Factorization of differential expansion for antiparallel double-braid knots

Continuing the quest for exclusive Racah matrices, which are needed for evaluation of colored arborescent-knot polynomials in Chern-Simons theory, we suggest to extract them from a new kind of a double-evolution — that of the antiparallel double-braids, which is a simple two-parametric family of two-bridge knots, generalizing the one-parametric family of twist knots. In the case of rectangular representations R = [rs] we found an evidence that the corresponding differential expansion miraculously factorizes and can be obtained from that for the twist knots. This reduces the problem of rectangular exclusive Racah to constructing the answers for just a few twist knots. We develop a recent conjecture on the structure of differential expansion for the simplest members of this family (the trefoil and the figure-eight knot) and provide the exhaustive answer for the first unknown case of R = [33]. The answer includes HOMFLY of arbitrary twist and double-braid knots and Racah matrices S¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{S} $$\end{document} and S — what allows to calculate [33]-colored polynomials for arbitrary arborescent (double-fat) knots. For generic rectangular representations fully described are only the contributions of the single-floor pyramids. One step still remains to be done.


Introduction
Construction of knot polynomials [1][2][3][4][5][6][7][8] is currently at the front-line of modern theoretical physics, because this is an exactly solvable problem in quantum field theory, which remains unsolved for years, despite tremendous effort by many distinguished researchers. Far ago it was reformulated in terms of Chern-Simons theory [9]- [17], where (at least for the knots in S 3 ) it is basically reduced to a free-field calculation, and a formal answer is provided [18]- [32] in terms of representation theory of quantum groups. This, however, does not help to obtain explicit answers, except for the simplest situations, covered by the well-known databases [33][34][35]. The problem is that group-theory methods themselves are undeveloped, moreover, the crucial quantities, starting from Racah matrices, are ambiguously defined in most interesting cases and thus do not attract attention of pure mathematicians. The ambiguities drop out of the final answers for knot polynomials, but intermediate steps involve less-invariant objects -what can in fact be interpreted as a kind of a new gauge invariance of some effective field theory [36,37], arising on the way from the fundamental

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Chern-Simons to Wilson-loop observables. This somewhat unexpected twist of the story makes it even more similar to "real" Yang-Mills theories, like QCD, and confirms expectations that the study of exactly-solvable models can shed light on the more complicated physical problems.
Recent revival of interest and new fast progress in 2d conformal theories (CFT) [38]- [57] attracted new attention to 3d Chern-Simons, which is its closest relative -and naturally caused a progress there, which is, however, not so spectacular yet. The difference is that knot polynomials are exact non-perturbative quantities, associated with modular transformations of non-perturbative conformal blocks, which are still far from being wellstudied in conformal theory as well. Fortunately, modular transformations are simpler than conformal blocks themselves, and one can proceed with knot polynomials even when conformal-block issues remain unsolved. Recent achievements in this direction are largely based on the new version [58]- [77] of the Reshetikhin-Turaev (RT) formalism, when quantum R-matrices in the space of representations, rather than in representation spaces, and Clebsh-Gordan coefficients are substituted by Racah and matrices (and their more complicated convolutions, known as mixing matrices). An early example of the strength of such approach was provided by the celebrated Rosso-Jones formula [78]- [93], which fully solves the problem of colored knot polynomials for a distinguished case of torus knots. However, despite a considerable progress, made in above references, nothing comparably impressive is yet achieved beyond torus knots -the problem turns to be extremely complicated. A new hope appeared with the introduction of the special class of double-fat knots in [94], where knot polynomials are presumably made from monodromy matrices of 4-point conformal blocks -and thus the CFT methods can be directly applied. This class is rather rich, in includes all the two-bridge and pretzel knots, moreover, it appeared to coincide with the arborescent knots, well known in mathematical literature [95,96]. In general such knots are made by contractions of "fingers" and "propagators", with just four lines/strands inside, and their knot polynomials can be considered as correlators in some new effective field theory [36,37], which is in fact a gauge theory, as we already mentioned. What is needed for arborescent knots is just a pair of Racah matrices, called exclusive, to distinguish them from exclusive ones, where the "final state" is arbitrary representation Q ∈ R⊗R⊗R, not just R. The problem, however, is that they are needed in arbitrary representation R, if one wants to calculate R-colored knot polynomials. Modern group theory is incapable to provide the answers beyond pure symmetric and antisymmetric representations [97][98][99][100] [103] and [104] for some inclusive Racah matrices for R = [3,1] and R = [2, 2] respectively). Further progress on these lines seems to be beyond the current computer capacities. However, the answers, when known, are pretty simple, and it is clear that they can be found/guessed by some other, indirect methods. One of the obvious ideas is to extract the

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Generalization to non-rectangular diagrams faces additional (perhaps, related) problems: the structure of DE is more sophisticated, even for trefoil and the figure eight, and non-trivial multiplicities arise, turning Racah matrices in more sophisticated operators, which do not possess any canonical matrix form. Moreover, additional care is needed [36,37] in this case to formulate the arborescent calculus of [94] for multi-finger knots -the interaction vertices in the corresponding effective "field theory" are also not canonically defined (or "non-local"). Thus non-rectangular case will be further elaborated on elsewhere.
Despite there is hardly any need to advertise the long-standing problem of calculating at least some non-symmetrically-colored HOMFLY for at least some knots -any result here is a breakthrough -we begin in section 2 from explaining that the particular twoparametric family of antiparallel double-braids is especially important at the present stage of knowledge: if found, it provides exclusive Racah matrices and thus allows to calculate for arbitrary arborescent knots, what is already quite something. The idea is further illustrated by the example of the fundamental representation in section 3. We switch to DE only in section 4, where the coefficients of the expansion, structured in [105], are found for the twist knots, by applying the evolution method [107] to the known [22]-colored-HOMFLY from [104]. The culmination comes in section 5, where with available examples we discover and illustrate the factorization property of DE for antiparallel double braidsand reproduce from the approach of section 2 the exclusive Racah matricesS 22 , recently found from a very different double evolution in [102]. In section 6 we make a formal conjecture, that this property holds for arbitrary rectangular representations. Given all the recent achievements after [94] this basically reduces rectangular arborescent calculus to finding the DE coefficients in [105] for the rather simple one-parametric family of twist knots. For them a separate conjecture is needed, of which we cook up just a prototype in section 7: we make it in full generality for the single-floor pyramids. At two floors we consider only the first unknown example of R = [33]. Further generalizations look rather straightforward, but are left for the future work.
Throughout the text we use the standard notation {x} = x − x −1 and [n] = {q n } {q} .

Exclusive Racah matrices from double evolutions
In [102] exclusive Racah matrices S were extracted from the double evolution family of 3-strand knots: they diagonalize the double evolution matrix. ThenS is obtained from with diagonal matrices T andT , made from the eigenvalues of the relevant R-matrices. Instead, the double evolution family of double-braid two-bridge knots definesS directly: in the notation of [94] H and S is then extracted from (2.1) as a diagonalizing matrix ofTST .
Thus a knowledge of rectangular HOMFLY for the double-braid family can be used to obtain these Racah matrices in rectangular representations. In particular, this provides an alternative derivation for R = [22] -and coincidence with the result of [102] can serve as a check of our factorization hypotheses about the double-braid.

Fundamental representation
For double braids of above type (two antiparallel braids with even crossing numbers the normalized fundamental HOMFLY are equal to H {m,n} with A-independent factor F (m,n) [1] . Thus Alexander polynomial Al (m,n) [1] = 1 + mn{q} 2 has degree one, and defect [110] of the differential expansion is zero for the entire family. This means that we can use the conjecture of [105] for the shape of differential expansions for rectangular representations of the defect-zero knots. What needs to be found are the A, qdependent coefficients (F -factors), which should be guessed from the limited knowledge of  U means the unknot (all F -factors are zero), the lines/columns with 2m or 2n = ±2 contain twist knots. The table has obvious symmetries m ↔ n and (m, n) ↔ (−m, −n) (in the latter case the knot turns into its mirror, and the change A, q −→ A −1 , q −1 should be made in knot polynomial), thus there are not too many different knots in the table. Fortunately, some colored HOMFLY are also available beyond 10 crossings due to powerful family method of [36,37].
For the fundamental representation R = [1] with d [1] = [N ] we get from (3.1) i.e.S 4 Twist knots for R = [22] According to [105], for defect-zero knots K (0) , The two F -factors in the first line are known from differential expansions for symmetric representations, the two F-factors in the second line are some q-deformations of the cube and the forth power of F [1] (A) -both symmetric under the change q ↔ q −1 (because of the transposition symmetry of R = [22]), but very different from, say, F [3] (A, q = 1)

(4.3)
From this we deduce the evolution formulas for twist knots: while for symmetric representations we have [107] (note that eigenvalues are now different, Of course, these formulas can be checked for other low-intersection twist knots 6 1 , 8 1 , 10 1 , respectively for m = −2, −3, −4.

Factorization of differential expansion for double-braids
Much more interesting are the non-twist knots from the double-braid family. Already in the first two examples we obtain: Clearly, the coefficients F factorize into products of the twist-family F-factors! Assuming that this amusing factorization is always true for all double braids, we conjecture: what can be tested in the other two available examples in table (3.2) -7 4 and 9 5 . This conjecture provides a double-evolution matrix, which through (2.2) gives the entries of dX dȲSXȲ . This actually proves the conjecture, because the result reproduces the matrix S, derived from a very different double evolution in [102]: The matrix has two symmetries:S i,j =S j,i andS i,j = ±S 7−j,7−i , the signs are shown explicitly in above table, where "12" at the place ofS 21 means that this element is equal toS 12 etc. Presented is the matrix, obtained by our new procedure: it differs a little (in signs and order of lines -as allowed by conjugation freedom) from the one in [102].
As already mentioned, another inclusive Racah matrix S is obtained from (2.1), by diagonalization ofTST with The eigenvalues are equal to and the entries of S XY are then given by Cramer rule as minors of the matrixTST − T , like in a similar procedure, described in [102]. Factorization property holds for symmetric and antisymmetric representations as well: and H (m,n) However, factorization is not extended to polynomials of the form with k = 2 -it fails already for the fundamental representations of the simplest knots of thus type, like 5 1 = (1, 1, 1, 1) and 7 3 = (1, 1, 1, 2). Instead we claim that it is extended to other rectangular representations -but only for double antiparallel braids. This is, however, exactly what we need to get the matricesS.

Other rectangular representations R = [r s ]
Now the plan is clear.
• We take general differential expansion of rectangular HOMFLY for defect-zero knots from [105]. It has the form where sum goes over peculiar multi-floor pyramids, labeled by pairs of the Young-like diagrams A, B, All the coefficients F 4 1 A,B = 1 for the figure-eight knot and F 3 1 A,B are just monomials (powers of q and A) for the trefoil.
• Guess a general formula for the coefficients F • Assume that antiparallel double braids satisfy factorization A,B = 1.
• Representing double-braid case (k = 2) in the form of a double evolution in m 1 and m 2 , use (2.2) to extract the Racah matrixS [r s ] .
• DiagonalizingTST with the help of known eigenvalues and Cramer rule, as explained in [102], obtain another Racah matrix S [r s ] .
• Calculate H [r s ] for any desired arborescent knot by the technique of [36,37,94] (some additional care/checks can be needed with effective vertices in the multi-finger case) and make possible checks. The most artful part for today is the second step: revealing the structure of F for twist knots from very special examples -symmetric and antisymmetric representations R = [r] and R = [1 r ] and the only double-floor example, provided by R = [22].

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More formally, complementing the conjecture of [105] by above suggestions, we look for the [r s ]-colored HOMFLY in the following form: and our new goal is to associate the evolution functions F

(m)
A,B with every non-empty pyramid These functions depend on the eigenvalues λ c , which are powers of A and q. However, for concrete pyramid {A, B} the function F

(m)
A,B depends on only some of these eigenvaluesand this subset depends on the pyramid and not on r and s (these parameters, however, select particular pyramides that contribute to H [r s ] ). Schematically, The first two tasks are to describe the sets I A,B and J A,B , and the third is the finding of the combinatorial coefficients ξ c . In this paper they are fulfilled only partly -for the single-floor diagrams {A, B}.
7 On the structure of F (m) A,B

Already known cases
To begin with let us remind this structure in the case of symmetric representations [r], where the relevant diagrams are just single-floor boxes a . . . 0 with b = 0. Then, from [107] we know the answer: In fact, it is already a challenge to unify (7.1) and (7.2) into a single formula. The knowledge of the R = [22] case adds to two more pyramids: Now there are ten irreps in [33]⊗ [33] and ten eigenvalues: six "old" ones, the same as (7.5), and four "new": (somewhat strange labels in λ + 0 c emphasize that the powers of q are smaller than c(c − 1) in λ +c , familiar from symmetric representation case (7.1), see section 7.4 below for a more systematic description). The "old" pyramids are described by the same formulas (7.3) and (7.4), involving only "old" eigenvalues (7.5). Of the four "new" pyramids one is still described by (7.1):   • requirement that each term in the sum over λ c depends on even powers q, • requirement, that entire F

(m)
A,B is a polynomial for any m, i.e. all poles cancel after summation over c, • vanishing F A,B = f (−q a f −b f A 2 ) a f +b f +1 for m = 1, i.e. for the trefoil, see [105]. For the particular case of R = [33] these are rather restrictive requirements and the above guess is actually less ambiguous, than it can seem. In any case, the real confirmation comes a posteriori -from the final expressions for Racah matrices and reasonable answers they provide for the [33]-colored knot polynomials.
With these guesses we can calculate H (m,n) [33] ?
for double braids and extract the 10 × 10 matrixS [33] , which is described in the appendix to this paper. Using this matrix we can calculate H [3,3] for some arborescent knots, which can be made without the use of the second exclusive matrix S [33] . In examples these polynomials are consistent with available Vassiliev invariants and pass other checks from the list in [103]. Building of S fromS with the help of (2.1) and thus extension to arbitrary arborescent knots by the method of [36,37,94] is also straightforward.
For further generalizations to other rectangular representations we need to look at the above requirements for F a little closer. Since it is still a guesswork, it is not really formalized -thus we provide just some sketchy comments, followed by new conjectures.
They are the first of the necessary ones for polynomiality of F at all m -for that purpose above combinations should be proportional to the product of D-factors in denominators of F, but these products have a higher degree in A, thus the proportionality coefficient is just zero. Identities (7.12) are linear combinations of . . . (7.13) which are the q-deformations of identities where µ (i+n+1) = µ (i) . These follow from the vanishing of det (n+2)×(n+2) P i (x i ′ ) for any n + 2 polynomials of degree n -because this determinant has degree n in each variable , to which it is obviously proportional, has bigger degree n+1. For our purposes we need peculiar collections of shifts a (i) j , when determinants in (7.14) are actually binomial coefficients.
Note, however, that nullification of F for the unknot is not by itself enough restrictive. For example, as indicated in (7.12), it would allow to change the two underlined terms in (7.8) for a single − [4] λ 2m 03 {Aq 4 }{Aq 3 }{A}{A/q 2 } -this, however, would give a wrong (and in fact, non-polynomial) answer already for the trefoil 3 1 .

Eigenvalues
In general the eigenvalues are parameterized as

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with 0 ≤ i f ≤ a f and 0 ≤ j f ≤ b f and an additional embedding constraint Also some upper floors f can be left empty, i.e. allowed is also i f +j f +1 = 0. In particular, λ 0 = 1 is associated with all empty floors, thus it is more natural to call it λ ∅ . In this sense all eigenvalues are associated with the ordered-by-(7.16) collection of boxes in the pyramids -two per each floor f , except for degenerate cases when i f + j f = 0 or i f + j f + 1 = 0, and the corresponding boxes coincide or are just absent. This implies an obvious change of notation for the eigenvalue labeling. In these pyramid notation the eigenvalues (7.5) and (7.6) fit into the big tower:

Single-floor pyramids
Looking at a few explicit examples that we already possess, one can assume the following structure for denominators, associated with particular eigenvalues: Boxes contain products k {Aq k } over k with no gaps. They turn into unity when the lower limit exceeds the upper one, like i + 1 > 2i for i = 0 or 2i + 2 > a + i + 1 for i = a. Generalization to arbitrary λ i,j , appearing in the single-floor pyramids, is now obvious. More complicated is adjusting the combinatorial coefficients, which guarantee cancelation of poles and matching with the unknot, figure eight and trefoil. By trial and error we get the conjectural answer for arbitrary single-floor pyramid: which fits all the expectations. The item in the first line contains λ 2m ∅ = 1 and can be considered as associated with the zeroth floor. This item is directly read from the pyramid itself, the eigenvalue leaves it intact.

Other floors
A natural generalization of peculiar A-dependent factor in front of the power of the eigenvalue λ ...
i f j f ...
in the multi-floor analogue of (7.18) is 19) This gives the underlined "inter-floor interaction" factors like (7.20) Note, that they all preserve parity of powers in q. Moreover, in empty cases, when i f ′′ + j f ′′ + 1 = 0 the underlined interaction factor drops away, because of the cancelations between the numerator and denominator of (7.19). Unfortunately, this is not exactly what needed -even if a floor in the eigenvalue is empty, it still affects the shape of the contribution. This is already seen in the fully reliable (not just conjectured) formulas (7.4) Another way to suspect that the empty floor matters, is to compare the first two terms in . . (7.21) and the naive The coefficient c in (7.22) should be a quantization of 5, but it should contain odd powers of q, thus it can not be [5]. In (7.21) its analogue is a quantization of 4, which is made out of even powers -and it is [2] 2 . However, there is no such simple way out for c in (7.22) -and already the second term in this formula is a less-naive combination of inverse differentials. Formula (7.9) is an example. In general, when one proceeds from This demonstrates that (7.19) is not the whole story in the multi-floor situation, that additional guesses are needed -and in fact easy to make. Still, we prefer to postpone further speculations about the higher-floor contributions, waiting for independent examination of the already made sequence of conjectures.

Conclusion
To conclude, in this paper we made a new series of conjectures, which hopefully lead to explicit formulas for exclusive Racah matrices S andS and thus to construction of arborescent knot polynomials in arbitrary rectangular representations. This is a long standing problem, and its solution seems now within reach. After the structure of differential expansion for defect-zero knots (6.3) is revealed in this case in [105] and after the discovery of further factorization (6.2) of its coefficients for double braids in the present paper, it remains to conjecture these coefficients for just a relatively simple family of twist knots. To demonstrate that this can actually be possible, we provided explicit expressions for all the JHEP09(2016)135 contributions (7.3)-(7.10) in the case of the previously unknown representation R = [33] and made a general conjecture (7.18) for the contributions for single-floor pyramids. Its further extension to the second floor would provide the answers for arbitrary arborescent knots in representations R = [rr] and R = [2 r ], while generic rectangular R = [r s ] requires min(r, s) floors, which will hopefully come as direct generalization. Hopefully, this last step will be made soon enough.

Acknowledgments
This work was performed at the Institute for the Information Transmission Problems with the support from the Russian Science Foundation, Grant No.14-50-00150.
A Exclusive Racah matrixS [3,3] The Young diagram R = [33] is not symmetric and there is no additional symmetry in Racah matrix: it is just symmetricS ij =S ji . The transformationS ij −→ ±S 11−j,11−i converts it into something else.
Standing in the first line/column are the square roots of quantum dimensions of the ten irreducible representations, which constitute the product [33] ⊗ [33], of which only six were present in the case of [22] ⊗ [22]: Most other entries are not fully factorizable and non-factorized pieces can be expressed through D i = {Aq i }/{q} in different ways. The choices below are economic (with the exception ofS 55 ), but not canonical : JHEP09(2016)135 The complementary matrix is T [3,3] = diag 1, −A, q 2 A 2 , A 2 /q 2 , −A 3 , −q 6 A 3 , A 4 , q 4 A 4 , −q 4 A 5 , q 6 A 6 (A.1) Then the eigenvalues of the productT [3,3]S [3,3]T [3,3] in (2.1) are: T −1 [3,3] = A 6 · diag q 18 , −q 14 , q 12 , q 8 , −q 6 , q 2 , −1, q −2 , −q −6 , q −12 (A.2) and the diagonalizing matrix S [3,3] can be obtained by the Cramer rule, i.e. its entries are minors of the matrixTST − T −1 . Cramer rule says, that any matrix M ij with eigenvalues λ i is diagonalized by a matrix V jk = Minor jm (M − λ k · I) with any m: to make S orthogonal (but not symmetric). Minors are determinants and they are relatively easy to calculate (most important -to simplify and factorize, what makes this diagonalization method most practical in our situation). With the matrix S the first thing to calculate is H 3 1 [3,3] = d [3,3] · S [33] T −3 [33] S ∅∅ for the trefoil. In this way we reproduce the right answer, which was the starting point for the entire consideration in [105] -thus closing the circle of reasoning, at least in the first previously unknown case of R = [33].

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