HOMFLY polynomials in representation [3, 1] for 3-strand braids

This paper is a new step in the project of systematic description of colored knot polynomials started in [1]. In this paper, we managed to explicitly find the inclusive Racah matrix, i.e. the whole set of mixing matrices in channels R⊗3 −→ Q with all possible Q, for R = [3, 1]. The calculation is made possible by the use of a newly-developed efficient highest-weight method, still it remains tedious. The result allows one to evaluate and investigate [3, 1]-colored polynomials for arbitrary 3-strand knots, and this confirms many previous conjectures on various factorizations, universality, and differential expansions. We consider in some detail the next-to-twist-knots three-strand family (n, −1 | 1, −1) and deduce its colored HOMFLY. Also confirmed and clarified is the eigenvalue hypothesis for the Racah matrices, which promises to provide a shortcut to generic formulas for arbitrary representations.


JHEP09(2016)134 1 Introduction
Colored knot/link polynomials [2][3][4][5][6][7][8][9] are characteristics of knots and links, which factorize into products for composite knots. In this sense, the knot polynomials for the prime knots play the role of prime numbers, only in the world of knots, and the most challenging is the question if they are indeed prime or there is some additional "more elementary" structure, which allows one to reconstruct them from some simpler constituents. Another source of interest to the colored knot/link polynomials is that they are the Wilson loop averages in Chern-Simons theory [10,11], the simplest of all Yang-Mills theories, and they can be exactly calculated non-perturbatively, because they are polynomials in the variable q = exp(g −2 ). This provides a very non-trivial check for our emerging understanding of non-perturbative methods in quantum field and string theory. Both these reasons make evaluation of colored knot polynomials extremely important problem, since it is the first step towards understanding their properties. Until recently, it was an unachievable task, but development of theoretical methods in [12]- [103], combined with the current computer power makes it nearly realistic.
This explains our reasons to make a try, and it is already partly successful. At the previous stages, the calculus for totally symmetric and antisymmetric representations was successfully developed [64][65][66], then extended in [74,91,92,96,97,100] to representation [2,1], and, in the present paper, we report a new achievement: results for 3-strand polynomials in representation [3,1]. The next big challenge is the first two-hook representation [4,2], this requires a new serious theoretical advance, but now it seems within reach. The powerful method needed for that purpose will be described in detail in a separate publication, here we just mention it in general description of our approach.
It involves the following steps: 1. Define the representation R by a highest weight in the space [1] ⊗R .

2.
Find the highest weight of representation Y ∈ R ⊗ R.
3. Distinguish between Y belonging to the symmetric and antisymmetric squares of R. If a given representation Y appears in these both, we treat Y + and Y − as different representations: they can have there own multiplicities, but they are never summed. The reason for this separation of Y ± is that the knot polynomials depend on the eigenvalues of quantum R-matrices: these are different (in sign) for Y + and Y − .

4.
Find highest weights of representations Q ∈ Y ⊗ R and Q ∈ R ⊗ Y . These weights h l Q (Y ) and h r Q (Y ) (superscripts stand for "left" and "right") are related by the Racah (mixing) matrix U Q : In the case of non-trivial multiplicities, U is defined modulo rotations in the multiplicity spaces of Q, Y and Y ′ , but these rotations leave R-matrix intact, and therefore do not affect knot polynomials. These rotations, however, can be essential for the eigenvalue hypothesis, expressing the entries of U through those of R, then this freedom should be somehow fixed.
Throughout the paper we use the notation Let us also note that we use throughout the text the term "k-strand knot/link" which implies the knot/link whose braid representation with minimal number of strands is k-strand.

Comments on the highest weight calculus
One of the crucial decisions that makes calculations doable is to extract the mixing matrices from the highest weights. We already described the basis of this technique in [100], nowadays step 2 from the above list is provided by a fast working computer program, which finds highest weight of Y ∈ R 1 ⊗ R 2 , the same program is used at step 4. We now briefly describe the new aspects, which were not clear enough at the time of [100], other details can be found in that paper. We distinguish between elementary and advanced levels of the method: most calculations for [3,1] were performed at the former level, but further work on higher representations can be hard without going to the latter one.
• Elements of the Verma modules are generated from the highest weights by action of the Chevalley generatorsT −a and their ordered productsT −A = iT −a i . The sets of ordered sequences A are not Young diagrams (like they were in the case of Heisenberg or Virasoro Fock modules), they can seem to be exponentially growing with the level |A| = i a i , but this is not true if one takes into account the conditionŝ and the Serre relationŝ • Highest weights are the vectors in [1] ⊗M annihilated by all rising operatorsT a , they are parameterized by the Young diagrams of size M .
• The highest weight V R is a linear combination of sequences which contain definite amounts of zeroes, units, twos etc. These quantities are directly dictated by the Young diagram: for R = {r 1 ≥ r 2 ≥ . . .} the number of zeroes is r 1 , the number of units is r 2 and so on. For example, V [5,1,1,1] is a combination of sequences with #(0) = 5, #(1) = 1, #(2) = 1, #(3) = 1. We call collection of these entries the type t R of representation R, e.g. t [5,1,1,1] = {00000123}. The highest weight V R is a certain linear combination of these sequences with different orderings.
• The highest weight V Y of representation Y ∈ R 1 ⊗ R 2 can be obtained by acting with the lowering operators on the tensor product of the highest weights of R 1 and R 2 : 3) The question is how to choose A and B.
• The first criterion is simple: one can look at the difference of types of Y and R 1 and

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• To fix the coefficients, we apply the second criterion: the highest weight condition If there are no multiplicities, i.e. Y appears in R 1 ⊗ R 2 exactly once, the coefficients are defined unambiguously modulo a common normalization factor.
• Find the highest weights of representations Q ∈ Y ⊗ R and Q ∈ R ⊗ Y . If Q has multiplicities, then the highest weights can be chosen in an arbitrary way. This freedom affects the form of the Racah/mixing matrix, but not the answer for the knot polynomial.
• The mixing matrices are unitary if the highest weights are normalized. Technically simplest is the norm • The issue of multiplicities does not matter for the knot polynomials, because R-matrix is diagonal in representation Y , so the basis in the multi-dimensional space of highest vectors can be chosen arbitrarily. There is, however, one important exception: for R 1 = R 2 = R one should distinguish between Y ± belonging to q-symmetric and q-antisymmetric squares of R, because the corresponding eigenvalues of R have different signs and do not coincide. This is not a simple task, but there is a simple solution: if one rotates the unitary mixing matrix U Y into symmetric (simply symmetric, with no reference to q-symmetry), it automatically separates Y + from Y − .

Advanced level
The method of the previous subsection provides highest weight vectors V Y as linear combinations of integer-number sequences from [1] ⊗M , which often contain enormously many terms (not exponentially growing, but still too much). The mixing matrices define linear dependencies between such combinations and they are difficult to find even on powerful computers, despite it is just a linear algebra problem.
The following procedure helps to tame these linear combinations by noting that they are made from certain standard pieces so that one can combine substantially smaller number pieces rather than the original number sequences: • Parametrization by the number sequences can be converted into the one by polynomials of M auxiliary x-variables by the rule (a 1 , a 2 , . . . a M ) ←→ x a 1 1 x a 2 2 . . . x a M M .
• The highest weights of totally antisymmetric representations [1 r ] are totally qantisymmetric polynomials in x, i.e. certain quantum deformations of the Vandermonde determinants ∆ (r) = ∆ 1...r = r i<j (x i − x j ). The point now is that the highest weights of arbitrary representations R are expressed through the same determinants.

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• The Young diagram R = {r 1 ≥ r 2 ≥ . . .} has r i as the heights of its columns. The lengths of its lines are similar parameters of the transposed diagram R ′ = {r ′ 1 ≥ r ′ 2 ≥ . . .}. These can be associated with totally antisymmetric representations [1 r ′ i ], and the highest weight V R (x) in x representation is just a linear combination of i ∆ (r ′ i ) : The sum goes over different distributions of |R| variables x i between different Vandermonde determinants, and the coefficients C σ are determined by the highest weight condition (that the sum is annihilated by all raising operatorsT a ).
• The Vandermonde decomposition is highly ambiguous, because there are many linear relations between products of the Vandermonde determinants of a given type R ′ , but this ambiguity does not change the highest weight itself. Of course, in the case of non-trivial multiplicities, there are several linear independent solutions to the highest weight condition.
The symmetrizations in (3 ↔ 4) and (7 ↔ 8) are done independently, i.e. each explicitly written term is substituted by the four ones. As already mentioned, there are many different ways to represent the r.h.s. because there are many linear relations between ∆-products of a given type and the choice of basis is somewhat arbitrary. Also we do not provide here the exact definitions of q-deformed ∆'s, thus these formulas are mostly for demonstrative purposes. They, however, can be directly used for q = 1, i.e. for studying the classical Racah matrices, which is by itself quite a non-trivial problem for non-symmetric representations.

Representations in the cube [3, 1] ⊗3
The representation content of the cube is now  and the highest weights are too numerous to be listed here, see [105].
Each block of lines here is associated with the corresponding intermediate representation: [6,2], [6, 1, 1], . . . from (3.1), and most representations appear in several blocks in the table below. The number of times the representation Q enters (3.4) is given in the first column of the table, and it is the size of the mixing matrix U Q which we need to calculate. Clearly, there are quite a few (40) matrices of non-unit size, some are quite big. Only 26 of them (up to size 6) can be found from the eigenvalue hypothesis of [72,73] and its recent generalization in [101], all the rest had to be calculated by the methods of section2 (actually we did so also for the matrices of sizes 5 and 6 to double check the eigenvalue hypothesis). Among the knots and links having 3-strand braid representation there are many composites consisting of 2-strand torus components. The reduced HOMFLY polynomials for the composites are just products of those for the constituents and, for the sake of completeness, we complement (1.2) by its simple 2-strand counterpart: and ǫ Y = ±1 depending on whether Y belongs to the symmetric or antisymmetric square of R.

Abundance of 3-strand prime knots in Rolfsen table
As to the prime knots given by 3-strand braids, their place in the entire set is clear from the following

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Clearly, in this form they are not too informative. A list, suitable for further studies is provided in the form of a txt-file at site [105]. Much more informative are expressions for the evolution families, of which we provide a couple of simple examples in the next section, and spectacularly elegant properties of these complicated expressions in the later sections.
5 Knot polynomials from evolution 5.1 (m 1 , n 1 |m 2 , n 2 )-evolution Studying the knot polynomials for families of knots is the most natural and effective approach, see [1,56,83,86,100,102] for motivation and examples. The simplest of this kind is the evolution method of [56] and [83], for the three-strand braids (m 1 , n 1 |m 2 , n 2 | . . .) it studies the dependence on parameters m i and n i .
The simplest family (m, n) fully consists of composites of the 2-strand knots, this means that their reduced polynomials are products of those for the constituents, which in this case are the 2-strand torus knots/links with the HOMFLY in arbitrary representation given in section 4.1: Since knots/links (m 1 , n 1 |m 2 , 0) = (m 1 + m 2 , n 1 ), the same remains true for threeparametric families. The first non-trivial family is (m 1 , n 1 |m 2 , n 2 ). According to the general rules of the evolution method where N R is the number of different representations in the square R ⊗ R (if the same representation appears in the symmetric and antisymmetric squares, it contributes twice) and λ a are the corresponding R-matrix eigenvalues, λ Q = ǫ Q q κ Q −4κ R A −|R| (the dependence on R is fixed by the topological framing). Thus, within the evolution method, the colored HOMFLY polynomials for our family are described by the 4-rank tensors h R . As a simple illustration, in the fundamental representation N [
Since the family under consideration belongs to the pretzel knots, the coefficients h R,Y (A, q) in the general evolution formula,  [95,106], as well as for representation [2,1] (see [91,97]): where S is the Racah matrix that relates the cases of differently placed brackets in the map of the product: R ⊗ R ⊗R → R, whileS for R ⊗R ⊗ R → R, and T is the diagonal matrix with entries being eigenvalues Thus, we list below the coefficients h R,Y (A, q) for the (anti)symmetric representations and representation [2,1] and calculate for the [3,1] case from the 3-strand representation of this paper, since the proper Racah matrices are not known yet.
After summation over X or from the coefficients h a 1 a 2 a 3 a 4 R of (5.2) after weighted summation (averaging) over three indices a 2,3,4 , formula (5.6) can be rewritten in the form of corrections to the 2-strand formula (4.1), in terms of the A-independent coefficients C R,Y i (q): where µ R = r − s for the single hook representations R = [r, 1 s−1 ].

Symmetric representations R = [r]
For symmetric R = [r] the sum in (5.5) goes over the representations [2r − a, a], and one can use formulas from [95,106] to write manifestly whered k is the quantum dimension of the representations arising in the decomposition

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and where we used the symmetric q-Pochhammer symbol (A; q) n = n−1 From (5.8) one can also read off the coefficients C i (r, a) in representation of h(A, q) in form (5.7) In particular, one can realize the following properties of these coefficients: • They have a symmetry i −→ r − i, involving also the change q −→ q −1 and rescaling.

Representation R = [2, 1]
In this case, the evolution coefficients can be calculated both from formula (5.6) and by the evolution method from several known knot polynomials [83]. Let us use the second possibility here, since only this approach can be used in the [3,1] case. For [2,1] ⊗2 the eigenvalues of R-matrix are and therefore the m-evolution is defined as follows: h [4,2] q 5n + h [4,1,1] +h [3,3] Note that two pairs of representations have the same eigenvalues, thus their contributions are not separated. The two representations [3, 2, 1] ± contribute equivalently to knot polynomials: for odd n only the difference h [3,2,1] + − h [3,2,1] − is seen, but link polynomials with even n separate these two contributions. The other two sums are inseparable within the (n, −1 | 1, −1) series, we, however, suggest a plausible decomposition, which can be checked in the analysis of richer evolution patterns. In fact, after proposing these decompositions we derived them using (5.6) and manifest formulas for the Racah matrices from [97] (notice an additional sign factors that have to be added in this case in accordance with [97, s.7] and [102, s.2.7]). Unfortunately, at the moment, these Racah matrices are not known for [3,1] case yet and, hence, such a method is unavailable in that case. Finally, the values of the coefficients are: h [4,2] = d [4,2] d [2,1]
6.2 Factorization at roots of unity, q 8 = 1 According to [99], H [3,1] = H [4] at q 8 = 1 (6.3) This is true in all our examples and this provides a new serious support to the factorization conjectures of [99].

Symmetric Jones polynomials
It is well-known that the representation R = [3,1] reduces to R = [2] for the Lie algebra sl 2 . Therefore, if we put A = q 2 , the HOMFLY polynomial H [3,1] reduces to the Jones polynomial in the first symmetric representation J [2] : Since the symmetric Jones polynomials are known for many knots, in particular, for all knots from the Rolfsen table, one can use them to check our results.

Universality and adjoint representation at A = q 4
By the general rule (rank-level duality [56,58,63]), for the transposition of Young diagram (an additional inversion A → −A −1 provides a mirror knot instead of transposing the representation). Thus, together with H [3,1] we simultaneously know For the particular case of sl 4 algebra, [2,1,1] is the adjoint representation and the adjoint HOMFLY polynomial satisfies [101,107] the universality hypothesis [108], unifying them with the adjoint polynomials for other groups, including the much simpler adjoint Kauffman polynomials (they are simpler because the adjoint representation of so N is just [1,1] for all N , while it is an N -dependent [2, q N −2 ] for sl N ). This allows one to compare our H [3,1] at A = q 4 with the universal formulas from [101]. They match in all examples that we looked at. JHEP09(2016)134

Loop expansion and Vassiliev invariants
The HOMFLY polynomials have an interesting well-defined expansion when → 1, A = exp N 2 , q = exp 2 . This expansion is known as loop expansion, and the coefficients are the celebrated Vassiliev invariants [109][110][111]. In the Vassiliev approach, the HOMFLY polynomial can be written as [114] where r (R) i,j are the polynomials of degree |i| in N corresponding to the trivalent diagrams [112][113][114], and N i is the dimension of the vector space formed by the trivalent diagrams. Here v K i,j are finite type knot invariants introduced by V.Vassiliev. Thus, what stands in (6.7) is the double series in powers of and N , such that the degree of exceeds or equals to the degree of N . The polynomials r (R) i,j are known up to degree 6 in arbitrary representation R, whereas the Vassiliev invariants up to order 6 are known for all knots with number of crossings less than 15. Thus, it provides a lot of explicit checks. To be more concrete, we consider one example of the knot 10 161 from section 4.3.
As an illustration, we look at the first terms of the expansion (6.7) in the case of R = [3,1] and for the knot 10 161 mentioned in section 4.3. Taking the knot-independent trivalent diagrams from [89] and the lowest Vassiliev invariants for 10 161 from [115], we obtain the first terms of the -expansion (6.7): Comparing with the coefficients of the expansion of (6.8), we find a complete agreement. At the present moment, we know the trivalent diagrams explicitly only up to degree 6 (they are available at [89] up to degree 4). The corresponding Vassiliev invariants are fully determined by the HOMFLY polynomials in representations R = [1] and R = [2]. To extract new Vassiliev invariants from more interesting higher symmetric, [2,1]-and the newly-calculated [3,1]-colored HOMFLY, more complicated diagrams are needed.

Genus expansion and Hurwitz τ -function
Another interesting expansion of the HOMFLY polynomials is the large N , or genus expansion, which relates knot theory with the Hurwitz enumeration problem. In this expansion, one expands the knot polynomials in powers of (q = e ) similarly to the previous subsection, but now A, i.e. · N rather that N is fixed. It turns out that the Ooguri-Vafa partition functions of the colored HOMFLY polynomials [47] is equal to certain generating functions of the Hurwitz numbers [78][79][80]89], hence, we also call this expansion Hurwitz, and the corresponding generalization of the KP/Toda τ -functions (a general solution to the AMM/EO topological recursion [116][117][118][119]) is called Hurwitz τ -function.
The genus expansion for colored HOMFLY is also interesting, because it separates the representation-and knot-dependencies: (in the leading order, this reproduces (6.1), also known as exponential growth property [82]). The sum here goes over all Young diagrams ∆, as usual, |∆| and l(∆) denote respectively the number of boxes and of lines in the diagram, ϕ R (∆) are the symmetric group characters: they do not depend on the knot and are common for all Hurwitz τfunctions, [78][79][80]. Dependent on the knot are the general special polynomials in the free energy expansion where σ K ∆ (g) are knot-dependent polynomials in A, presumably related by the AMM/EO topological recursion for a knot-dependent spectral curve.
For us of importance is that, since ϕ R (∆)'s are non-diagonal, one and the same special polynomial σ K ∆ (n) affects the HOMFLY polynomial in different representations. Since the lowest polynomials were already extracted from study of the symmetric representations, we can now use them to test our new results for R = [3,1]. Indeed, since ϕ [3]  one can compute the first special polynomials σ K ∆ (g) with the help of only symmetric representations R = [1], [2], [3], [4] using [72,73].
Let us briefly consider our sample example of knot 10 161 . In this case, (1) = 2A 12  Now one can find the values of ϕ R (∆) in the table in [120,121], and, with the help of formula (6.11), compare the results with the corresponding genus expansion of H 10 161 [3,1] . They coincide, which provides yet another nontrivial check of our results for H [3,1] .
To understand if the newly-calculated [3,1]-colored HOMFLY provides new special polynomials σ ∆ , as one could naturally expect, a better understanding of non-linear relations between different ϕ R (∆) is needed.

Differential expansion
The naive genus (or loop) expansion in powers of does not respect the polynomial property of HOMFLY. This is cured in a far less trivial "differential expansion" [63,83,84,98], which also reflects the hidden "differential structure" [48] lying in the base of the Khovanov approach [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46] to knot polynomials, very different from the R-matrix one exploited in the present paper. In variance with naive genus expansion, the differential expansion contains only a finite number of terms up to the r + s power of Z's, where Z (k) The expansion in the variable h defined as q = e = (1 + h), A = e N = (1 + h) N can also be made finite for each natural N . What breaks the polynomiality in h is the analytical continuation from integer values of N . While reasonably understood for the symmetric representations, the differential expansion remains a complete mystery beyond them, simply because almost nothing has been known so far about the generic colored HOMFLY polynomial. Our new results for R = [3, 1] allow one to make a new small step as compared to [74,97,100], where only the information about R = [2, 1] could be used.
From [74] we know the Z-linear term of the differential expansion for the hook diagrams: Z r|2s−r−2j · G [1] (A, q) + O(Z 2 ) (6.13) with the knot-dependent polynomial G 1 (A, q), which is one and the same for all representations, including the fundamental one with r = s = 1. Therefore, for representation [3,1] JHEP09(2016)134 their consistency with representation theory. Less trivial (but more speculative) conjectures about the non-symmetric differential expansion will be presented elsewhere.
Since we devoted a piece of this paper to implications of the evolution method, it deserves mentioning that description of the knot polynomials in terms of the differential expansion parameters G R is badly consistent with the evolution considered in section 5.2. This is clear already from the fact that unity, the first term in the differential expansion is not an eigenvalue of the R-matrix in the R ⊗2 channel (it is in the channel R ⊗R, and this explains a nice consistency of the evolution and differential expansion for the peculiar family of twist knots [63,83,92]). In general the interplay between the two hidden structures, the evolution and differential expansion coefficients leads to a very interesting and important puzzle in knot theory.

Conclusion
In this paper we report the results of tedious calculation of the Racah (mixing) U Q matrices for [3,1] ⊗3 −→ Q and their application to study of the 3-strand knots and links. These results confirm previous conjectures and slightly extend our understanding of complicated issues like evolution method [56,83] and differential expansion [48,84]. This advance was made possible by application of the highest weight method, which we developed in [100]. The size of this paper does not reflect the actual complexity and amount of calculations, we just commented briefly on various problems encountered and solved on the way, and provided simple illustrations for the results at different steps of calculations. For the full set of newly-derived mixing matrices (additionally converted to the block form) see [105]: they are too big for a paper, but are nicely handled by the eigenvalue hypothesis [72,73]. Also at [105] there are the [3, 1]-colored HOMFLY polynomials for the 3-strand knots up to 10 crossings and the evolution coefficients for the entire infinite next-to-twist family (n, −1 | 1, 1) = {(n + 3) 2 , (2 − n) 3 }. In this paper, we also updated the list of properties in section 6, including the improved differential expansion for asymmetric representations.
This list of 5 boldfaced items describes the main results reported in the present paper. The next steps in study could be: • Finding the inclusive Racah matrices R ⊗3 −→ all for higher representations R. This can look like impossible dream, but there is certain evidence that general formulas can exist. Still, this requires an essential progress in the highest weight method, we devote a separate publication to these perspectives.
• Finding the Racah matrices S R for R ⊗R ⊗R −→ R andS R for R ⊗ R ⊗R −→ R. These are much simpler, because only one outgoing representation Q = R is requested. They are more complicated for the highest weight method, becauseR depends on N (of sl N ) and so do the highest weights. As explained in [75-77, 97, 102], the knowledge of these matrices allows one to handle all arborescent (double fat) knots [9,122,123], and merging that knowledge with the results of the present paper extends this to the fingered 3-strand knots [1,100,102] which include at least the entire Rolfsen table up to 10 intersections and, after applying the power family method from these papers, many more.
• In fact, a simple trick allows one to extract S from comparison of two different expressions, 3-strand and Pretzel ones for a peculiar two-parametric evolution family (m, −1|n, −1) = Pr(m, n,2). The second matrixS can then be built from S by equation (63) from [97]. The result, requiring a generalization of our analysis in section5, is presented in [124].
• A serious shortcut to the Racah matrix calculus can be provided by the eigenvalue hypothesis of [72,73], see also [101], expressing these matrices through the much simpler eigenvalues of quantum R-matrices (for which there is a general formula through the eigenvalues of the Casimir/cut-and-join-operator [56,120,121]). Even if the very hypothesis is true, it suffers from two uncertainties: the sign choice (well defined are the squares of matrix elements) and the problem of degenerate R-matrix eigenvalues. One of the results of our studies in this paper is a confirmation of a natural assumption: that this degeneracy just implies a decomposition of mixing matrices into simpler (smaller-size) blocks, which are provided by the same eigenvalue hypothesis. Further work in this direction looks quite promising, and, perhaps, can provide general formulas for the Racah matrices much sooner than any alternative approach.