Colored HOMFLY polynomials of knots presented as double fat diagrams

Many knots and links in S^3 can be drawn as gluing of three manifolds with one or more four-punctured S^2 boundaries. We call these knot diagrams as double fat graphs whose invariants involve only the knowledge of the fusion and the braiding matrices of four-strand braids. Incorporating the properties of four-point conformal blocks in WZNW models, we conjecture colored HOMFLY polynomials for these double fat graphs where the color can be rectangular or non-rectangular representation. With the recent work of Gu-Jockers, the fusion matrices for the non-rectangular [21] representation, the first which involves multiplicity is known. We verify our conjecture by comparing with the [21] colored HOMFLY of many knots, obtained as closure of three braids. The conjectured form is computationally very effective leading to writing [21]-colored HOMFLY polynomials for many pretzel type knots and non-pretzel type knots. In particular, we find class of pretzel mutants which are distinguished and another class of mutants which cannot be distinguished by [21] representation. The difference between the [21]-colored HOMFLY of two mutants seems to have a general form, with A-dependence completely defined by the old conjecture due to Morton and Cromwell. In particular, we check it for an entire multi-parametric family of mutant knots evaluated using evolution method.


Introduction
R-colored HOMFLY polynomials [1] are defined as the Wilson loop averages in 3d Chern-Simons theory [2] H L⊂M and for simply connected embedding space M = R 3 or S 3 it is an integer polynomial (i.e. all the coefficients are integer) of the variables q = exp 2πi κ+N and A = q N , where N parameterizes the gauge group SU (N ). Moreover, it can often be promoted to a positive integer superpolynomial [4]- [11], if one more parameter t is introduced: deviation from t = −1 can be considered as a β-deformation [12]. However, construction of superpolynomials is still a mystery, especially for non-rectangular representations R (i.e. when the corresponding Young diagram R is not rectangular), see [13,14,15]. In fact, even the colored HOMFLY polynomials are very difficult to find, and this enhances the difficulties of conceptual considerations.
Two recent achievements open a way to explicit calculation of the colored HOMFLY polynomials beyond rectangular representations. The first of them is evaluation of the 3-strand knot polynomials for representation 1 arXiv:1504.00371v1 [hep-th] 1 Apr 2015 [21] in [13,16] and of the underlying Racah matrices S andS, in [17]. The second is a conjectured expression [18,19,20] for the peculiar double fat tree diagrams, expressing the unreduced colored HOMFLY polynomials through the Racah matrices:  (2) where the sum goes over representations X, which belong either to the product R ⊗R or to R ⊗ R, depending on the configuration and orientation of the diagram, and over additional indices a (µ) i = 1, . . . , m Xµ , appearing when X enters the product with a non-unit multiplicity m X . The weight factor is the product of quantum dimensions d X of X. Additional factors are just signs, σ = ±1, they appear only for non-trivial multiplicities and we ignore them for a while leaving until sec.7, though in fact σ are crucially important for non-rectangular R including R = [21].
We define the double fat tree diagrams as ordinary trees with vertices µ of arbitrary valencies k µ , e.g.
Let us consider a typical example of the double fat diagram of the knot. Suppose the usual knot diagram can be presented in the form i.e. in this concrete case it is three four-strand braids (propagators, or edges) connected by double lines into the vertex X, we call it the double fat diagram. In the field theory form, this particular diagram has a peculiar shape of a starfish with three fingers: d d X X µ X a1,a2,a3=1 σ X|a1,a2,a3 · d R 3 We always consider each edge in (3) being a 4-strand braid: Boxes in this picture denote vertical 2-strand braids of given lengths, which can be parallel or antiparallel, depending on the directions of arrows (which in their turn depend on parity of the lengths). The number of boxes in each propagator can be arbitrary. Each vertex µ in (3) is the cyclic junctions of k µ 4-strand braids: . . . X, a 1 X, a 2 X, a k−1

X, a k
With each such vertex µ one associates a representation X µ . The crucial feature of this construction is the selection rule for propagators: the two representations X,X at one end of the four-strand braid (5) differ by conjugation, while the additional indices a and b can be different. Thus, the propagator has two multi-indices Xab and Y cd: i.e. is a matrix element A Y cd,Xab, , while the matrix itself is made by the usual conformal block rule from the R-matrix T and the Racah matrix S.
Interchanging m α and l α at each level α inside every fingeris a mutation transform (provided one considers only the tree diagrams), and in fact one can perform a mutation to change m α , l α → m α + k α , l α − k α with any k α . In result, for symmetric representations R, the HOMFLY polynomial depends only on the sums m α + l α , however, for generic R dependencies on m α and l α are separated.
Beyond the rectangular R, additional indices a and b are added to X, still there are just two independent S-matrices, S andS, from which all other are made by various conjugations. Whatever is R, the first lines in the Racah matrices S andS are always made from the quantum dimensions: This property is crucial for self-consistency of (2) when an edge is a tadpole. Then the role of the end-vertex (with the sum over the corresponding X) is to imitate gluing of the caps like with a singlet representation 0 ∈ R ⊗R, or XX = δ 1X (12) Relations (10) ensure that each tadpole (end-edge of a branch in the tree, which we call finger) is nothing but the matric element A ab 1X . Because of this in what follows we omit the cycles at the end-edges of the branches. Additional universally applicable simplifications emerge from the elementary S, T -matrix identities, like the celebrated (ST ) 3 = 1 in the case of R = [1] for SU q (2), for its generalization to arbitrary representations of SU q (N ) see (62).
The set of fat-tree diagrams is very big. According to [19], it contains at least all the knots from the Rolfsen table [21] (with the possible exception of 10 161 ), we do not know any proved example of knot beyond this family. 4 Most importantly, many complicated knots (with many intersections) look quite simple when represented as double fat graphs, in particular a lot of interesting examples already get into the three-finger set, i.e. have k = 3. As already mentioned, there are plenty of obvious mutants already within the starfish family (defined in (31), see below) and even among the pretzel knots [22], what makes it especially interesting for the studies of [21]-colored polynomials, which are the simplest to distinguish some mutants.
At the same time, eq.(2) describing the colored HOMFLY polynomials for the fat tree diagrams is by no means a trivial formula. Its origins are not immediately clear neither from the conformal block [3,23] nor from the Reshetikhin-Turaev [24,25,26] approaches to knot/link polynomials, despite (2) is clearly very much in the spirit of these both. Anyhow, there is a lot of evidence, coming from direct applying the RT/conformal block method [18,19,20] and its advanced versions like the evolution [27], cabling [16] and differential expansion [15] methods, which supports this formula in the case of symmetric representations, and we add more in this paper for the case of representation [21]. Moreover, eq.(2) looks typical for topological field theories (like the Hurwitz model in [28,29]) and it can serve as a basis of a new intuition and calculus in knot theory, involving a kind of pant decomposition of link diagrams.
When we write "vertices" and "propagators" in (2) we actually make a choice between two dual interpretations of this kind of formulas: one could instead do the opposite and treat the circles in (3) as loops and assign the edges to vertices. Within this framework, the tree diagrams are no longer trees but contain loops, however, the overlapping loops are still absent. Technically this is not very convenient, because we get unit propagators, simple universal loop factors d −1/2 X and an infinite variety of complicated vertices (5). This is why in the present, largely technical text we make use of the first interpretation. However, conceptually the second one can be more much closer in spirit to the common language of string theory, and provides a useful intuitive vision.
The paper is organized as follows. In the first part we describe the elementary building blocks for double far graphs and explain how to construct knots out of them. In the second part we first describe some peculiarities of representation theory especially related to the first non-trivial mixed representation [21], and then consider the colored HOMFLY polynomials in representation [21] for various knots. In particular, we check that these HOMFLY polynomials do differ between the notorious Conway-Kinoshita-Terasaka (KTC) mutant pair, in accordance with expectations [30], moreover, the manifest difference between their HOMFLY polynomials is in complete agreement with [31]. We also discuss other mutant pairs, in particular, find a difference between the HOMFLY polynomials of a whole 2-parametric family of mutants. However, representation [21] turns out not to be enough to differ between another set of mutants. Some more explicit HOMFLY polynomials in representation [21] for various knots is listed in the Appendix, where one can find also a table that describes the features of all knots with not more than 10 intersections relevant for this paper.
Note that all answers for the HOMFLY polynomials in representation [21] in this paper for the knots that have a three braid representation we compared with the results obtained by the cabling method of [13,16]. Besides, we made a few self-consistency checks, see s.7.4.
Throughout the paper we use the notation [n] both for the quantum number and for the representation (e.g., [21]). Hopefully this would not lead to a misunderstanding. Other our notations are and d R is the quantum dimension of the representation R.
Part I Diagrammar 2 Chern-Simons evolution Expression (6) for each propagator (5) in (2) is usually interpreted as the ordinary time evolution in Chern-Simons theory, which is provided by monodromies of conformal block made from R-matrices T of three types (depending on the pair of adjacent strands that it acts on) and Racah matrices S, see [3] for the original idea and [23,19,20] for recent applications. In the case of arbitrary representation R there is a whole variety of S matrices and eq.(6) needs to be formulated more accurately. This section describes our notation, which will be used in the rest of the text. S is a matrix, with two peculiar multi-indices Xab and Y cd, which are collections of square matrices of different sizes, depending on irreducible representations X and Y . Two more potentially convenient notation are The punctured line helps to illustrate the meaning of indices, it indicates that, after applying the Racah matrix S, the representation R 1 becomes close to R 4 . Note that the initial state Xab stands to the right/bottom of the final Y cd, but representations are ordered oppositely: from the left to the right and from the top to the bottom. When we write explicit matrices in sec.6, the left/upper indices (like Y cd) label rows, while the right/bottom (Xab) -columns.
The composition of two evolutions gives the identity: If combined with unitarity of S (orthogonality when S is real), this implies: Note that S converts two parallel braids into two antiparallel, whileS converts antiparallel into antiparallel: Operators T describe crossing of two adjacent strands and are much simpler than S. The only delicate point is that there are three different pairs of adjacent strands in the 4-strand braid and therefore there are three different T -matrices, which we denote T + , T 0 and T − . Also, the two intersecting strands can be either parallel or antiparallel, what we denote by T andT respectively. This brings the number of different T -insertions in (6) to six.
One can promote these operators to matrices of the same type as S: The eigenvalues are the standard ones in the theory of cut-and-joinŴ -operators [29] and the same which appear in the Rosso-Jones formula [32,6]: All X 's take values ±1. We usually omit the indices ±, 0 of T later on, since these T 's are all the same as matrices.

Pretzel fingers and propagators
Our next goal is to classify different types of expressions (6) for the propagator (5) -according to the possible number and parities of parameters n α , m α and l α and to directions of strands (arrows). A special role is played by the terminal branches in the tree (3), which we call fingers. They are represented by matrix elements A ab 1X = A 0,Xab , and all the T -matrices standing to the very left of all S can be omitted. According to (18), the last (most left) S in such matrix element can be eitherS or S but not S † .
In this section we begin from the simplest possible types of propagators and fingers, belonging to the pretzel type. For pretzels all m α = l α = 0 and there is just one parameter n in each of the k of the fingers. Still, there are propagators and thus fingers of three different kinds, depending on directions of arrows and the parity of n. In the case of pretzels, the notation with bars is sufficient to distinguish between all these cases, still we add explicit indices par, ea, oa to avoid any confusion: The choice of S-matrices in above examples is easy to understand from (18), one should remember that S converts parallel states (from R ⊗ R) into antiparallel (from R ⊗R), while S † acts in the opposite direction, andS relates antiparallel to antiparallel.
In the final expressions (boxed) we suppressed most indices, thus implying that we deal with matrices in extended space, where basis is labeled by the multi-index Xab. We explain the details of this formalism in sec.6, while in sec.5 we use the symbolical notation: just X for the multi-index and X for appropriate contractions, including non-trivial sign-factors σ (which is symbolized by the bar over the sum).
In the case of propagators there are three more option, but they can not be closed non-trivially and do not show up in the pretzel family and in the remaining parts of the present paper.

Other building blocks 4.1 Non-pretzel fingers
Generalization to arbitrary non-pretzel fingers and propagators (5) with arbitrary parameters m α , l α , n α is exhaustively described by (6), one just needs to put appropriate S and T matrices for the given choice of arrows, like we did for the pretzel fingers in sec.3. Since this is straightforward we do not provide this additional list.
What is more important, some diagrams, which do not a priori look like (5), actually belong to this family, and this is the reason for the double fat graph description to cover so many different knots and links. In this section, we mention just two examples of this kind, exploited in our further considerations.

Horizontal loop
The first example is simple: It is nearly obvious that this is just the non-pretzel finger with parameters n 1 = m 2 = l 2 = ±1 (signs and the types of S and T matrices depend on the types of intersection and directions of arrows). No S-matrix identities are needed to get an expression for it. One could easily insert a horizontal braid of arbitrary length and consider a sequence of such horizontal loops:

Horizontal braid
The isolated horizontal braid is nothing but the pretzel finger. The type of the finger depends on orientation of lines and the parity of m.
More interesting is the horizontal braid located in between the two middle lines in the double fat propagator: This is a contraction of three blocks, where the middle one is exactly (4.3). Schematically, it is In fact there are seven versions of this relation, for different arrow directions and different parities of m. Four of them describe even m:

Odd antiparallel pretzel knots
More generally, if there is any odd number of odd antiparallel fingers, If the number of fingers is even, we get links rather than knots.

Pure parallel pretzel knots
If the pretzel knot is made from the parallel fingers only, their number should be even and exactly one length should be even, otherwise we get a link rather than a knot. The answer for the HOMFLY polynomial is

Mixed parallel-antiparallel pretzel knots
The remaining family of the pretzel links/knots contains even number of parallel fingers and odd number of antiparallel, which have even lengths. The corresponding HOMFLY polynomial is

Non-pretzel fingers
Now let us consider a few non-pretzel examples.
Knot 10 71 . Its HOMFLY polynomial is With the help of (63) the last matrix element can be changed for The last two factors in the sum are already familiar A par (p) and A par (q) from (22), while the new one is (like all the sums inside particular fingers, this is an ordinary matrix multiplication without any sign σ-factors).

The third finger is
and Configuration (44) is not always a knot: it can also be a link with two or even three components: p q r # of link components even even odd 1 odd even even 1 odd odd odd 1 even even even 2 even odd odd 2 odd even odd 2 odd odd even 2 even odd even 3 Among knots, the notable members of the three-finger family are the thick knots 10 124 and 10 139 , which also have pretzel realizations: 10 124 = (5, 1, 3, 1) and 10 139 = (4, 2, 3, 1).

Generic four-box 3-strand braids beyond starfish family
For s = 1 the braid (42) can not be converted by the Reidemeister moves to a starfish configuration. Instead, it is equivalent to where the propagator is actually performing an operation like (14) on antiparallel strands, i.e. is represented by the matrixS. According to (2), the corresponding HOMFLY polynomial is

Diagrams with non-tadpole propagators
In the case of (51) and (52) not all the pretzel fingers are contracted directly, an additional propagator appears on the way and our basic formula (2) involves two independent summations.

The ppS block
A new building block appearing in (52) is This expression remains the same ifS is transposed,S XY −→S Y X . The number of parallel fingers in this building block can exceed two (but needs to be even), however, for our purposes below, the two will be enough.
For the 4-box 3-strand braid one can rewrite (52) as

Double braids
In fact, propagator can be less trivial than in (52). Important generalization is provided by the horizontal braid (27): The right two lines in the second block do not change the representation, the left part of this block is just the Pretzel finger A eg 1Z . The type of S-matrices and the Pretzel finger depends on the directions of arrows and on the parity of the braid length m.
The simplest application of (54) is to the double braids of [27], which has also the pretzel representation (−1, 2k, m).

Pretzel fingers connected via horizontal braid
More interesting is the situation when pretzel fingers appear from both sides of the horizontal braid: The central block is just the horizontal braid, antiparallel of length 2, described by the corresponding version of (54). Therefore, the reduced HOMFLY polynomial for the diagram (55) with even m is The four external fingers involve parallel braids, while that of the even length m in the propagator is antiparallel. The fat tree is the same (51), only the internal propagator is more sophisticated. Mutation is a permutation of p and q or of r and s.
At s = 0 the parallel finger A par If additionally q = 0, we obtain a composite knot, made out of two 2-strand constituents: All formulas in this part work immediately for the fundamental and symmetric representations. They were tested in various representations from this set. Since the Racah matrices are explicitly known for all symmetric representations [34], [20, 2nd paper], there is no problem to make further tests in this direction. In all these cases all the σ-factors are unities, and there is no need to deal specifically with overlined sums. The same formulas continue to work for representation [21], but then some σ-factors are −1. Expressions for R = [21] are rather involved, because such are the Racah matrices, explicitly calculated in [17], and we describe them in the next section.
Part II Examples 6 Racah matrices 6

.1 Generalities
The Racah matrices relate the two expansions and that is, it is a linear operator When the vector spaces W are one-dimensional ("no multiplicities" case), one can represent this linear operator as a matrix with indices P ∈ R 1 ⊗R 2 and S ∈ R 2 ⊗R 3 . When W are multidimensional, there is no distinguished any basis and the concrete form of the Racah matrix depends on conventions, and on four additional indices labeling the bases in the four W -spaces for each given pair P and S. For the purposes of knot (rather than link) theory, all the four representations R 1 , R 2 , R 3 , Q are either R or its conjugateR. In result, there are two essentially different Racah matrices: • S S,c,d|P,a,b with P ∈ R ⊗ R, but S ∈ R ⊗R, and •S S,c,d|P,a,b with both P, S ∈ R ⊗R. These are the only two kinds of the Racah matrices that show up in our discussion of the double fat tree diagrams.
Note that S is essentially asymmetric, whileS can be symmetric, and actually is in the multiplicity-free case.
Another important fact is that the singlet representation 0 ∈ R ⊗ R, but 0 / ∈ R ⊗ R (except for the case of N = 2), therefore, the matrix elements 1X in (2) can not have S † at the very left, but only S orS. In general, as we already saw earlier, (18) S converts the parallel strands into antiparallel, whileS antiparallel to antiparallel. Our fingers never contain three parallel strands, thus these two types of relations are sufficient for our consideration. However, beyond this paper the third Racah matrix connecting the parallel strands to parallel (it was called "mixing matrix" in [25]) plays a big role. Actually, it is much simpler: it does not depend on N (i.e. on A). For symmetric representations, it coincides with restrictions of both S andS to N = 2, but for more complicated representations the story is a little more involved.
The Racah matrices S are always unitary and satisfy (10). There are also additional non-trivial relations like [35,17] which are the implication of In more detail, with indices restored: An important application of this identity is the possibility of shifting any pretzel finger of length 1 to any position [22] at any representation R, though permutations of arbitrary fingers are not a symmetry of the knot.

The simplest symmetric representations
For symmetric representations R, there are no multiplicities, no indices a, b, c, d and the Racah matrices are canonically defined.
• When R is the fundamental representation R = [1], they are given by (8) and (9), or in a generalizable notation, • Similarly, for representation R = [3] and so on.

Representation [21]
In the case of R = [21] there are seven different items X in the decomposition 19 The corresponding dimensions and eigenvalues are (68) with non-trivial multiplicity m Adj = 2, however this makes the Racah matrices 10 × 10, since X m 2 X = 6 · 1 + 1 · 2 2 = 10. They are explicitly found in [17]. The pure antiparallel matrixS is Here we again encounter one item with multiplicity two, and the parallel → antiparallel Racah matrix S is This matrix is obtained from that of [17] by transposition to make it consistent with (10) and thus with our representation of knots. Also minor (always allowed) conjugations are performed in (70) and (73) to get rid of unnecessary minuses and imaginary units. Diagonalized R-matricesT and T are read off from the last columns of (69) and (72): and In these S-and T -matrices the index X runs from 1 to 7, and for X = 7 there are additional indices a i which take two values, which we substitute as 7 11 Then contractions in (2) for the simplest double-fat graph (31) can be rewritten as follows: It is the underlined term that breaks the permutation symmetry between different A (i) in the original product down to just cyclic symmetry, and it is the one that distinguishes mutants. When all A (i) 9 or A (i) 10 are vanishing, the symmetry is preserved and mutants remain indistinguished by the [21]-colored HOMFLY.

σ-factors and other technicalities
Since eq.(2) is a kind of a conjecture/educated guess, and was so far well tested only for symmetric representations where no multiplicities (indices a, b, c, . . .) arise, there is no reason to insist that contractions (76) are exactly correct. In fact, they are not: for complicated enough knots, (2) with rule (76) does not produce polynomials in representation [21] and, in result, fails to reproduce the known answers. Both these problems are cured by switching to σ, which is allowed to take the values ±1 only, but is not obligatory identical unity.
The procedure at the moment is to adjust theses sign factors for a given knot so that the answer for HOMFLY is a Laurent polynomial in A and q. This choice is either unique, or the answer does not depend on allowed choices. In our examples, the sign ambiguity appears only when one of the fingers is pretzel of length one and some matrix elements simply vanish, thus the sign in front of them is just irrelevant. No general rule is found yet for a priori determination of σ, except for some simple families of double fat diagrams it is available, in this technical section we describe what is already known.

3 fingers
In the simplest case of 3 fingers we use the following notation: First of all, it is always the case that σ 111 = σ 222 = 1. Now, it turns out that the two relevant sets of signs of the remaining σ's, appearing in the right formulas are either all equal to -1: or only two σ abc equal to -1. Depending on the order of fingers, it can be either σ 112 and σ 221 or two other pairs obtained by cyclic permutations from these, we will always choose the order in such a way that the first pattern is realized: However, the choice between the two possibilities depends on the type of knot. As to the naive option it also sometimes happens to provide right answers, but only when some of the pretzel lengths are unit and identities (63) relate (80) to (78)-(79). It deserves mentioning that (79) explicitly breaks the cyclic symmetry between A, B and C, thus one can expect that it is relevant when such a symmetry is absent, say in the case of two parallel and one antiparallel pretzel fingers. Accidentally or not, three parallel pretzel fingers is an impossible configuration, while for three antiparallel pretzel fingers the non-diagonal terms are just vanishing.

More fingers
Similarly, for 4-finger parallel pretzel knots the relevant choice is At last, for the 5-finger pretzel knot with one antiparallel finger (for the sake of definiteness, it is the first finger) the proper choice is Thus, we learn, first, that always σ 11...1 = σ 22...2 = 1 and, second, there is a symmetry of simultaneous replace 1 ↔ 2 in all indices. Specifically for the parallel pretzel knots (they exist only for the even number of fingers), there is also an additional symmetry: all sign factors obtained by a cyclic permutation are the same.

Sign dependence on knots
This list of relevant sign choices is not necessarily complete, but it is necessary and sufficient for all the examples that we studied so far. With these sign prescriptions, H [21] for knots are polynomials (otherwise denominators can occur) and coincide with the previously known answers from [13,16,17,33], when they are available. If talking about the concrete knots, among the starfish configurations, all the 3-finger configurations we checked so far are described by (79) (with the distinguished of the three fingers put third) for exception of knot 10 71 , when (78) is realized, and all the 4 parallel finger and (one antiparallel + 4 parallel) finger configurations satisfy the rules described in ss.7.1.2.
As for the double sum configurations (see the Appendix), 10 152 is described by (79) in the both sums, 10 153 is described by (79) in the sum over pretzel fingers and (78) in the second sum. At last, 10 154 is described by (78) in the both sums.
At last, the three sum configuration, (56) is described by (78) in the internal sum in (55) and (79) in the two sums within the ppS-blocks.

Explicit calculations in the pretzel case
After the matrices S, T and the sign factors σ are explicitly known, it is straightforward to insert them into formulas of sec.5 and obtain the colored HOMFLY polynomials in representation [21] for an enormous set of knots. Numerous tricks are needed to make software running fast enough on ordinary computers, but by now this is a routine in knot theory and we do not dwell upon these details. In fact, the answers themselves are huge, thus it makes no sense to list them all, especially given the (large) number of examples. Therefore, we just enumerated the classes of diagrams, and in the Appendix we list explicit examples. Here, for illustrative purposes, we present some general discussion of the pretzel case. A vast set of concrete examples can be found again in the Appendix.

Even antiparallel
This finger depends on the even parameter n and is given by the matrix element Again, the unitarity A Y X (0) = δ Y X makes this choice natural as compared with those with additional insertions of Λ.
For n = 0, this matrix element is just δ X,1 . For n = ±2, the matrix elements of A ea 1X with X + 9, 10 are very simple, similarly to the parallel case: The simplest check to be made is for the Hopf link, which can be represented both through parallel and antiparallel braid: This is the reduced HOMFLY invariant, and since it describes a link, it is not a polynomial. Any combinations of even antiparallel fingers only provide links, not knots. The simplest knot family involving such fingers is P r(n 1 , n 2 ,m), where two of the three fingers are parallel. Then = X d 4 [12] d

Odd antiparallel
This finger depends on the odd parameter n and is given by the matrix element This time n is odd and cannot vanish, therefore there is no obstacle for inserting Λ and writing S † T n ΛS 1X instead, however, this does not actually affect the answers, at least in the cases discussed here.
Elementary calculation or application of identity (62) gives that at n = ±1 where X belongs to the parallel sector, X ∈ R ⊗ R. Once again, this immediately provides the answers for the trefoil: For other values of n these matrix elements are multiplied by polynomials: In particular, this means that the odd antiparallel fingers vanish at X = 9, 10 for all values n, which means that odd antiparallel Pretzel mutants are undistinguishable by the colored HOMFLY polynomials in representation [21]. The polynomials P themselves are, however, somewhat involved. For instance, The twist knots are pretzel T w k = P r(1,1, 2k − 1) i.e. are made from three odd antiparallel fingers: Because the off-diagonal terms in (76) vanish for odd antiparallel fingers, there is no difference between 8 X=1 and X . The answers for twist knots reproduce those from [16,17] and [33]. Other members of the same three-finger family are:

Checks
All answers for the HOMFLY polynomials in representation [21] in this paper for the knots that have a three braid representation we compared with the results obtained by the cabling method of [13,16]. Besides, there are five types of self-consistency checks one can currently make to test our answers. The first four are true for all our examples, the fifth one is more tedious and was only partly verified.

Alexander polynomials for 1-hook representations
As conjectured in [36] a "dual" factorization property holds at A = 1, i.e. for the Alexander polynomials, but only for representations R described by single-hook diagrams: Representation [21] belongs to this class. Neither any meaning, nor generalizations of this remarkable factorization property is currently available.

Jones polynomials
For SU q (2), i.e. for A = q 2 all two-line Young diagrams [r 1 r 2 ] get equivalent to the single-line [r 1 − r 2 ], in particular, [21] gets indistinguishable from [1]. This means that

The weak form of differential expansion
Expressions for colored knot polynomials are extremely complicated, but in fact they have a lot of hidden structure and satisfy a lot of non-trivial relations. Understanding of these structures nicknamed "differential expansions" [36,27,15] because their first traces were observed in [4] devoted to the study of Khovanov-Rozansky "differentials" is still very poor. In its weakest possible form, the differential expansion conjecture implies for representation [21] that [13,33] H and H R are the reduced HOMFLY polynomials. In other words certain linear combination of H [21] and H [1] should vanish at A = q ±2 , i.e. in the case of sl (2).
Despite the restriction to SU q (2) is the same as in the previous test, this one looks independent.

Quasiclassical expansion and Vassiliev invariants
Differential expansions from the previous paragraph are examples of cleverly-structured quasiclassical expansions in parameters like , where q = e and A = q N or z = {q} = q − 1/q. The most structured expansion of this type, known so far is the Hurwitz-style formula [39] for reduced colored HOMFLY: Here ϕ R (∆) are characters of the universal symmetric group (defined as in [29]), l(∆) is the number of lines in the Young diagram ∆ and the coefficients of generalized special polynomials Σ K ∆,k (A) are made from the Vassiliev invariants.
This formula includes (102) as a particular case and is also closely related to (105). It imposes non-trivial restrictions on the coefficients of colored polynomials. As already mentioned, we made only a few simple checks of our answers with this formula, and it is very interesting to extend them in order to understand what are the really independent "degrees of freedom" in H [21] , as compared to those already captured by the colored HOMFLY polynomials in symmetric representations. This study can be also helpful for superpolynomial extensions of the [21]-colored knot polynomials, which still remains a complete mystery.

Mutants
Of special interest among the newly available answers are those for the pairs of mutants, i.e. knots related by the mutation transformation, which are inseparable by knot polynomials in symmetric representations.

Generalities
Mutation in knot theory is the transformation of link diagram, when one cuts a sub-diagram with exactly four external legs, rotate and glue it back to the original position. Within the Reshetikhin-Turaev approach, it is clear that cutting corresponds to decomposition of knot polynomial in the channel R 1 ⊗ R 2 and mutation is a rotation in the spaces of intertwining operators W Q R1R2 : R 1 ⊗ R 2 → Q. If these spaces are one-dimensional, like in the case of rectangular representations R 1 = R 2 orR 1 = R 2 , the mutation does not affect the corresponding HOMFLY polynomial. Thus, mutants can be distinguished by colored HOMFLY polynomials in non-rectangular representations, the first of which is R = [21]. The difference H mutant1 [21] − H mutant2 [21] = {q} 11 [21] (q 2 ) = (108) = {q} 4 · {Aq 3 } 2 {Aq 2 }{A}{A/q 2 }{A/q 3 } 2 · A γ · M mutant [21] (q 2 ) has the universal prefactor. The differentials D ±2 and D 0 appear in it because the Alexander and the Jones polynomials at A = 1 and A = q 2 are not affected by the mutation and, thus, the difference should vanish for SU q (N ) with N = 0 and N = 2, i.e. at the corresponding values of A = q N and, since the Young diagram [21] is symmetric, at A = q −N (due to the level-rank duality, [6,8,36]. Vanishing for SU q (3) follows from a more involved argument of ref. [30]. There are no factors D 1 D −1 , because we consider the reduced HOMFLY polynomial, which is equal to the original unreduced one divided by d 21 = D1D0D−1 [3] . The additional A-independent factor {q} 4 seems to be typical for all non-diagonal terms considered in this paper, and this provides an explanation for power 11 in (108). In all examples, which we managed to analyze, M mutant (q 2 ) is, indeed, a function of q only, with no A-dependence.

Examples of mutants
Of these, the KTC pair is most famous. However, the simplest are pretzel ones: there are two such pairs close to the bottom of above list, see also eq.(111) below. For enumeration of mutants with up to 16 intersections see [40].

KTC mutants
The Kinoshita-Terasaka (11n42) and Conway (11n34) knots (KTC mutants) are respectively M (3, −2|2| − 3, 2) and M (3 − 2|2|2, −3). Both knots have representations with 11-intersection, but for our purposes the realization with 12 intersections is more convenient, These diagrams are already easy to bring to the form (55) with (p, q, r, s) = (3, −2, −3, 2) and (3, −2, 2, −3) for 11n42 and 11n34 respectively. Their HOMFLY are: 2 2 We use here the notation from [19]. Matrix lists the coefficients of a polynomial in A 2 and q 2 by the following rule: Changing m = 2 −→ m = −2 simply switches between these two polynomials. At q = 1 both these polynomials are cubes of the special polynomial, H [21] (q = 1) = H [1] (q = 1) At A = q 2 both reproduce the same Jones and at A = 1 the Alexander polynomial, which in this particular case is just unity, The first terms of the differential expansion (see ss.7.4) in the classical limit, A = q N , q = e , are In accordance with [30], the difference between the two polynomials shows up in the order 11 , which is related to the power 11 of {q} in (108). The complete difference is [2] [7] and z = q − q −1 = {q}. It perfectly matches the result of [31]. The difference could actually be calculated for three-cabled knots (because symmetric and antisymmetric representations [3] and [111] do not contribute to it), what allows one to make calculations with the help of the ordinary skein relations. However, the answers for individual H [21] could not be obtained in that way.

Other mutant pairs
In the list (109) of 11-intersection mutant pairs there are four more pairs, for which the fat tree description is already provided. In fact, from this point of view they belong exactly to the same class as the KTC mutants, just some intersection signs are different. The differences are H 11n36 [21] − H 11n44 [21] = A −9 [ Somewhat surprisingly, some differences are the same for different pairs: for 11n36/11n44 and 11n41/11n47, as well as for 11n39/11n45 and 11n151/11n152. The same phenomenon one could observe for the pretzel mutants: the differences are the same for pairs: 11n71/11n75 and 11n76/11n78; 11a47/11a44, 11a57/11a231 and 11n73/11n74 (in the latter case the three differences coincide), see ss.8.3. The entire HOMFLY are, of course quite different, see the Appendix.
It remains to be seen, if the remaining pairs in (109) possess a double fat tree description.

Evolution method in application to mutant families
Of course, most interesting are not just particular knots, but entire families, depending on various parameters. The way to study this kind of problems is provided by the evolution method of [6,27]. For its application to knot polynomials in representation [21] see [33], but there only the simplest family of twist knots was considered (the torus knots were described in this way in arbitrary representation [32,5,6,7], but this is a much simpler exercise, because of the clear algebraic nature of the torus family). We briefly remind this story and then extend consideration to the first mutant-containing family.
9.1 Twist knots [16,17,33] The [21]-colored HOMFLY for generic twist knots was obtained by the evolution method in [33]: This example is already quite interesting for the study of differential expansion [15] beyond (anti)symmetric representations, however, this family does not contain mutant pairs and, hence, is not representative enough.

4-finger pretzel knots
The simplest multi-parametric family containing mutant pairs is the four-parallel-finger pretzels P r(n 1 , n 2 , n 3 , n 4 ) with even n 1 and n 2 , n 3 , n 4 odd. In this case, one has H P r(n1,n2,n3,n4) [21] − H P r(n1,n2,n4,n3) and For generic even values of n 1 expression is more complicated: The r.h.s. of (126) is actually a Laurent polynomial in q. This formula is obtained by the evolution method of [27] w.r.t. the variables n 2 , n 3 , n 4 , and it has the structure predicted by the general expression (108) for mutants. Also it vanishes whenever any of n 2 , n 3 , n 4 is unity, because for the pretzel knots unit length always commutes with any other length due to identities like (63). Eqs.(111)-(112) in ss.8.3 are particular cases of (126).

Conclusion
In this paper we continued the study of the double fat tree realization of link diagrams, for which HOMFLY polynomials are described by an absolutely new and impressively effective formula (2). This formula looks like coming from some new quantum field theory, which should still be found and explored.
In the present paper we focus on another property of (2), which actually reduces evaluation of nearlyarbitrary knot polynomials to that for the two-bridge knots. It goes without saying that this overcomes most calculational complexities in knot/link theory. As an immediate illustration, we calculate a number of [21]colored polynomials, and actually can now do this for almost arbitrary given example. Among other things this is used to explicitly illustrate/validate old expectations about the mutant pairs: they are indeed sometimes (but not always!) distinguished/separated by the [21]-colored polynomials. For instance, the KTC mutant pair can be resolved by these polynomials, but the pretzel mutants made from antiparallel fingers of odd lengths can not.
Of greatest importance for knot theory are now three questions: • How large is the double fat tree family: what are the knots/links beyond it (if any?), and what is the way to find a double fat tree realization of a given knot/link?
• What is the origin of (2) and its effective field theory interpretation? What could be the meaning of quantization and loop diagrams in this effective theory.
• What is the β-deformation of (2), i.e. can this powerful formula be lifted to the super-and Khovanov-Rozansky polynomials?
Added to these could be two obvious technical next steps: • to extend the double fat trees made from 4-strand braids propagators to to arbitrary braids. The need for this for knot theory depends on the answer to the very first question, however, the problem itself can have its own value, especially because of its relation to multi-point conformal blocks. Since (2) is going to provide a new look at all this set of subjects, its generalizations are valuable in arbitrary directions.
• to develop a systematic calculus for colored knots in arbitrary representations, beyond symmetric and [21]. As clear from the present text, this is actually a problem of calculating the simplest Racah matrices describing the 2-bridge knots. If one really does not need the multi-bridge extension for most of knot theory, this technical problem acquires a new value, and hopefully will be solved in the near future, at least to a certain extent.
Calculations in the case of representation [21] are quite involved and can be performed in slightly different ways. In particular, such technically independent exercise for the KTC mutants is reported in a parallel paper [41].
Here we list examples of knots from the Rolfsen table of [21], for which the [21]-colored HOMFLY are now available.

A list of knots with up to 10 intersections
We start from the table which contains some relevant information about the knots up to 10 intersections.
The left part of the table lists the previously known cases: • For the torus knots, the arbitrary colored HOMFLY polynomial is given by the Rosso-Jones formula [32].
• For three-strand knots, the [21]-colored HOMFLY polynomials can be calculated by the cabling method of [16] on ordinary computers. When the number of strands is three, we explicitly give a braid word, only instead of τ a1 1 τ a2 2 τ a3 1 . . . we write the sequence a 1 , a 2 , a 3 , . . . • For two-bridge knots the knowledge of Racah matrices S andS from [17] is sufficient: eq.(2) in this case reduces to an obvious matrix element A 11 . The corresponding answers are available from [17]. When the number of bridges is two, we explicitly give an S − T word.
The right part of the table lists the cases which are available only now, by the method of the present paper, these knots are boldfaced in the first column. Numerous intersections between the left and right parts of the table are important for checking our conjecture (2).
An exhaustive list of the pretzel knots up to 10 intersections is borrowed from the third paper of ref. [20]. They are distinguished from generic double fat tree knots only by simplicity of computer calculations, what is actually quite important.
The starfish cases are also usually simple enough: they still involve just one sum over representations (all pretzel knots are automatically starfish). The cases with two and three propagators involve two and three such sums and are considerably more difficult for computers. A search for maximally simple representations are therefore important from this point of view, and hopefully the table can be significantly improved.
The right part of the table is currently incomplete, but it seems all the knots with 10 or less intersections to fit into it (with the possible exception of 10 161 , which is anyhow present in the left part of the table).
In addition to this table, answers are available for some explicitly identified mutants with 11 intersections.

Some thin knots
A lot of explicit examples with less than nine intersections can be found in [16] and [17]. Here we add the only 8-intersection knot 8 15 which was not present in those lists (it is pretzel, but possesses also a simpler triple-finger realization), and a couple of 10-intersection knots: and

The first thick knots
Thick are the knots, for which the fundamental superpolynomials are not obtained from HOMFLY by a change of variables and Khovanov homologies have non-trivial entries off critical diagonals (marked in red in [21]). In most cases, these superpolynomials have more terms than the HOMFLY polynomial (though this discrepancy can often be eliminated by switching to differential expansion a la [15]). The first thick knots in the Rolfsen The next have eleven and more intersections.

3-strand cases
Five of these are 3-strand and therefore the answers for H [21] are easily available by the methods of [16]: Moreover, two of these are torus knots, 8 19 = T orus [3,4] and 10 124 = T orus [3,5], and therefore arbitrary colored HOMFLY polynomials for them are available: provided by the Rosso-Jones formula [32,6].

4-parallel pretzel finger cases
Three of the above thick knots are of the pretzel type and are described by the simple formulas: and H P r(n1,n2,n3,n4) R These three cases are 3-strand (and thus already known)

Realizations from [19]
According to [18] and [19], seven thick knots from [4] can be realized just as triple-finger starfish diagrams: H 942 10 145 is the only example in our thick-knot set, where summation is over representations X ∈ R ⊗ R, i.e. in the parallel sector. This summation is insensitive to the choice of parameters ξ in sec.7, thus there is no overline in this case.
H 10145 [21] = Three knots are more complicated, their HOMFLY polynomials are represented by double sums, at best: