Review Article | Open Access

A. Anokhina, A. Mironov, A. Morozov, And. Morozov, "Colored HOMFLY Polynomials as Multiple Sums over Paths or Standard Young Tableaux", *Advances in High Energy Physics*, vol. 2013, Article ID 931830, 12 pages, 2013. https://doi.org/10.1155/2013/931830

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

**Academic Editor:**Kadayam S. Viswanathan

#### Abstract

If a knot is represented by an -strand braid, then HOMFLY polynomial in representation is a sum over characters in all representations . Coefficients in this sum are traces of products of quantum -matrices along the braid, but these matrices act in the space of intertwiners, and their size is equal to the multiplicity of in . If is the fundamental representation , then is equal to the number of paths in representation graph, which lead from the fundamental vertex to the vertex . In the basis of paths the entries of the relevant -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 -matrices consist of just and blocks, given by very simple explicit expressions. If cabling method is used to color the knot with the representation , then the braid has as many as strands; have a bigger size , but only paths passing through the vertex are included into the sums over paths which define the products and traces of the relevant -matrices. In the case of , 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: 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: and they depend on a closed contour in a three dimensional manifold , on the representation of the gauge group and on the coupling constant . Since the theory is topological, the dependence is actually only on the topological class of ; that is, the contour can be considered as a knot. For the simply connected space or the average is actually a polynomial in and , 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 -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 -duality between the 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 -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 -functions, probably related to the generalized -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 -Matrices

The method may begin with choosing the temporal gauge [88–90] in Chern-Simons theory; then the theory becomes quadratic with the ultralocal propagator . 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 -graded trace of the product of quantum -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 strands, then the product involves different -matrices: stands at the intersection of strands and , and . For instance, for the -strand braid one has In the pattern picture , , , , and (see Figure 1).

Similarly, for arbitrary The trace here is defined with additionally inserted element so that are the quantum dimensions of the representation (the characters of the group at the special values ). In this formula the matrices are of the huge size and this expression can seem absolutely hopeless to evaluate for generic group and representation .

However, things are actually much simpler. The product can be expanded into a sum of irreducible representations, generically, with nontrivial multiplicities. The crucial observation is that act as unity in each irreducible representation, so that the matrices in (4) can be reduced to of a much smaller size, equal to just the multiplicities of in [70, 85]: Here is a normalization factor which emerges due to our choice of nonstandard normalization of -matrices and is the writhe number. The standard normalization of -matrix in the vertical framing is restored with the factor , where and the sum runs over the coordinates of the boxes in the Young diagram that corresponds to the representation . In order to restore the topological invariance, one has to change framing with a factor of which totally gives .

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

To deal with this formula one needs an explicit expression for the -matrices; see [85, 91, 92]. Those papers contain many various observations about the structure of -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 , 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 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. -Matrices via Paths in the Representation Graph

Since the cabling reduces the problem from 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).

Implicit in [86] is representation of the coefficients 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 of the representation in is obviously equal to the number of directed paths in the representation graph, connecting and . More generally, is equal to the number of directed paths between and . The matrices , can be represented in the basis of paths between and and according to [86, 87] they have extremely simple form in this basis.

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

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

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 and form a block in . This block is described as follows. First, that of the paths and which lies to the left of the other, corresponds to the left column and to the first row of the block. Second, the Young diagrams is obtained by adding two boxes to the diagram , and the two paths correspond to doing this in two different orders, thus providing at the intermediate stage the two adjacent vertices and . The two added boxes are connected by a hook in the Young diagram, which has length (measured between the *centers* of the two boxes). Then the block is equal to
Figure 5 shows that for , , , and , the parameter .

This provides a complete and very explicit description of the matrices , , which appear under the trace in the expression for . Note that in this form these matrices depend on . 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 in the representation graph are labeled by the standard Young tableaux, that is, the Young diagram with the numbers 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 . The number of paths is therefore equal to the well-known number of the standard Young tableaux, where is the length of the hook in associated to the box.

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

#### 4. Examples

To illustrate the use of these formulas, we calculate the contributions of simple diagrams 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 in the fundamental representation , , then (20) for an -strand braid involves of the size , that is, at the level in the representation graph. To make use of (20), one needs to know the matrices .

. There are two possible and and one matrix in each case.

For there is a single path from to ; that is, ; it coincides with the segment ; thus, according to our rules the corresponding matrix .

Similarly for one gets .

Thus the generic expression (19) in this case is The coefficient at the l.h.s. takes into account the normalization factor .

*.* For there are unique paths from and the corresponding matrices are again and are equal to and , respectively.

However, for 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 in the sector is ; it is diagonal with the entries and . The matrix is again , 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 , our rules imply that and the formula for the arbitrary 3-strand braid is [85] (see Figure 6)

. For all the three matrices are and equal to and , respectively.

For there are three paths. Ordered from the left to the right they are At level 2 the flip relates and and at level 3 relates and . This implies that in sector one has [85]

For the answer is similar; the three paths are now

For , there are just two paths: and they are related by the flip at level 2. This means that both and are and diagonal, while is not:

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 appears in the product of fundamental representations, ; thus, the answer for can be extracted from the answer for by the projection: 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 -matrices and integrable systems [103–106] (where it is called the fusion procedure).

Here is an -strand braid, obtained from by substituting each line (strand) by a bunch (cable) of strands so that the intersection of two strands is now a peculiar combination of original -matrices, but in the fundamental representations (see Figure 7). In other words, is the projection from the reducible representation onto . It can actually be inserted everywhere in between the matrices, but for the case of knots (not links) a single insertion is sufficient.

Now, (20) can be written explicitly and calculated with the help of Section 3, but in terms of the -strand braid and the corresponding matrices replaced with . 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 ** is nearly trivial,* at least in the case of knots: one should include only the directed paths from to , which pass through the vertex , this effectively decreases the size of the matrices , from to .

*NB.* The constituent matrices , , cannot be reduced in this way: one cannot insert the projector *inside* .

#### 6. Cabling for and

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:

According to our rules, if we consider, for example, for a -strand knot, we should leave in only the paths passing through the vertex .

This means that the single path leading to remains intact, while the one to is now eliminated; this means that there will be no contribution of to .

From the three paths which led to , only two remain, and ; thus the third line and the third column should be omitted from the matrix : This matrix has two eigenvalues: 0 and −1.

Similarly from the three paths to only one survives, , and matrix is reduced to : Thus, despite a path survives in the [211] sector, its contribution is actually vanishing, as it should be, because .

Also for only one path of the two remains, , and

Putting everything together one gets for odd (i.e., for the knot): 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 *do not* commute with 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 do not contribute to the middle line for somewhat different reasons; does not appear, because there are no paths, going from to [1111] via [2], while for such path exists, just its contribution to the relevant element of 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 , , or, the time-variables , 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 , the sizes of -matrices are considerably bigger than theoretically possible (though the entries are simpler). Of course, *the path sum representation exists for the arbitrary *, but it is in terms of another representation graph, describing powers of the representation , and the corresponding -matrices are more involved (their constituents are no longer just and blocks). At the same time that graph is the subset of the full one, , which we considered in the present letter, and the sum over the paths from to , passing through the vertex , can be also considered as a sum over just the paths from to , more in the spirit of (7). However, there are many such paths in with the same image in , and they contribute to the matrix elements of the composite . The quantities (20) contain also multiple sums over the paths from 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 , and the sum path formula should be straightforwardly generalized to arbitrary groups, in particular, from the HOMFLY (for ) to Kauffman (for ) 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.

#### References

- S. S. Chern and J. Simons, “Characteristic forms and geometric invariants,”
*Annals of Mathematics*, vol. 99, pp. 48–69, 1974. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. W. Alexander, “Topological invariants of knots and links,”
*Transactions of the American Mathematical Society*, vol. 30, no. 2, pp. 275–306, 1928. View at: Publisher Site | Google Scholar | MathSciNet - J. H. Conway, “Algebraic properties,” in
*Proceedings of the AIP Conference on Computational Problems in Abstract Algebra*, J. Leech, Ed., pp. 329–358, Pergamon Press, Oxford, UK, 1970. View at: Google Scholar - V. F. R. Jones, “Index for subfactors,”
*Inventiones Mathematicae*, vol. 72, no. 1, pp. 1–25, 1983. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. H. Kauffman, “State models and the jones polynomial,”
*Topology*, vol. 26, no. 3, pp. 395–407, 1987. View at: Google Scholar - P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett, and A. Ocneanu, “A new polynomial invariant of knots and links,”
*American Mathematical Society Bulletin*, vol. 12, no. 2, pp. 239–246, 1985. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. H. Przytycki and P. Traczyk, “Invariants of links of Conway type,”
*Kobe Journal of Mathematics*, vol. 4, no. 2, pp. 115–139, 1988. View at: Google Scholar | MathSciNet - A. S. Schwarz, “New topological invariants arising in the theory of quantized fields,” in
*Proceedings of the International Topological Conference, Part II*, Baku, Azerbaijan, 1987. View at: Google Scholar - E. Witten, “Quantum field theory and the Jones polynomial,”
*Communications in Mathematical Physics*, vol. 121, no. 3, pp. 351–399, 1989. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. Guadagnini, M. Martellini, and M. Mintchev, “Chern-Simons field theory and quantum groups,” in
*Quantum Groups*, vol. 370, pp. 307–317, Springer, Berlin, Germany, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. Guadagnini, M. Martellini, and M. Mintchev, “Chern-Simons holonomies and the appearance of quantum groups,”
*Physics Letters B*, vol. 235, no. 3-4, pp. 275–281, 1990. View at: Google Scholar - N. Y. Reshetikhin and V. G. Turaev, “Ribbon graphs and their invaraints derived from quantum groups,”
*Communications in Mathematical Physics*, vol. 127, no. 1, pp. 1–26, 1990. View at: Publisher Site | Google Scholar - M. Alvarez, J. M. F. Labastida, and E. Pérez, “Vassiliev invariants for links from Chern-Simons perturbation theory,”
*Nuclear Physics B*, vol. 488, no. 3, pp. 677–718, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Moore and N. Seiberg, “Taming the conformal zoo,”
*Physics Letters B*, vol. 220, no. 3, pp. 422–430, 1989. View at: Google Scholar - V. V. Fock and Y. I. Kogan, “Generating function for $2+1$d WZW model correlators and the Sugawara construction from the $N$ Chern-Simons theory,”
*Modern Physics Letters A*, vol. 5, no. 17, pp. 1365–1371, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. M. F. Labastida and A. V. Ramallo, “Operator formalism for Chern-Simons theories,”
*Physics Letters B*, vol. 227, no. 1, p. 92, 1989. View at: Publisher Site | Google Scholar - J. M. F. Labastida and A. V. Ramallo, “Chern-Simons and conformal field theories,”
*Nuclear Physics B*, vol. 16, pp. 594–596, 1990. View at: Google Scholar - L. Alvarez-Gaumé, C. Gomez, and G. Sierra, “Quantum group interpretation of some conformal field theories,”
*Physics Letters B*, vol. 220, no. 1-2, pp. 142–152, 1989. View at: Google Scholar - J. Maldacena, “The large ${A}_{N-1}$ limit of superconformal field theories and supergravity,”
*Advances in Theoretical and Mathematical Physics*, vol. 2, no. 2, pp. 231–252, 1998. View at: Google Scholar | Zentralblatt MATH | MathSciNet - S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, “Gauge theory correlators from non-critical string theory,”
*Physics Letters B*, vol. 428, no. 1-2, pp. 105–114, 1998. View at: Publisher Site | Google Scholar | MathSciNet - E. Witten, “Anti de Sitter space and holography,”
*Advances in Theoretical and Mathematical Physics*, vol. 2, no. 2, pp. 253–291, 1998. View at: Google Scholar | Zentralblatt MATH | MathSciNet - L. F. Alday, D. Gaiotto, and Y. Tachikawa, “Liouville correlation functions from four-dimensional gauge theories,”
*Letters in Mathematical Physics A*, vol. 91, no. 2, pp. 167–197, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - N. Wyllard, “$N=2$ conformal Toda field theory correlation functions from conformal $\mathrm{SU}(N)$$N=2$ quiver gauge theories,”
*Journal of High Energy Physics*, no. 11, p. 002, 2009. View at: Publisher Site | Google Scholar | MathSciNet - A. Mironov and A. Morozov, “The power of Nekrasov functions,” http://arxiv.org/abs/0908.2190. View at: Google Scholar
- A. Mironov and A. Morozov, “On AGT relation in the case of U(3),” http://arxiv.org/abs/0908.2569. View at: Google Scholar
- A. Mironov, S. Mironov, A. Morozov, and A. Morozov, “Resolving puzzles of massive gravity with and without violation of Lorentz symmetry,”
*Classical and Quantum Gravity*, vol. 27, no. 12, Article ID 125005, 2010. View at: Publisher Site | Google Scholar - G. Bonelli and A. Tanzini, “Hitchin systems, $N=2$ gauge theories and W-gravity,” http://arxiv.org/abs/0909.4031. View at: Google Scholar
- G. Giribet, “On AGT description of ${N}_{f}=4$ SCFT with $N=2$,”
*Journal of High Energy Physics*, no. 1, p. 097, 18, 2010. View at: Google Scholar | Zentralblatt MATH | MathSciNet - K. Maruyoshi and M. Taki, “Deformed prepotential, quantum integrable system and Liouville field theory,” http://arxiv.org/abs/1006.4505. View at: Google Scholar
- A. Marshakov, A. Mironov, and A. Morosov, “On AGT relations with surface operator insertion and a stationary limit of beta-ensembles,”
*Journal of Geometry and Physics*, vol. 61, no. 7, pp. 1203–1222, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Kozcaz, S. Pasquetti, and N. Wyllard, “A & B model approaches to surface operators and Toda theories,” http://arxiv.org/abs/1004.2025. View at: Google Scholar
- J.-E. Bourgine, “Large N techniques for Nekrasov partition functions and AGT conjecture,” http://arxiv.org/abs/1212.4972. View at: Google Scholar
- N. Seiberg and E. Witten, “Electric-magnetic duality, monopole condensation, and confinement in $N=2$ supersymmetric Yang-Mills theory,”
*Nuclear Physics B*, vol. 426, no. 1, pp. 19–52, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - N. Seiberg and E. Witten, “Monopoles, duality and chiral symmetry breaking in $N=2$ supersymmetric QCD,”
*Nuclear Physics B*, vol. 431, no. 3, pp. 484–550, 1994. View at: Google Scholar - A. Hanany and Y. Oz, “On the quantum moduli space of vacua of $N=2$ supersymmetric $N=2$ gauge theories,”
*Nuclear Physics B*, vol. 452, no. 1-2, pp. 283–312, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Minahan and D. Nemeschansky, “Hyperelliptic curves for supersymmetric Yang-Mills,” http://arxiv.org/abs/hep-th/9507032. View at: Google Scholar
- J. Minahan and D. Nemeschansky, “$N=2$ Super Yang-Mills and Subgroups of SL(2,Z),” http://arxiv.org/abs/hep-th/9601059. View at: Google Scholar
- O. Aharony and S. Yankielowicz, “Exact electric-magnetic duality in $N=2$ supersymmetric QCD theories,”
*Nuclear Physics B*, vol. 473, no. 1-2, pp. 93–120, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P. Argyres, “S-duality and global symmetries in $N=2$ supersymmetric field theory,” http://arxiv.org/abs/hep-th/9706095. View at: Google Scholar
- P. Argyres, “Dualities in supersymmetric field theories,” http://arxiv.org/abs/hep-th/9705076. View at: Google Scholar
- J. A. Minahan, “Duality symmetries for $N=2$ supersymmetric QCD with vanishing $R$-functions,”
*Nuclear Physics B*, vol. 537, no. 1–3, pp. 243–259, 1999. View at: Publisher Site | Google Scholar | MathSciNet - D. Galakhov, A. Mironov, and A. Morozov, “S-duality as a beta-deformed Fourier transform,”
*Journal of High Energy Physics*, vol. 2012, p. 67, 2012. View at: Publisher Site | Google Scholar - M. Billo, M. Frau, L. Gallot, A. Lerda, and I. Pesando, “Deformed $N=2$ theories, generalized recursion relations and S-duality,” http://arxiv.org/abs/1302.0686. View at: Google Scholar
- N. Nemkov, “S-duality as Fourier transform for arbitrary e_1,e_2,” http://arxiv-web3.library.cornell.edu/abs/1307.0773. View at: Google Scholar
- V. S. Dotsenko and V. A. Fateev, “Conformal algebra and multipoint correlation functions in 2D statistical models,”
*Nuclear Physics B*, vol. 240, no. 3, pp. 312–348, 1984. View at: Google Scholar - A. Marshakov, A. Mironov, and A. Morozov, “Generalized matrix models as conformal field theories. Discrete case,”
*Physics Letters B*, vol. 265, no. 1-2, pp. 99–107, 1991. View at: Google Scholar - S. Kharchev, A. Marshakov, A. Mironov, A. Morozov, and S. Pakuliak, “Conformal matrix models as an alternative to conventional multi-matrix models,”
*Nuclear Physics B*, vol. 404, no. 3, pp. 717–750, 1993. View at: Google Scholar - R. Dijkgraaf and C. Vafa, “Toda theories, matrix models, topological strings, and $N=2$ gauge systems,” http://arxiv.org/abs/0909.2453. View at: Google Scholar
- H. Itoyama, K. Maruyoshi, and T. Oota, “The Quiver matrix model and 2d-4d conformal connection,”
*Progress of Theoretical Physics*, vol. 123, no. 6, pp. 957–987, 2010. View at: Google Scholar - T. Eguchi and K. Maruyoshi, “Penner type matrix model and Seiberg-Witten theory,” http://arxiv.org/abs/0911.4797. View at: Google Scholar
- T. Eguchi and K. Maruyoshi, “Seiberg-Witten theory, matrix model and AGT relation,” http://arxiv.org/abs/1006.0828. View at: Google Scholar
- R. Schiappa and N. Wyllard, “An A_r threesome: matrix models, 2d CFTs and 4d $N=2$ gauge theories,” http://arxiv.org/abs/0911.5337. View at: Google Scholar
- A. Mironov, A. Morozov, and Sh. Shakirov, “Matrix model conjecture for exact BS periods and Nekrasov functions,” http://arxiv.org/abs/0911.5721. View at: Google Scholar
- A. Mironov, A. Morozov, and Sh. Shakirov, “Conformal blocks as Dotsenko-Fateev integral discriminants,” http://arxiv.org/abs/1001.0563. View at: Google Scholar
- A. Mironov, A. Morozov, and A. Morozov, “Conformal blocks and generalized Selberg integrals,”
*Nuclear Physics B*, vol. 843, no. 2, pp. 534–557, 2011. View at: Publisher Site | Google Scholar - V. A. Alba, V. A. Fateev, A. V. Litvinov, and G. M. Tarnopolskiy, “On combinatorial expansion of the conformal blocks arising from AGT conjecture,”
*Letters in Mathematical Physics*, vol. 98, no. 1, pp. 33–64, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Belavin and V. Belavin, “Agt conjecture and integrable structure of conformal field theory for c=1,”
*Nuclear Physics B*, vol. 850, no. 1, pp. 199–213, 2011. View at: Publisher Site | Google Scholar - M. Khovanov, “A categorification of the Jones polynomial,”
*Duke Mathematical Journal*, vol. 101, no. 3, pp. 359–426, 2000. View at: Google Scholar - D. Bar-Natan, “On Khovanov's categorification of the Jones polynomial,”
*Algebraic & Geometric Topology*, vol. 2, pp. 337–370, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Khovanov and L. Rozhansky, “Matrix factorizations and link homology,” http://arxiv.org/abs/math/0401268. View at: Google Scholar
- M. Khovanov and L. Rozhansky, “Matrix factorizations and link homology II,” http://arxiv.org/abs/math/0505056. View at: Google Scholar
- N. Carqueville and D. Murfet, “Computing Khovanov-Rozansky homology and defect fusion,” http://arxiv.org/abs/1108.1081. View at: Google Scholar
- V. Dolotin and A. Morozov, “Introduction to Khovanov homologies I. Unreduced Jones superpolynomial,”
*Journal of High Energy Physics*, no. 1, p. 065, front matter + 46, 2013. View at: Google Scholar | MathSciNet - S. Gukov, A. Schwarz, and C. Vafa, “Khovanov-Rozansky homology and topological strings,”
*Letters in Mathematical Physics*, vol. 74, no. 1, pp. 53–74, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - N. M. Dunfield, S. Gukov, and J. Rasmussen, “The superpolynomial for knot homologies,”
*Experimental Mathematics*, vol. 15, no. 2, pp. 129–160, 2006. View at: Google Scholar - P. Dunin-Barkowski, A. Mironov, A. Morozov, A. Sleptsov, and A. Smirnov, “Superpolynomials for toric knots from evolution induced by cut-and-join operators,” http://arxiv.org/abs/1106.4305. View at: Google Scholar
- M. Aganagic and Sh. Shakirov, “Knot Homology from Refined Chern-Simons theory,” http://arxiv.org/abs/1105.5117. View at: Google Scholar
- M. Aganagic and Sh. Shakirov, “Refined Chern-Simons theory and topological string,” http://arxiv.org/abs/1210.2733. View at: Google Scholar
- A. Morozov, “Challenges of beta-deformation,” http://arxiv.org/abs/1201.4595. View at: Google Scholar
- A. Mironov, A. Morozov, and A. Morozov, “Character expansion for HOMFLY polynomials. I. Integrability and difference equations,” http://arxiv.org/abs/1112.5754. View at: Google Scholar
- A. Mironov, A. Morozov, and A. Sleptsov, “Genus expansion of HOMFLY polynomials,” http://arxiv.org/abs/1303.1015. View at: Google Scholar
- A. Morozov, “Integrability in non-perturbative QFT,” http://arxiv.org/abs/1303.2578. View at: Google Scholar
- A. Mironov, A. Morozov, and S. Natanzon, “Complete set of cut-and-join operators in Hurwitz-Kontsevich theory,” http://arxiv.org/abs/0904.4227. View at: Google Scholar
- A. Mironov, A. Morozov, and S. Natanzon, “Algebra of differential operators associated with Young diagrams,” http://arxiv.org/abs/1012.0433. View at: Google Scholar
- A. Mironov, A. Morozov, and S. Natanzon, “Integrability properties of Hurwitz partition functions. II. Multiplication of cut-and-join operators and WDVV equations,” http://arxiv.org/abs/1108.0885. View at: Google Scholar
- A. Alexandrov, A. Mironov, A. Morozov, and S. Natanzon, “Integrability of Hurwitz partition functions. I. Summary,” http://arxiv.org/abs/1103.4100. View at: Google Scholar
- A. Gerasimov, S. Khoroshkin, D. Lebedev, A. Mironov, and A. Morozov, “Generalized Hirota equations and representation theory. I. The case of SL(2) and SL_q(2),” http://arxiv.org/abs/hep-th/9405011. View at: Google Scholar
- S. Kharchev, A. Mironov, and A. Morozov, “Non-standard KP evolution and quantum t-function,” http://arxiv.org/abs/q-alg/9501013. View at: Google Scholar
- A. Mironov, “Quantum deformations of t-functions, Bilinear identities and representation theory,” http://arxiv.org/abs/hep-th/9409190. View at: Google Scholar
- A. Mironov, “Group theory approach to the t-function and its quantization,”
*Theoretical and Mathematical Physics*, vol. 114, pp. 127–183, 1998. View at: Google Scholar - “Knot Atlas,” Page (by D.Bar-Natan), http://katlas.org/wiki/Main. View at: Google Scholar
- R. K. Kaul and T. R. Govindarajan, “Three-dimensional Chern-Simons theory as a theory of knots and links,”
*Nuclear Physics B*, vol. 380, no. 1-2, pp. 293–333, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P. Ramadevi, T. R. Govindarajan, and R. K. Kaul, “Three dimensional Chern-Simons theory as a theory of knots and links III: compact semi-simple group,”
*Nuclear Physics B*, vol. 402, pp. 548–566, 1993. View at: Publisher Site | Google Scholar - P. Ramadevi, T. R. Govindarajan, and R. K. Kaul, “Knot invariants from rational conformal field theories,”
*Nuclear Physics B*, vol. 422, no. 1-2, pp. 291–306, 1994. View at: Google Scholar - A. Mironov, A. Morozov, and A. Morozov, “Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid,”
*Journal of High Energy Physics*, vol. 03, p. 034, 2012. View at: Google Scholar - A. Anokhina, A. Mironov, A. Morozov, and A. Morozov, “Racah coefficients and extended HOMFLY polynomials for all 5-, 6- and 7-strand braids,”
*Nuclear Physics B*, vol. 868, no. 1, pp. 271–313, 2013. View at: Publisher Site | Google Scholar - A. Anokhina and An. Morozov, “Cabling procedure for the colored HOMFLY polynomials,” http://arxiv-web3.library.cornell.edu/abs/1307.2216. View at: Google Scholar
- A. Morozov and A. Smirnov, “Chern-Simons theory in the temporal gauge and knot invariants through the universal quantum $A$-matrix,”
*Nuclear Physics B*, vol. 835, no. 3, pp. 284–313, 2010. View at: Publisher Site | Google Scholar | MathSciNet - A. Smirnov, “Notes on Chern-Simons Theory in the Temporal Gauge,” Proceedings of International School of Subnuclar Physics. Erice, Italy, 2009. View at: Google Scholar
- D. Galakhov, A. Mironov, A. Morozov, and A. Smirnov, “On 3d extensions of AGT relation,” http://arxiv.org/abs/1104.2589. View at: Google Scholar
- H. Itoyama, A. Mironov, A. Morozov, and A. Morozov, “Character expansion for HOMFLY polynomials. III. All 3-Strand braids in the first symmetric representation,” http://arxiv.org/abs/1204.4785. View at: Google Scholar
- H. Itoyama, A. Mironov, A. Morozov, and A. Morozov, “Eigenvalue hypothesis for Racah matrices and HOMFLY polynomials for 3-strand knots in any symmetric and antisymmetric representations,” http://arxiv.org/abs/1209.6304. View at: Google Scholar
- A. Mironov, A. Morozov, and A. Morozov, “Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid,”
*Journal of High Energy Physics*, no. 3, p. 034, 2012. View at: Google Scholar | MathSciNet - P. Ramadevi and T. Sarkar, “On link invariants and topological string amplitudes,”
*Nuclear Physics B*, vol. 600, no. 3, pp. 487–511, 2001. View at: Publisher Site | Google Scholar - Zodinmawia and P. Ramadevi, “SU(N) quantum Racah coefficients & non-torus links,” http://arxiv.org/abs/1107.3918. View at: Google Scholar
- Zodinmawia and P. Ramadevi, “Reformulated invariants for non-torus knots and links,” http://arxiv.org/abs/1209.1346. View at: Google Scholar
- H. Fuji, S. Gukov, and P. Sulkowski, “Volume conjecture: refined and categorified,” http://arxiv.org/abs/1203.2182. View at: Google Scholar
- S. Nawata, P. Ramadevi, Zodinmawia, and X. Sun, “Super-A-polynomials for twist knots,” http://arxiv.org/abs/1209.1409. View at: Google Scholar
- H. Fuji, S. Gukov, M. Stosic, and P. Sulkowski, “3d analogs of Argyres-Douglas theories and knot homologies,” http://arxiv.org/abs/1209.1416. View at: Google Scholar
- S. Nawata, P. Ramadevi, and Zodinmawia, “Colored HOMFLY polynomials from Chern-Simons theory,” http://arxiv.org/abs/1302.5144. View at: Google Scholar
- C. C. Adams,
*The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots*, W. H. Freeman and Company, New York, NY, USA, 1994. View at: MathSciNet - A. Anokhina, A. Mironov, A. Morozov, and An. Morozov, “Knot polynomials in the first non-symmetric representation,” http://arxiv.org/abs/1211.6375. View at: Google Scholar
- P. P. Kulish and N. Yu. Reshetikhin, “On GL3-invariant solutions to the Yang-Baxter equation and the assosiated quantum systems,”
*Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta*, vol. 120, pp. 92–121, 1982 (Russian). View at: Google Scholar - P. P. Kulish and N. Yu. Reshetikhin, “GL3-invariant solutions of the Yang-Baxter equation and associated quantum systems,”
*Journal of Soviet Mathematics*, vol. 34, pp. 1948–1971, 1986. View at: Google Scholar - P. P. Kulish and E. K. Sklyanin, “Quantum spectral transform method. Recent developments,” in
*Integrable Quantum Field Theories*, vol. 151 of*Lecture Notes in Physics*, pp. 61–119, Springer, Berlin, Germany, 1982. View at: Google Scholar | Zentralblatt MATH | MathSciNet - M. Jimbo, T. Miwa, and M. Okado, “Solvable lattice models whose states are dominant integral weights of Ait-1(1),”
*Letters in Mathematical Physics*, vol. 14, no. 2, pp. 123–131, 1987. View at: Publisher Site | Google Scholar - M. Rosso and V. Jones, “On the invariants of torus knots derived from quantum groups,”
*Journal of Knot Theory and its Ramifications*, vol. 2, no. 1, pp. 97–112, 1993. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - X.-S. Lin and H. Zheng, “On the Hecke algebras and the colored HOMFLY polynomial,”
*Transactions of the American Mathematical Society*, vol. 362, no. 1, pp. 1–18, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Stevan, “Chern-Simons invariants of torus links,”
*Annales Henri Poincaré*, vol. 11, no. 7, pp. 1201–1224, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. Gorsky, “q,t-Catalan numbers and knot homology,” http://arxiv.org/abs/1003.0916. View at: Google Scholar
- I. Cherednik, “Jones polynomials of torus knots via DAHA,” http://arxiv.org/abs/1111.6195. View at: Google Scholar
- A. Oblomkov, J. Rasmussen, and V. Shende, “The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link,” with an Appendix by E.Gorsky, http://arxiv.org/abs/1201.2115. View at: Google Scholar
- A. Mironov, A. Morosov, and Sh. Shakirov, “Towards a proof of AGT conjecture by methods of matrix models,”
*International Journal of Modern Physics A*, vol. 27, no. 1, p. 1230001, 32, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Mironov, A. Morozov, and Sh. Shakirov, “Torus HOMFLYPT as the Hall-Littlewood polynomials,”
*Journal of Physics A*, vol. 45, no. 35, p. 11, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. Gorsky, A. Oblomkov, J. Rasmussen, and V. Shende, “Torus knots and the rational DAHA,” http://arxiv.org/abs/1207.4523. View at: Google Scholar
- A. Negut, “Moduli of Flags of Sheaves on P
^{2}and their K-theory,” http://arxiv.org/abs/1209.4242. View at: Google Scholar - E. Gorsky and A. Negut, “Refined knot invariants and Hilbert schemes,” http://arxiv.org/abs/1304.3328. View at: Google Scholar
- A. Mironov, A. Morozov, and A. Morozov, “Conformal blocks and generalized Selberg integrals,”
*Nuclear Physics B*, vol. 843, no. 2, pp. 534–557, 2011. View at: Publisher Site | Google Scholar - S. Arthamonov, A. Mironov, and A. Morozov, “Differential hierarchy and additional grading of knot polynomials,” http://arxiv.org/abs/1306.5682. View at: Google Scholar
- A. Morozov, “Special colored superpolynomials and their representation-dependence,”
*Journal of High Energy Physics*, no. 12, p. 116, 2012. View at: Google Scholar | MathSciNet - A. Morozov, “The first-order deviation of superpolynomial in an arbitrary representation from the special polynomial,” http://arxiv.org/abs/1211.4596. View at: Google Scholar
- A. Brini, B. Eynard, and M. Mariño, “Torus knots and mirror symmetry,”
*Annales Henri Poincaré*, vol. 13, no. 8, pp. 1873–1910, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. Gelca, “Non-commutative trigonometry and the A-polynomial of the trefoil knot,”
*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 133, no. 2, pp. 311–323, 2002. View at: Publisher Site | Google Scholar | MathSciNet - R. Gelca and J. Sain, “The noncommutative A-ideal of a (2, 2p + 1)-torus knot determines its Jones polynomial,”
*Journal of Knot Theory and its Ramifications*, vol. 12, no. 2, pp. 187–201, 2003. View at: Publisher Site | Google Scholar - S. Gukov, “Three-dimensional quantum gravity, Chern-Simons theory, and the A-polynomial,”
*Communications in Mathematical Physics*, vol. 255, no. 3, pp. 577–627, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Garoufalidis, “On the characteristic and deformation varieties of a knot,” in
*Proceedings of the Casson Fest*, vol. 7 of*Geometry & Topology Monographs*, pp. 291–309, Geometry & Topology, 2004. View at: Publisher Site | Google Scholar | MathSciNet - A. Mironov and A. Morozov, “Equations on knot polynomials and 3d/5d duality,”
*AIP Conference Proceedings*, vol. 1483, pp. 189–211, 2012. View at: Google Scholar

#### Copyright

Copyright © 2013 A. Anokhina et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.