HOMFLY for twist knots and exclusive Racah matrices in representation [333]

Next step is reported in the program of Racah matrices extraction from the differential expansion of HOMFLY polynomials for twist knots: from the double-column rectangular representations R=[rr] to a triple-column and triple-hook R=[333]. The main new phenomenon is the deviation of the particular coefficient $f_{[332]}^{[21]}$ from the corresponding skew dimension, what opens a way to further generalizations.


Introduction
Calculation of Racah matrices is the long-standing, difficult and challenging problem in theoretical physics [1]. It is further obscured by the basis-dependence of the answer in the case of generic representations, but this "multiplicity problem" is absent in the case of rectangular representations. The modern way [2]- [7] to evaluate the most important "exclusive" matricesS R µν , is based on the combination of two very different expressions for R-colored HOMFLY polynomials [8] of the double-braid knots, coming one from the arborescent calculus of [9] and another from the differential expansion theory [10][11][12] in the case of rectangular R = [r s ] with s columns of length r: .

❲ ✗
Here the sums go over sub-diagrams of the Young diagram R, and χ * (N ) denote the corresponding dimensions for the algebra sl N , i.e. the values of Schur functions χ{p k } at the topological locus p k = p * k = {A k } {q k } with {x} = x − x −1 and A = q N . The other ingredients of the formula come from the evolution method [11,13] applied to the family of twist knots (double braids with n = 1): as functions of the "evolution parameter" m knot polynomials are then decomposed into sums of representation µ ∈ R ⊗R (which for rectangular R can be labeled by sub-diagrams of the R itself) with dimensions D µ , and m-dependence is then provided by m-th power of the "eigenvalue" Λ µ of the R-matrix, a q-power of the Casimir or cut-and-join operator [13]. In arborescent calculus the weights are made from the elements of Racah matrixS while in the theory of differential expansions they are composed into amusing generating functions [11] with f ∅ λ = 1. Each term in the sum has a non-trivial denominator, however the full sum is a Laurent polynomial in A and q for all m. Moreover, it vanishes for m = 0 (unknot), equals one for m = −1 (figure eight knot 4 1 ) and is a monomial at m = 1 (trefoil). According to [3] and [5] the F -functions are best described in a peculiar hook parametrization of Young diagrams: In particular, the overall coefficients and Clearly, c λ drops away from the r.h.s. of (1). The shape of the coefficients f µ λ strongly depends on the number of hooks in λ and µ. Currently they are fully known for λ = (a 1 , b 1 |a 2 , 0) -what is enough to get the Racah matricesS for the case R = [r, r] (actually, for this purpose b 1 = 0, 1 is sufficient).
• For two-hook λ = (a 1 , b 1 |a 2 , b 2 ) the formulas are far more involved, and they are different for different number of hooks in µ: Non-trivial are the correction factors, true for a 2 · b 2 = 0: and where δ x = 1 for x = 0 0 for x = 1 and Thus corrections involve a natural modification of K-factors and somewhat strange shifts of the argument N , i.e. multiplicative shift of A by powers of q. These formulas were found in [3,5] for the case when a 2 · b 2 = 0 (i.e. when either b 2 = 0 or a 2 = 0). Sufficient for all the simplest non-symmetric rectangular representations R = [r, r] and R = [2 r ] are respectively b 2 = 0 and a 2 = 0. Note that underlined expression are the arguments of K-functionsnot additional algebraic factors. Boxes contain projectors on sectors with particular values of i 1 and j 1 .
Our goal in this paper is to make the first step towards lifting the restriction a 2 · b 2 = 0. Namely, we consider the case of the simplest 3-hook R = [333], which has 20 Young sub-diagrams, of which there are two, The diagram [332] = (22|11) is still two-hook, but both a 2 = b 2 = 1 are non-vanishing. If we apply just the same formulas (14)-(19) in this case, the answer will be non-polynomial. However, one can introduce additional correction factors η µ λ for all the items in the sum over µ and adjust them to cancel all the singularities. Of 19 factors non-trivial (different from unity) are just 8 (we omit the subscript λ = (22|11) to simplify the formulas): and the resulting expression is It nicely satisfies the sum rules (5).
3 Extension to F (a 1 b 1 |11) We can now develop the success with F (22|11) and extend it to other 2-hook diagrams with a 2 · b 2 = 0. We actually restrict our attention to the case of a 2 · b 2 = 1, i.e. a 2 = b 2 = 1.
In the next case of F (33|11) the correction factors are (again we write just η µ instead of η µ (33|11) ): This implies a simple extension of (16) and (17) to arbitrary diagrams (a 1 , b 1 |1, 1), i.e. true for a 2 · b 2 = 0, 1 are: and ξ (i1,j1|i2,j2) Formula (23) means that the coefficient f (11) λ is no longer proportional to the skew character χ * λ/ (11) . Interpretation of this deviation remains to be found. Note that for a 2 · b 2 = 0 we have just instead of (23) -as one more manifestation of discontinuity of the formulas, expressed in terms of hook variables.

Extension to F
Again, we can easily extend this result to arbitrary a 1 and b 1 : the substitute of (23), true for a The shift N −→ N + (i 1 + i 2 + i 3 + 3) · δ b3 − (j 1 + j 2 + j 3 + 3) · δ a3 in the last line is not actually tested by these formulas, because the associated K (i1j1|11|00) (a1b1|11|00) do not depend on A. The quantity u (a1b1|11|00) is given by a literal analogue of (23): Dimensions D µ of these representations are obtained from the terms with ν = ∅ in (1), becauseS µ∅ = √ Dµ dR : in obvious notation After thatS The simplest test of the result is thatS is orthogonal matrix, It is also symmetric. The second exclusive matrix S [333] is then the diagonalizing matrix ofTST [9]: with the known diagonal T andT , made from the q-powers of Casimir. This is actually a linear equation for S, The polynomial in brackets reduces to D(0) 6 = {A} 6 at q = 1 and to − ([4] [3][2]) 3 {q} 6 at A = 1. A better quantity for practical calculations is unnormalizedσ µν =S µν · D µ D ν , which does not contain square roots.

Conclusion
The main result of the present letter is explicit expression for the two previously unknown F -functions F (m) (22|11) and F (m) (22|11|00) . Most important is the deviation from the coefficient f (11) (22|11) from the skew dimension, even shifted -what is expressed by eq.(23), see also (31). This new phenomenon explains the failure of previous naive attempts to write down an explicit general expression for F in arbitrary representation: an adequate substitute of the skew characters and appropriate generalization of the corresponding conjecture in [5] is needed for this. The next step in this study should be further extension to a 2 · b 2 > 1.
The two newly-found functions, if combined with the other 18, associated with 0,1,2-hook diagrams λ with the property a 2 · b 2 = 0, provide explicit expression for [333]-colored HOMFLY for all twist and double braid knots. Moreover, from (1) one can read all the elements of the Racah matrixS [333] , while S [333] is then found from (36). Thus this paper solves the long-standing problem to evaluateS [333] and S [333] . Explicit expressions for these Racah matrices as well as for the [333]-colored HOMFLY for the simplest twist and double-braid knots are available at [14].
It still remains to evaluate the twist-knot polynomials and Racah matrices for generic rectangular representations -the new step, made in the present paper, provides the essential new knowledge about this problem which can help to overcome the existing deadlock.
For additional peculiarities of non-rectangular case see [6]. The main point there is that representations in R⊗R are no longer in one-to-one correspondence with the sub-diagrams of non-rectangular R. Still, factorization of the coefficients in the differential expansion for double braids persists, and thus the Racah matricesS can still be extracted from knot polynomials -though the procedure becomes more tedious [7].