the problem for all positive values of n, but then all the polynomials with negative n will be pure negative -violating also the invariance (1.6). Sometime this anomaly can cause serious problems for construction of superpolynomials [61], but for the torus knots per se this is a rather innocent detail, which, however, should be remembered and taken into account. It leads to a minor modification in the case of reduced super-and KR polynomials, while for the unreduced ones it is also minor, but can look somewhat unfamiliar, see below.
Expressions (3.1) for the superpolynomials have the form 2) The last line in this table is to remind that for N ≤ m only the eigenvalues from the first N columns do contribute to the answer, while the coefficients V of the remaining ones vanish JHEP04(2018)066 in these cases. Such polynomials satisfy the recursion relations Since for N ≤ m the polynomials P n,m in fact depend only on the first few eigenvalues, they actually satisfy the shortened recursions, which are generated by the difference operators, which survive in the last line in the if we multiply each P on the normalisation factor that makes the eigenvalue for Q 0 = [m] to be equal 1. Recursions for m = 3 and m = 4 here are the short ones, true for A = t 2 , when {A/t 2 } = 0, i.e., they are valid for (super)Jones polynomials at N = 2: for generic N the degrees of the equations would be 3 and 5 (or 4 for HOMFLY in the case of m = 4, when representation [22] does not contribute). Note that for m = 4 one does not expect difference equations with∇ (m/2) , because the coefficients in the superpolynomials depend on the residue n mod m. Lyapunov exponents for the torus n-evolution are made from MacDonald split of the Casimir eigenvalues: for the Young diagram The choice of the coefficient in front of the "differential" is, however, not specified. In other words, and additional principles are needed to separate positive ξ and positive K, which also contains positive terms divisible by 1 + t -which could be, but are not moved to ξ. The search for these principles is a big challenge, but it is beyond the scope of the present text.

Recursion for torus Khovanov polynomials at m = 2
In this case Khovanov polynomials are known from direct calculation at all odd n [30,32,[35][36][37][38][39] and they actually coincide with the result of the substitution of N = 2 into The standard definition in the literature actually includes additional factor: We, however, omit it because this leads to significant simplifications in the logic, while it is always easy to restore the factor in the answers. Note that, like the superpolynomials (3.1) our K [m,n] can be pure negative rather than pure positive. Also the reduced N = 2 polynomials computed with the standard programs [33,71] contain one more extra factor q −1 , which we also omit.

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for reduced and for unreduced superpolynomials. As explained in [40], relation between the two is implied by the differential expansion of the reduced fundamental superpotential: and for N -reduction {A/t} −→ (1+q 2N−2 t). Naively unreduced polynomial is obtained by multiplication with the quantum number [N ]. However, one can use the usual identification rule [40] (1 + -and this is how (4.2) is obtained from (4.1).
Putting N = 2, we obtain from (4.1) for reduced case and from (4.2) for the unreduced one. Both of them satisfy non-homogeneous first-order equations: Note that these are exactly the same recursions as the one in the first line of (3.5): q 2 t 2 = q 4 t 2 , only now we provided explicit expressions for the right hand sides. One can eliminate these right hand sides by applying one more difference operator, then the equation is the same for reduced and unreduced polynomials: with Invariance w.r.t. inversion of n would state: It is easy to check that the first recursion for K r in (4.8) is indeed invariant under this transformation, i.e. (4.11) is indeed true -what is not a surprise, because K r is obtained by a change of variables from the topologically invariant reduced superpolynomial (4.1). However, this is not the case for the unreduced K ur -like the unreduced superpolynomial (4.2), it changes non-trivially under the n-inversion (unless t = −1), because of the underlined term in (4.8). The second branch differs from (4.7) by a slight modification of the first term: Instead of (4.11) the evolution formula for the unreduced Khovanov polynomial satisfies We can now inverse the logic: begin from the recursion relation (3.5) and derive the evolution formula for arbitrary K [2,n] . Short and long recursions for m = 2 mean that K [2,n] = α ′ + β ′ · λ 2n ⇐⇒ K [2,n] = α · λ n + β · λ 3n (4.14) with λ = (−q 2 t) 1/2 and some n-independent α and β. Note that at N = 2 the two Lyapunov exponents differ by a factor of 3, which is independent of parameters q and t.
To solve the recursion we need initial conditions and the knowledge of just two should be enough. It can seem that we always have these two: n = ±1 correspond to the unknot. The problem, however, is that we need (4.11) to use the both. If (4.11) is true -like it is for reduced KR polynomial, we immediately get: i.e., reproduce (4.6). Note that the input as initial conditions was only the unknot (!) -this illustrates the usual power of evolution method. However, for unreduced KR polynomial we rather need (4.13), not (4.11):

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what coincides with (4.7). For the dual branch we havẽ are fully positive (or fully negative) respectively for n > 0 and n < 0, while in the "foreign" domains they contain terms with different signs -still they remain related by (4.13).
There is also a direct way from (4.15) to (4.16) and (4.17): instead of just multiplying by [2] = q + q −1 one should apply the rule (4.4): Then we get from the first version of (4.15) i.e., exactly (4.16), while the second version turns into (4.17).

Recursion for Khovanov (N = 2) polynomials at m = 3
In this case recursion is almost as simple as at m = 2. For N = 2 the r.h.s. with

{A/t 2 }
A=t 2 = 0 in (3.5) can be omitted, but at the l.h.s. one should substitute a = q 2 , i.e., (A/t) 6 = −q 6 t 3 : this is a manifestation of the double-face reduction rule (3.8). Thus, though this does not yet follow from any first principle argument, one can expect that Khovanov polynomials at N = 2 satisfy the second order homogeneous equation −q 12 t 7 K [3,n] (N = 2) = 0 (5.1) while Khovanov-Rozansky at N = 3 should satisfy the full-fledged third order one, the same as the full superpolynomials with a = q 3 and (A/t) 6 = −q 12 t 3 : We return to KR at N = 3 in section 9 below, and concentrate in this section on Khovanov polynomials with N = 2.
Making use of explicit expressions [32,71] we can check that this short-evolution hypothesis is indeed true and restore entire evolution formulas:

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With the help of (4. 15 However, there is a non-trivial factor in the mirror-symmetry relation: and, accordingly, K Similarly, for unreduced Khovanov polynomials we get: Again, these formulas can be obtained directly from (5.3) by application of the rule (4.18), e.g., Unreduced polynomials satisfy topological identity and mirror relation    This means that they should be linear combinations of three Lyapunov exponentialsand indeed they are.

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For n = 1 mod 4 Underlined structures enter with the coefficient (1+t), because it does not contribute in the case of HOMFLY polynomial (the coefficient in front of D [22] is "accidentally" vanishing when t = q). Despite it inters with negative sign, it does not spoil the positivity of the entire expression at n > 0. However, at n < 0 the polynomial fail to be positive or negative and consists of monomials with different signs. Thus there are no chances for any relation like (1.6). For n = −1 mod 4 Despite this is not evident from the formulas, for n > 0 (but not for n < 0) these polynomials are pure positive or pure negative, depending on parity of n+1 2 . As already mentioned, this time there is no relation like (5.5): neither in unreduced nor in reduced case, moreover, the l.h.s. is not even a positive Laurent polynomial. The valid topological identity is K [4,3] = K [3,4] (6.7)

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Also, it is unclear how the rule (4.4) can be applied to derive unreduced polynomials from the reduced ones. This drawback is cured by an improved (better structured) version of above formulas -which one day should be directly deduced from an adequately structured expression for the superpolynomials.
For n = 1 mod 4: Together the two items in the second line form a positive polynomial, but this positivity is still not explicit. But now the unreduced polynomial is obtained by direct application of (4.18): The analogues of these formulas for n = −1 mod 4 are a little more complicated: Residue 1.
Formulas are written in the form, where positivity of the series is explicitly seen.

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However, this is not a positive series. The true unreduced expression arises by application of the rules [3](1 + q 2 t) −→ 1 q 2 (1 + q 6 t), to decomposition in the second line: (1−q 6 t 4 )(1−q 8 t 6 ) The same rules convert above expressions for the full reduced polynomials K [4,n] into unreduced ones -this is illustrated by expansions of particular coefficients, not captured by projective limit. In the product of two (or more) differentials reduction acts on the lowest one, e.g., (1+q 2 t)(1+q 6 t) −→ 1 q 2 (1+q 6 t) 2 -otherwise we would get [3](1+q 2 t)(1+q 6 t) which contains three time more items.

Remnant of differential expansion
Differential expansions (DE) of [57] and [58][59][60] play an increasingly important role in modern theory of knot polynomials. Essential for KR reductions will be a complementary nested DE [72]. Somewhat amusingly, the ordinary DE partly survives the Khovanov reduction at N = 2 as well. Namely, while

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mirror anomaly is concentrated in the framing factor and thus is indeed comprehensible, however it becomes far more complicated for m ≥ 4. Technically it is related to the presence of negative items in the coefficients in front of the fractions 1−λ n 1−λ . Actually these negative items are absorbed into structures, proportional to (1−λ), and thus do not spoil positivity of entire polynomial -but this works only for n > 0, while at n < 0 such combinations fail to be sign-definite and can not pretend to be super-and KR polynomials. This is an unpleasant property of the evolution formulas, already encountered in a similar situation in [61].
(d) The knowledge of the evolution recursion in n allows one to easily calculate Khovanov polynomials with N = 2 for arbitrary torus knots, beginning actually from the unknot(!): one does it first for arbitrary n at m = 2 and then proceeds recursively in m, using the topological identity between [m, n] and [n, m]. Important here is the claim (b): dramatic reduction of recursion degree, which at N = 2 grows linearly with m, what makes above identity sufficient for the m-recursion (e) Though differential expansion [58][59][60], implying that P − 1 ∼ {Aq}{A/t} (11.3) for fundamental HOMFLY and superpolynomials in topological framing, is violated for KR polynomials, it is substituted by a weaker, still powerful structure: a linear decomposition of properly normalized K into differentials: As a simplest application, reduction rule (4.4) applies nicely to Khovanov polynomials and it allows to directly obtain unreduced polynomials from reduced ones. In result unreduced Khovanov polynomial always satisfies ur K (2) − [2] ∼ (1 + q 4 t) (11.5) However, this kind of rule are not sufficient to further deduce reduced Khovanov per se from superpolynomials and more effort is needed for this purpose. More relevant can be the nested structure, which helps to fix the ambiguities in linear decomposition, -but this story is beyond the scope of this letter.
(f ) A more traditional version of differential expansion is found to survive for Khovanov polynomials (N = 2) in a somewhat surprising form of (10.2), with additional powers of q, specific for torus knots. It would be interesting to see what happens to these relations beyond the present scope -of torus knot and fundamental representations.