Colored HOMFLY Polynomials as Multiple Sums over Paths or Standard Young Tableaux

Abstract

If a knot is represented by an 𝑚-strand braid, then HOMFLY polynomial in representation 𝑅 is a sum over characters in all representations 𝑄∈𝑅^⊗𝑚. Coefficients in this sum are traces of products of quantum ℛ-matrices along the braid, but these matrices act in the space of intertwiners, and their size is equal to the multiplicity 𝑀_𝑅𝑄 of 𝑄 in 𝑅^⊗𝑚. If 𝑅 is the fundamental representation 𝑅=[1]=□, then 𝑀_□𝑄 is equal to the number of paths in representation graph, which lead from the fundamental vertex □ to the vertex 𝑄. In the basis of paths the entries of the 𝑚−1 relevant ℛ-matrices are associated with the pairs of paths and are nonvanishing only when the two paths either coincide or differ by at most one vertex, as a corollary ℛ-matrices consist of just 1×1 and 2×2 blocks, given by very simple explicit expressions. If cabling method is used to color the knot with the representation 𝑅, then the braid has as many as 𝑚|𝑅| strands; 𝑄 have a bigger size 𝑚|𝑅|, but only paths passing through the vertex 𝑅 are included into the sums over paths which define the products and traces of the 𝑚|𝑅|−1 relevant ℛ-matrices. In the case of 𝑆𝑈(𝑁), this path sum formula can also be interpreted as a multiple sum over the standard Young tableaux. By now it provides the most effective way for evaluation of the colored HOMFLY polynomials, conventional or extended, for arbitrary braids.

1. Introduction

Knot polynomials are currently among the central objects of interest in quantum field theory; they are exactly at the border between the known and unknown. The knot polynomials can be defined as Wilson loop averages: 𝐻_𝑅^𝒦⊂ℳ𝑞∣𝐺=tr_𝑅𝑃exp_𝒦𝒜_𝐶𝑆, (1) that is, generic gauge-invariant observables in the simplest version of the 3-dimensional Yang-Mills theory and the topological Chern-Simons model [1] with the action: 𝑘4𝜋_ℳ2tr𝒜𝑑𝒜+3𝒜³, (2) and they depend on a closed contour 𝒦 in a three dimensional manifold ℳ, on the representation 𝑅 of the gauge group 𝐺=𝑆𝑈(𝑁) and on the coupling constant 𝑞=𝑒^{2𝜋𝑖/(𝑘+𝑁)}. Since the theory is topological, the dependence is actually only on the topological class of 𝒦; that is, the contour can be considered as a knot. For the simply connected space ℳ=𝑅³ or 𝑆³ the average 𝐻(𝑞∣𝐺) is actually a polynomial in 𝑞 and 𝐴=𝑞^𝑁, hence, the name “knot polynomial.”

The study of knot polynomials in topology goes back to [2–7], and they were put into the context of quantum field theory in the seminal works by Schwarz [8] and Witten [9], further developed in [10–13]. Since the theory is topological, there are no dynamical phenomena like confinement, instead a close relation exists to the 2-dimensional conformal theories [9, 14–18], very much in the spirit of AdS/CFT correspondence [19–21]. In this way the knot polynomials are related to the most difficult part of conformal field theory, to modular transformations, which through the AGT relations [22–32] are connected to the 𝑆-duality between the 𝒩=2 supersymmetric Yang-Mills theories [33–44]. This makes the study of knot polynomials the next task after the structure of conformal blocks themselves more or less understood in terms of the Dotsenko-Fateev matrix models [45–55] and other similar representations [56, 57].

An additional interest is induced by existence of nontrivial deformations of the knot polynomials: to superpolynomials [58–66] and refined Chern-Simons theory [67, 68] (see also [69]), to extended knot polynomials [70], which puts them into the class of 𝜏-function like objects, and so forth. At least naively [71, 72], they belong to the family of Hurwitz partition functions [73–76], more general than the conventional KP/Toda 𝜏-functions, probably related to the generalized 𝜏-functions of [77–80].

However, for any kind of generic investigation and application of knot polynomials, they should be first effectively calculated and represented in a theoretically appealing form, allowing evaluation of these polynomials for particular knots and representations. There are different competitive approaches to do this, for example, [81–84]. The goal of this letter is to summarize the results of our method [70, 85–87], which provides a complete, nice, and practically efficient solution to this problem.

2. HOMFLY Polynomials via Quantum ℛ-Matrices

The method may begin with choosing the temporal gauge 𝐴₀=0 [88–90] in Chern-Simons theory; then the theory becomes quadratic with the ultralocal propagator 𝜃(𝑡)𝛿(𝑥). Then the original knot in 3-dimensions is substituted by a 2-dimensional knot diagram (a 4-valent oriented graph), and the Wilson average reduces to a 𝑞-graded trace of the product of quantum ℛ-matrices, standing at the vertices of the graph [10–12].

It is most convenient to choose the knot diagram in the form of a closure of a braid. If the braid has 𝑚 strands, then the product involves 𝑚−1 different ℛ-matrices: ℛ_(𝑖) stands at the intersection of strands 𝑖 and 𝑖+1, and 𝑖=1,…,𝑚−1. For instance, for the 3-strand braid one has 𝐻^{(𝑎1,𝑏1|𝑎2,𝑏2}_𝑅^|,…)=tr^grad_𝑅⊗3ℛ^𝑎1₍₁₎ℛ^𝑏1₍₂₎ℛ^𝑎2₍₁₎ℛ^𝑏2₍₂₎⋯. (3) In the pattern picture 𝑎₁=0, 𝑏₁=−2, 𝑎₂=2, 𝑏₂=−1, and 𝑎₃=3 (see Figure 1).

Figure 1 

Similarly, for arbitrary 𝑚𝐻^{(𝑎11,…,𝑎1,𝑚−1|𝑎21,…,𝑎2,𝑚−1}_𝑅^|,…)=tr^grad_𝑅⊗𝑚ℛ^𝑎11₍₁₎⋯ℛ^{𝑎1,𝑚−1}_()𝑚−1ℛ^𝑎21₍₁₎⋯ℛ^{𝑎2,𝑚−1}_()𝑚−1⋯. (4) The trace here is defined with additionally inserted element 𝑞^𝑅⊗𝑚 so that 𝜒_𝑄=tr^grad_𝑄𝐼=dim^𝐺_𝑞(𝑄) (5) are the quantum dimensions of the representation 𝑅 (the characters of the group 𝐺 at the special values 𝑝_𝑘=(𝐴^𝑘−𝐴^−𝑘)/(𝑞^𝑘−𝑞^−𝑘)). In this formula the ℛ matrices are of the huge size dim(𝑅)²×dim(𝑅)² and this expression can seem absolutely hopeless to evaluate for generic group 𝐺 and representation 𝑅.

However, things are actually much simpler. The product 𝑅^⊗𝑚 can be expanded into a sum of irreducible representations, generically, with nontrivial multiplicities. The crucial observation is that ℛ_(𝑖) act as unity in each irreducible representation, so that the matrices in (4) can be reduced to ℛ_(𝑖) of a much smaller size, equal to just the multiplicities 𝑀_𝑅𝑄 of 𝑄 in 𝑅^⊗𝑚 [70, 85]: 𝐻^𝒦_𝑅(𝐺)=𝔑^𝑤(𝒦)_{𝑄∈𝑅⊗𝑚}𝐶^𝒦_𝑅𝑄𝜒_𝑄(𝐺),𝐶^{(𝑎11⋯𝑎1,𝑚−1∣⋯)}_𝑅𝑄=tr_𝑀𝑅𝑄ℛ^𝑎11₍₁₎⋯ℛ^{𝑎1,𝑚−1}_(𝑚−1)ℛ^𝑎21₍₁₎⋯ℛ^{𝑎2,𝑚−1}_(𝑚−1)⋯ (6)(7) Here 𝔑 is a normalization factor which emerges due to our choice of nonstandard normalization of ℛ-matrices and 𝑤(𝒦) is the writhe number. The standard normalization of ℛ-matrix in the vertical framing is restored with the factor 𝑞^−2𝜘𝑅, where 𝜘_𝑅=∑_{𝑖,𝑗∈𝑅}(𝑗−𝑖) and the sum runs over the coordinates (𝑖,𝑗) of the boxes in the Young diagram that corresponds to the representation 𝑅. In order to restore the topological invariance, one has to change framing with a factor of 𝐴^−|𝑅|𝑞^−2𝜘𝑅 which totally gives 𝔑=𝐴^−|𝑅|𝑞^−4𝜘𝑅.

What is important, in formula (7) the knot and group dependencies are separated and one can consider 𝐻_𝑅 as a function of 𝐴=𝑞^𝑁 rather than 𝑁; the parameter 𝑁 enters only the quantum dimensions 𝜒_𝑄 and can be easily substituted by 𝐴. Moreover, this formula actually introduces the extended HOMFLY polynomial 𝐻_𝑅{𝑝}, if 𝜒_𝑄 are interpreted as characters, which are functions of infinitely many time-variables {𝑝_𝑘} instead of 𝑁 or 𝐴 [70, 85]. The topological invariance is, however, lost beyond the topological locus 𝑝_𝑘=𝑝^∗_𝑘=(𝐴^𝑘−𝐴^−𝑘)/(𝑞^𝑘−𝑞^−𝑘).

To deal with this formula one needs an explicit expression for the ℛ-matrices; see [85, 91, 92]. Those papers contain many various observations about the structure of ℛ-matrices, and they were used to calculate many nontrivial knot polynomials; still a complete description is not yet found on that way, except for the fundamental representation case of 𝑅=[1]=□, fully described in [86], and for the (anti)symmetric representation case [91–93] (some results in this latter case are also reproduced by alternative methods [94–100]).

It is therefore natural to attack the case of arbitrary 𝑅 with the help of the cabling approach [101] and apply the results of [86]. This is successfully done in [87, 102]; this letter is a short summary of [86, 87], purified from all the details and extensive list of examples evaluated there.

Note also that here we restrict our discussion to the knots only. The links can be dealt with similarly; however, they require some technical complications; thus, for the sake of brevity, we skip this extension (see details in [87]).

3. ℛ-Matrices via Paths in the Representation Graph

Since the cabling reduces the problem from 𝑅 to the case of the fundamental representation, the results of [86] are directly applicable [87] and we begin from repeating them in a concise and pictorial form (see Figure 2).

Figure 2 

Implicit in [86] is representation of the coefficients 𝐶_□𝑄 in the form of a sum over paths in the representation graph [87]. The first four levels of the representation graph of [85] are shown in Figure 2; in an obvious way it describes the multiplication of fundamental representations [1]. The multiplicity 𝑀_□𝑄 of the representation 𝑄 in □^⊗|𝑄| is obviously equal to the number of directed paths in the representation graph, connecting □ and 𝑄. More generally, 𝑀_𝑅𝑄 is equal to the number of directed paths between 𝑅 and 𝑄. The matrices ℛ_(𝑖), 𝑖=1,…,𝑚−1 can be represented in the basis of paths between 𝑄 and □ and according to [86, 87] they have extremely simple form in this basis.

First of all, with each index 𝑖 of the matrix ℛ_(𝑖) one associates a level 𝑖 in the graph. A given path 𝒫 is passing through exactly one vertex 𝑃_𝑖 at level 𝑖 and through some two adjacent vertices 𝑃_𝑖−1 and 𝑃_𝑖+1 at levels 𝑖−1 and 𝑖+1. The structure of the representation graph is such that these 𝑃_𝑖−1 and 𝑃_𝑖+1 are connected either by a single two-segment path (singlet) (then it is a fragment of our 𝑃) or by two such paths (doublet), the segments of our path 𝒫 and another path 𝒫^′. We call the transformations 𝒫↔𝒫^′ a flip (see Figure 3).

Figure 3 

In the former case (singlet), our path 𝒫 provides a diagonal element in ℛ_(𝑖) and it is equal to either 𝑞 or −1/𝑞. In the language of Young diagrams the singlet appears when the two boxes added to the diagram 𝑃_𝑖−1 in order to form 𝑃_𝑖+1 lie either in the same row, then we put 𝑞 at the diagonal of ℛ_𝑖, or in the same column, then we put −1/𝑞 (see Figure 4).

Figure 4 

Example of singlets.

In the latter case (the doublet) (see Figure 5), the two boxes are neither in the same row nor in the same column, and the two paths 𝒫 and 𝒫^′ form a 2×2 block in ℛ_(𝑖). This block is described as follows. First, that of the paths 𝒫 and 𝒫^′ which lies to the left of the other, corresponds to the left column and to the first row of the 2×2 block. Second, the Young diagrams 𝑃_𝑖+1 is obtained by adding two boxes to the diagram 𝑃_𝑖−1, and the two paths correspond to doing this in two different orders, thus providing at the intermediate stage the two adjacent vertices 𝑃_𝑖 and 𝑃^′_𝑖. The two added boxes are connected by a hook in the Young diagram, which has length 𝑛 (measured between the centers of the two boxes). Then the 2×2 block is equal to −𝑞^−𝑛𝑐_𝑛𝑠_𝑛𝑠_𝑛𝑞^𝑛𝑐_𝑛,𝑐_𝑛=1[𝑛]_𝑞=𝑞−𝑞⁻¹𝑞^𝑛−𝑞^−𝑛,𝑠_𝑛=1−𝑐²_𝑛=[𝑛−1]_𝑞[𝑛+1]_𝑞[𝑛]_𝑞. (8) Figure 5 shows that for 𝑃_𝑖−1=[5521], 𝑃_𝑖+1=[6522], 𝑃_𝑖=[6521], and 𝑃^′_𝑖=[5522], the parameter 𝑛=7.

Figure 5 

Example of doublets.

This provides a complete and very explicit description of the matrices ℛ_(𝑖), 𝑖=1,…,𝑚−1, which appear under the trace in the expression for 𝐶_□𝑄. Note that in this form these matrices depend on 𝑄. This description can seem rather lengthy and tricky, but in practice it is very algorithmic, easily programmable, and very effective for practical calculations [87].

The paths from □ to 𝑄 in the representation graph are labeled by the standard Young tableaux, that is, the Young diagram 𝑄 with the numbers 1,…,|𝑄| assigned to its boxes in such a way that the emerging sequences are increasing along each row and each column. The number in a given box is just the number of steps, at which the box was attached to the diagram when moving from □ to 𝑄. The number of paths is therefore equal to the well-known number of the standard Young tableaux, 𝑀_□𝑄=|𝑄|!∏_boxes∈𝑄𝐿(),box (9) where 𝐿(box) is the length of the hook in 𝑄 associated to the box.

Thus the central formula (7) can be considered as a multiple sum over the standard Young tableaux of 𝑄, and the number of summations is equal to the number ∑_𝑖𝑗|𝑎_𝑖𝑗| of vertices in the braid.

4. Examples

To illustrate the use of these formulas, we calculate the contributions of simple diagrams 𝑄 to the simple HOMFLY polynomials. There is nothing new in these formulas, they are present here just to illustrate how the method works. For numerous new examples see, [87].

If one considers only 𝐻_□ in the fundamental representation 𝑅=□=[1], |□|=1, then (20) for an 𝑚-strand braid involves 𝑄 of the size 𝑚, that is, 𝑄 at the level 𝑚 in the representation graph. To make use of (20), one needs to know the 𝑚−1 matrices ℛ_(𝑖).

𝑚=2. There are two possible 𝑄=[2] and 𝑄=[11] and one matrix ℛ₍₁₎ in each case.

For 𝑄=[2] there is a single path from □=[1] to [2]; that is, 𝑀_□[2]=1; it coincides with the segment ([1],[2]); thus, according to our rules the corresponding 1×1 matrix ℛ₍₁₎=𝑞.

Similarly for 𝑄=[11] one gets ℛ₍₁₎=−1/𝑞.

Thus the generic expression (19) in this case is 𝐴^𝑛𝐻_□^(𝑛)=𝑞^𝑛𝜒_[2](𝐺)+−𝑞^−𝑛𝜒_[11](𝐺),𝑛odd. (10) The coefficient at the l.h.s. takes into account the normalization factor 𝔑.

𝑚=3. For 𝑄=[3]and[111] there are unique paths from □ and the corresponding matrices ℛ₍₁₎=ℛ₍₂₎ are again 1×1 and are equal to 𝑞 and −1/𝑞, respectively.

However, for 𝑄=[21] the situation is already different. There are two paths between □ and [21], and the left one [1] → [2] → [21] contains the segment ([1], [2]), while the second path [1] → [11] → [21] contains the segment ([1], [11]). This means that the matrix ℛ₍₁₎ in the sector 𝑄=[21] is 2×2; it is diagonal with the entries 𝑞 and −1/𝑞. The matrix ℛ₍₂₎ is again 2×2, but it is not diagonal, because the two paths are connected exactly by the flip. Since the length of the hook in this case is 𝑛=2, our rules imply that ℛ₍₂₎=1[2]_𝑞−1𝑞²[3]_𝑞[3]_𝑞𝑞², (11) and the formula for the arbitrary 3-strand braid is [85] (see Figure 6) 𝐴^{𝑎1+𝑏1+𝑎2+𝑏2+⋯}𝐻^{(𝑎1𝑏1|𝑎2𝑏2}_□^|⋯)=𝑞^{𝑎1+𝑏1+𝑎2+𝑏2+⋯}𝜒_[3](𝐺)+−1𝑞^{𝑎1+𝑏1+𝑎2+𝑏2+⋯}𝜒_[111](𝐺)+tr_2×21𝑞00−𝑞^𝑎1×−1𝑞²[2]_𝑞[3]_𝑞[2]_𝑞[3]_𝑞[2]_𝑞𝑞²[2]_𝑞^𝑏11𝑞00−𝑞^𝑎2×−1𝑞²[2]_𝑞[3]_𝑞[2]_𝑞[3]_𝑞[2]_𝑞𝑞²[2]_𝑞^𝑏2⋯, (12)

Figure 6 

𝑚=4. For 𝑄=[4]and[1111] all the three matrices ℛ_1,2,3 are 1×1 and equal to 𝑞 and −1/𝑞, respectively.

For 𝑄=[31] there are three paths. Ordered from the left to the right they are [1]⟶[2]⟶[3]⟶[],[1]⟶[2]⟶[]⟶[],[1]⟶[]⟶[]⟶[].𝛼=31𝛽=2131𝛾=112131 (13) At level 2 the flip relates 𝛽 and 𝛾 and at level 3 relates 𝛼 and 𝛽. This implies that in sector [31] one has [85] ℛ₍₁₎=𝑞𝑞−1𝑞,ℛ₍₂₎=1𝑞000−𝑞²[2]_𝑞[3]_𝑞[2]_𝑞0[3]_𝑞[2]_𝑞𝑞²[2]_𝑞,ℛ₍₃₎=−1𝑞³[3]_𝑞[2]_𝑞[4]_𝑞[3]_𝑞0[2]_𝑞[4]_𝑞[3]_𝑞𝑞³[3]_𝑞0.00𝑞 (14)

For 𝑄=[211] the answer is similar; the three paths are now 𝛼=[1]⟶[2]⟶[21]⟶[211],𝛽=[1]⟶[11]⟶[21]⟶[211],[1]⟶[]⟶[]⟶[],ℛ𝛾=11111211₍₁₎=𝑞−1𝑞−1𝑞,ℛ₍₂₎=−1𝑞²[2]_𝑞[3]_𝑞[2]_𝑞0[3]_𝑞[2]_𝑞𝑞²[2]_𝑞0100−𝑞,ℛ₍₃₎=−1𝑞1000−𝑞³[3]_𝑞[2]_𝑞[4]_𝑞[3]_𝑞0[2]_𝑞[4]_𝑞[3]_𝑞𝑞³[3]_𝑞. (15)(16)

For 𝑄=[22], there are just two paths: 𝛿=[1]⟶[2]⟶[21]⟶[22],𝛿=[1]⟶[11]⟶[21]⟶[22], (17) and they are related by the flip at level 2. This means that both ℛ₍₁₎ and ℛ₍₃₎ are 2×2 and diagonal, while ℛ₍₂₎ is not: ℛ₍₁₎=ℛ₍₃₎=1𝑞00−𝑞,ℛ₍₂₎=−1𝑞²[2]_𝑞[3]_𝑞[2]_𝑞[3]_𝑞[2]_𝑞𝑞²[2]_𝑞. (18)

The counterpart of (10) and (12) is now obvious, but rather lengthy, so we do not write it down here.

5. Cabling Method

Cabling is based on the fact that the representation 𝑅 appears in the product of |𝑅| fundamental representations, 𝑅∈□^|𝑅|; thus, the answer for 𝐻^𝒦_𝑅 can be extracted from the answer for 𝐻^𝒦|𝑅|_□⊗|𝑅| by the projection: 𝐻^𝒦_𝑅(𝐺)=𝔑_{𝑐𝑄∈□⊗𝑚|𝑅|}ℭ^𝒦|𝑅|_□𝑄𝜒_𝑄(𝐺), (19) where the normalization factor is slightly corrected as compared with (7), see [87] for details. Cabling is widely used not only in the knot theory [101], but also in the theory of ℛ-matrices and integrable systems [103–106] (where it is called the fusion procedure).

Here 𝒦^|𝑅| is an 𝑚|𝑅|-strand braid, obtained from 𝒦 by substituting each line (strand) by a bunch (cable) of |𝑅| strands so that the intersection of two strands is now a peculiar combination ℜ of |𝑅|² original ℛ-matrices, but in the fundamental representations (see Figure 7). In other words, ℭ^𝒦|𝑅|_□𝑄=tr_𝑀□𝑄𝔓_𝑅ℜ^𝑎11₍₁₎⋯ℜ^{𝑎1,𝑚−1}_(𝑚−1)ℜ^𝑎21₍₁₎⋯ℜ^{𝑎2,𝑚−1}_(𝑚−1)⋯ (20)𝔓_𝑅 is the projection from the reducible representation □^⊗|𝑅| onto 𝑅. It can actually be inserted everywhere in between the ℜ matrices, but for the case of knots (not links) a single insertion is sufficient.

Figure 7 

Now, (20) can be written explicitly and calculated with the help of Section 3, but in terms of the 𝑚|𝑅|-strand braid and the corresponding ℛ matrices replaced with ℜ. This is a rather cumbersome expression even for the simplest knots (therefore we do not rewrite (20) in this way) but absolutely straightforward and adequate for practical computations.

Moreover, here comes a bonus of the path sum representation: the projector 𝔓_𝑅 is nearly trivial, at least in the case of knots: one should include only the directed paths from 𝑄 to □, which pass through the vertex 𝑅, this effectively decreases the size of the matrices ℜ_(𝐼), 𝐼=1,…,𝑚−1 from 𝑀_□𝑄 to 𝑀_□𝑅⋅𝑀_𝑅𝑄.

NB. The constituent matrices ℛ_(𝑖), 𝑖=1,…,𝑚|𝑅|−1, cannot be reduced in this way: one cannot insert the projector inside  ℜ.

6. Cabling for 𝑚=2 and |𝑅|=2

Now we can use the last example in Section 4 to demonstrate how the cabling works in our formulas. The 4-strand braids are enough to describe only the 2-strand (torus) knots in representations [2] and [11]. In this case there is just a single combined matrix: ℜ₍₁₎=ℛ₍₂₎ℛ₍₁₎ℛ₍₃₎ℛ₍₂₎. (21)

According to our rules, if we consider, for example, 𝐻^(𝑛)_[2] for a 2-strand knot, we should leave in ℜ₍₁₎ only the paths passing through the vertex [2].

This means that the single path leading to 𝑄=[4] remains intact, while the one to [1111] is now eliminated; this means that there will be no contribution of 𝑄=[1111] to 𝐻^(𝑛)_[2].

From the three paths which led to 𝑄=[31], only two remain, 𝛼 and 𝛽; thus the third line and the third column should be omitted from the matrix ℜ: 1ℜ=𝑞000−𝑞²[2]_𝑞[3]_𝑞[2]_𝑞0[3]_𝑞[2]_𝑞𝑞²[2]_𝑞𝑞𝑞−1𝑞×−1𝑞³[3]_𝑞[2]_𝑞[4]_𝑞[3]_𝑞0[2]_𝑞[4]_𝑞[3]_𝑞𝑞³[3]_𝑞0×100𝑞𝑞000−𝑞²[2]_𝑞[3]_𝑞[2]_𝑞0[3]_𝑞[2]_𝑞𝑞²[2]_𝑞=−1[3]_𝑞−[2]_𝑞[4]_𝑞[2]_𝑞[3]_𝑞𝑞²[2]_𝑞[3]_𝑞[4]_𝑞𝑞²[2]_𝑞[3]_𝑞−[2]_𝑞[4]_𝑞[2]_𝑞[3]_𝑞−[4]_𝑞[2]_𝑞[3]_𝑞−𝑞²[3]_𝑞𝑞²[2]_𝑞[3]_𝑞[4]_𝑞[2]_𝑞[3]_𝑞−𝑞²[3]_𝑞0⟶𝔓_[2]ℜ𝔓_[2]=110ℜ110⟶−1[3]_𝑞−[2]_𝑞[4]_𝑞[2]_𝑞[3]_𝑞−[2]_𝑞[4]_𝑞[2]_𝑞[3]_𝑞−[4]_𝑞[2]_𝑞[3]_𝑞. (22) This 2×2 matrix has two eigenvalues: 0 and −1.

Similarly from the three paths to 𝑄=[211] only one survives, 𝛼, and ℜ matrix is reduced to 1×1: −1ℜ=𝑞²[2]_𝑞[3]_𝑞[2]_𝑞0[3]_𝑞[2]_𝑞𝑞²[2]_𝑞0100−𝑞𝑞−1𝑞−1𝑞×−1𝑞1000−𝑞³[3]_𝑞[2]_𝑞[4]_𝑞[3]_𝑞0[2]_𝑞[4]_𝑞[3]_𝑞𝑞³[3]_𝑞×−1𝑞²[2]_𝑞[3]_𝑞[2]_𝑞0[3]_𝑞[2]_𝑞𝑞²[2]_𝑞0100−𝑞=01𝑞²[3]_𝑞[2]_𝑞[3]_𝑞[4]_𝑞𝑞²[2]_𝑞[3]_𝑞1𝑞²[3]_𝑞−[4]_𝑞[2]_𝑞[3]_𝑞[2]_𝑞[4]_𝑞[2]_𝑞[3]_𝑞[2]_𝑞[3]_𝑞[4]_𝑞𝑞²[2]_𝑞[3]_𝑞[2]_𝑞[4]_𝑞[2]_𝑞[3]_𝑞−1[3]_𝑞⟶𝔓_[2]ℜ𝔓_[2]=100ℜ100⟶(0). (23) Thus, despite a path survives in the [211] sector, its contribution is actually vanishing, as it should be, because [211]∉[2]⊗[2].

Also for 𝑄=[22] only one path of the two remains, 𝛿, and −1𝑞²[2]_𝑞[3]_𝑞[2]_𝑞[3]_𝑞[2]_𝑞𝑞²[2]_𝑞1𝑞00−𝑞²×−1𝑞²[2]_𝑞[3]_𝑞[2]_𝑞[3]_𝑞[2]_𝑞𝑞²[2]_𝑞=1𝑞²00𝑞²⟶𝔓_[2]ℜ𝔓_[2]=10ℜ10⟶1𝑞². (24)

Putting everything together one gets for odd 𝑛 (i.e., for the knot): 𝑞^2𝑛𝐴^2𝑛𝐻^(𝑛)_[2]=tr^grad_[1]⊗4𝔓_[2]ℜ^𝑛=tr^grad_[1]⊗4𝔓_[2]ℜ𝔓_[2]^𝑛=𝑞^4𝑛𝜒_[4](𝐺)+tr_2×2×−1[3]_𝑞−[2]_𝑞[4]_𝑞[2]_𝑞[3]_𝑞−[2]_𝑞[4]_𝑞[2]_𝑞[3]_𝑞−[4]_𝑞[2]_𝑞[3]_𝑞^𝑛𝜒_[31](𝐺)+𝑞^−2𝑛𝜒_[22](𝐺)=𝑞^−2𝑛𝑞^6𝑛𝜒_[4](𝐺)−𝑞^2𝑛𝜒_[31](𝐺)+𝜒_[22](𝐺). (25) The formula at the last line is the standard Rosso-Jones expression [107–109] for the colored HOMFLY polynomial in the case of a torus knot.

NB. Note that the matrices ℜ  do not commute with 𝔓_𝑅 and that the projector is as simple as described above only for the first cable in the braid.

Note also that two of the five 𝑄∈[1]⁴ do not contribute to the middle line for somewhat different reasons; 𝜒_[1111] does not appear, because there are no paths, going from □ to [1111] via [2], while for 𝜒_[211] such path exists, just its contribution to the relevant element of ℜ is zero.

These examples illustrate all the peculiarities of our formulas. Once they are understood, the use of the path sum formalism is straightforward.

7. Open Questions

The path sum formula provides a complete solution for the evaluation problem of arbitrary colored HOMFLY polynomials. It is nice looking and theoretically attractive, and it is algorithmic and very effective for practical computations (a vast list of examples is provided in [87, 102]). It represents the knot polynomials in the form of a character expansion [70, 85] and therefore expresses them directly in terms of 𝑁, 𝐴, or, the time-variables 𝑝_𝑘, whatever one prefers. Still there are a few obvious directions to study.

First, our final formula for the colored knots heavily relies on the cabling approach and therefore is in a certain sense more involved than (7). In particular, when 𝑅≠□, the sizes of ℛ-matrices are considerably bigger than theoretically possible (though the entries are simpler). Of course, the path sum representation exists for the arbitrary 𝑅, but it is in terms of another representation graph, Γ_𝑅 describing powers of the representation 𝑅, and the corresponding ℛ-matrices are more involved (their constituents are no longer just 1×1 and 2×2 blocks). At the same time that graph is the subset of the full one, Γ_□, which we considered in the present letter, and the sum over the paths from 𝑄 to □, passing through the vertex 𝑅, can be also considered as a sum over just the paths from 𝑄 to 𝑅, more in the spirit of (7). However, there are many such paths in Γ_□ with the same image in Γ_𝑅, and they contribute to the matrix elements of the composite ℜ. The quantities (20) contain also multiple sums over the paths from 𝑅 to □, but in fact they do not depend on this, and one can simply fix one such path arbitrarily (as a kind of a gauge fixing). It would be very interesting to find a relevant modification of the path sum formula which would not refer to the cabling procedure, and [91, 92] strongly implies that this can be possible and that such a formula should possess its own beauties.

Second, the representation graph exists for arbitrary Lie algebras, not only for 𝑆𝑈(𝑁), and the sum path formula should be straightforwardly generalized to arbitrary groups, in particular, from the HOMFLY (for 𝑆𝑈(𝑁)) to Kauffman (for 𝑆𝑂(𝑁)) polynomials.

Third, it is now obvious that the Khovanov-Rozansky approach [58–63] and the superpolynomials [64, 65] should have a similar representations in terms of the multiple sums over paths (i.e., over the standard Young tableaux). There are certain advances in this direction for the torus knots [110–115], particularly close should be the results by [116, 117]. We emphasize once again that the sums over Young tableaux provide the answers for arbitrary knots, not only torus; this is a theorem for the HOMFLY polynomials and a plausible conjecture for the superpolynomials.

To this list one should of course add a massive calculation of the colored HOMFLY polynomials for nontorus knots, especially in representations with many lines and rows in the Young diagram, that is, not just (anti)symmetric representations. Such examples are crucially important for understanding the structure of generic knot polynomials, for example, a la [93–100, 102, 118, 119] or [71, 72, 120, 121] or [122], and various relations between them [123–127].

Acknowledgments

The authors are indebted to S. Artamonov, H. Itoyama, and A. Popolitov for fruitful discussions, which helped them to improve the presentation of the results from [86, 87] for the purposes of this paper. Thier work is partly supported by the Ministry of Education and Science of the Russian Federation under contract 8207, by the Brazil National Counsel of Scientific and Technological Development (A. Morozov), by the Dynasty Foundation (And. Morozov), by NSh-3349.2012.2, by RFBR Grants 13-02-00457 (A. Mironov), 13-02-00478 (Al. Morozov), and 12-02-31078 mol-a (A. Anokhina; And. Morozov), by joint Grant 12-02-92108-Yaf-a, 13-02-90459-Ukr-f-a, and 13-02-91371-ST-a.