Abstract
Continuing the quest for exclusive Racah matrices, which are needed for evaluation of colored arborescent-knot polynomials in Chern-Simons theory, we suggest to extract them from a new kind of a double-evolution — that of the antiparallel double-braids, which is a simple two-parametric family of two-bridge knots, generalizing the one-parametric family of twist knots. In the case of rectangular representations R = [r s] we found an evidence that the corresponding differential expansion miraculously factorizes and can be obtained from that for the twist knots. This reduces the problem of rectangular exclusive Racah to constructing the answers for just a few twist knots. We develop a recent conjecture on the structure of differential expansion for the simplest members of this family (the trefoil and the figure-eight knot) and provide the exhaustive answer for the first unknown case of R = [33]. The answer includes HOMFLY of arbitrary twist and double-braid knots and Racah matrices \( \overline{S} \) and S — what allows to calculate [33]-colored polynomials for arbitrary arborescent (double-fat) knots. For generic rectangular representations fully described are only the contributions of the single-floor pyramids. One step still remains to be done.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928) 275.
V.F.R. Jones, Index for subfactors, Invent. Math. 72 (1983) 1.
V.F.R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. 12 (1985) 103.
V.F.R. Jones, Hecke algebra representations of braid groups and link polynomials, Annals Math. 126 (1987) 335 [INSPIRE].
L. Kauffman, State models and the Jones polynomial, Topology 26 (1987) 395.
P. Freyd et al., A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. 12 (1985) 239.
J.H. Przytycki and K.P. Traczyk, Invariants of links of Conway type, Kobe J. Math. 4 (1987) 115.
J.H. Conway, Algebraic properties, in Computational problems in abstract algebra, Proc. Conf. Oxford U.K. 1967, J. Leech ed., Pergamon Press, Oxford U.K. and New York U.S.A. (1970), pg. 329.
S.-S. Chern and J. Simons, Characteristic forms and geometric invariants, Annals Math. 99 (1974) 48 [INSPIRE].
E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
M. Atiyah, The geometry and physics of knots, Cambridge University Press, Cambridge U.K. (1990).
R. Gopakumar and C. Vafa, On the gauge theory/geometry correspondence, Adv. Theor. Math. Phys. 3 (1999) 1415 [hep-th/9811131] [INSPIRE].
H. Ooguri and C. Vafa, Knot invariants and topological strings, Nucl. Phys. B 577 (2000) 419 [hep-th/9912123] [INSPIRE].
J.M.F. Labastida, M. Mariño and C. Vafa, Knots, links and branes at large-N, JHEP 11 (2000) 007 [hep-th/0010102] [INSPIRE].
K. Liu and P. Peng, On a proof of the Labastida-Marino-Ooguri-Vafa conjecture, Math. Res. Lett. 17 (2010) 493 [arXiv:1012.2635] [INSPIRE].
K. Liu and P. Peng, New structure of knot invariants, J. Diff. Geom. 85 (2010) 479 [arXiv:1012.2636] [INSPIRE].
S. Gukov, A.S. Schwarz and C. Vafa, Khovanov-Rozansky homology and topological strings, Lett. Math. Phys. 74 (2005) 53 [hep-th/0412243] [INSPIRE].
N. Yu. Reshetikhin and V.G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Commun. Math. Phys. 127 (1990) 1 [INSPIRE].
E. Guadagnini, M. Martellini and M. Mintchev, Chern-Simons field theory and quantum groups, in Proc. Clausthal, (1989), pg. 307 [INSPIRE].
E. Guadagnini, M. Martellini and M. Mintchev, Chern-Simons holonomies and the appearance of quantum groups, Phys. Lett. B 235 (1990) 275 [INSPIRE].
V.G. Turaev and O.Y. Viro, State sum invariants of 3 manifolds and quantum 6j symbols, Topology 31 (1992) 865 [INSPIRE].
R.K. Kaul and T.R. Govindarajan, Three-dimensional Chern-Simons theory as a theory of knots and links, Nucl. Phys. B 380 (1992) 293 [hep-th/9111063] [INSPIRE].
R.K. Kaul and T.R. Govindarajan, Three-dimensional Chern-Simons theory as a theory of knots and links. 2. Multicolored links, Nucl. Phys. B 393 (1993) 392 [INSPIRE].
P. Rama Devi, T.R. Govindarajan and R.K. Kaul, Three-dimensional Chern-Simons theory as a theory of knots and links. 3. Compact semisimple group, Nucl. Phys. B 402 (1993) 548 [hep-th/9212110] [INSPIRE].
P. Ramadevi, T.R. Govindarajan and R.K. Kaul, Knot invariants from rational conformal field theories, Nucl. Phys. B 422 (1994) 291 [hep-th/9312215] [INSPIRE].
P. Ramadevi, T.R. Govindarajan and R.K. Kaul, Representations of composite braids and invariants for mutant knots and links in Chern-Simons field theories, Mod. Phys. Lett. A 10 (1995) 1635 [hep-th/9412084] [INSPIRE].
P. Ramadevi, T.R. Govindarajan and R.K. Kaul, Chirality of knots 942 and 1071 and Chern-Simons theory, Mod. Phys. Lett. A 9 (1994) 3205 [hep-th/9401095] [INSPIRE].
P. Ramadevi and T. Sarkar, On link invariants and topological string amplitudes, Nucl. Phys. B 600 (2001) 487 [hep-th/0009188] [INSPIRE].
Zodinmawia and P. Ramadevi, SU(N) quantum Racah coefficients & non-torus links, Nucl. Phys. B 870 (2013) 205 [arXiv:1107.3918] [INSPIRE].
Zodinmawia and P. Ramadevi, Reformulated invariants for non-torus knots and links, arXiv:1209.1346 [INSPIRE].
A. Morozov and A. Smirnov, Chern-Simons theory in the temporal gauge and knot invariants through the universal quantum R-matrix, Nucl. Phys. B 835 (2010) 284 [arXiv:1001.2003] [INSPIRE].
A. Smirnov, Notes on Chern-Simons theory in the temporal gauge, in Proc. of International School of Subnuclear Phys., Erice Italy (2009) [Subnucl. Ser. 47 (2011) 489] [arXiv:0910.5011] [INSPIRE].
D. Bar-Natan and S. Morrison, The Knot atlas webpage, http://katlas.org.
C. Livingston and J.C. Cha, Table of knot invariants webpage, http://indiana.edu/~knotinfo/.
Knotebook webpage, http://knotebook.org.
A. Mironov and A. Morozov, Towards effective topological field theory for knots, Nucl. Phys. B 899 (2015) 395 [arXiv:1506.00339] [INSPIRE].
A. Mironov, A. Morozov, A. Morozov, A. Sleptsov, P. Ramadevi and V.K. Singh, Tabulating knot polynomials for arborescent knots, arXiv:1601.04199 [INSPIRE].
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
A. Zamolodchikov and Al. Zamolodchikov, Conformal field theory and critical phenomena in 2d systems, (2009).
V.S. Dotsenko and V.A. Fateev, Conformal algebra and multipoint correlation functions in two-dimensional statistical models, Nucl. Phys. B 240 (1984) 312 [INSPIRE].
A. Gerasimov, A. Morozov, M. Olshanetsky, A. Marshakov and S.L. Shatashvili, Wess-Zumino-Witten model as a theory of free fields, Int. J. Mod. Phys. A 5 (1990) 2495 [INSPIRE].
L. Álvarez-Gaumé, Random surfaces, statistical mechanics and string theory, Helv. Phys. Acta 64 (1991) 359 [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Springer, Germany (1996).
A. Mironov, S. Mironov, A. Morozov and A. Morozov, CFT exercises for the needs of AGT, Theor. Math. Phys. 165 (2010) 1662 [Teor. Mat. Fiz. 165 (2010) 503] [arXiv:0908.2064] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
R. Flume and R. Poghossian, An algorithm for the microscopic evaluation of the coefficients of the Seiberg-Witten prepotential, Int. J. Mod. Phys. A 18 (2003) 2541 [hep-th/0208176] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].
N. Nekrasov and V. Pestun, Seiberg-Witten geometry of four dimensional N = 2 quiver gauge theories, arXiv:1211.2240 [INSPIRE].
N. Nekrasov, V. Pestun and S. Shatashvili, Quantum geometry and quiver gauge theories, arXiv:1312.6689 [INSPIRE].
L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].
N. Wyllard, A N − 1 conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [INSPIRE].
A. Mironov and A. Morozov, On AGT relation in the case of U(3), Nucl. Phys. B 825 (2010) 1 [arXiv:0908.2569] [INSPIRE].
B. Feigin, E. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum continuous \( \mathfrak{g}{\mathfrak{l}}_{\infty } \) : semiinfinite construction of representations, Kyoto J. Math. 51 (2011) 337 [arXiv:1002.3100].
B. Feigin, E. Feigin, M. Jimbo, T. Miwa and E. Mukhin, Quantum continuous \( \mathfrak{g}{\mathfrak{l}}_{\infty } \) : tensor products of Fock modules and \( {\mathcal{W}}_n \) -characters, Kyoto J. Math. 51 (2011) 365 [arXiv:1002.3113] [INSPIRE].
A. Mironov, A. Morozov, S. Shakirov and A. Smirnov, Proving AGT conjecture as HS duality: extension to five dimensions, Nucl. Phys. B 855 (2012) 128 [arXiv:1105.0948] [INSPIRE].
J.-E. Bourgine, Y. Matsuo and H. Zhang, Holomorphic field realization of SH c and quantum geometry of quiver gauge theories, JHEP 04 (2016) 167 [arXiv:1512.02492] [INSPIRE].
H. Awata et al., Explicit examples of DIM constraints for network matrix models, JHEP 07 (2016) 103 [arXiv:1604.08366] [INSPIRE].
A. Mironov, A. Morozov and An. Morozov, Character expansion for HOMFLY polynomials. I. Integrability and difference equations, in Strings, Gauge Fields, and the Geometry Behind: The Legacy of Maximilian Kreuzer, A. Rebhan, L. Katzarkov, J. Knapp, R. Rashkov and E. Scheidegger eds., World Scientific Publishing Co. Pte. Ltd., Singapore (2013), pg. 101 [arXiv:1112.5754].
A. Mironov, A. Morozov and A. Morozov, Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid, JHEP 03 (2012) 034 [arXiv:1112.2654] [INSPIRE].
A. Mironov, A. Morozov and A. Morozov, On colored HOMFLY polynomials for twist knots, Mod. Phys. Lett. A 29 (2014) 1450183 [arXiv:1408.3076] [INSPIRE].
A. Anokhina, A. Mironov, A. Morozov and A. Morozov, Racah coefficients and extended HOMFLY polynomials for all 5-, 6- and 7-strand braids, Nucl. Phys. B 868 (2013) 271 [arXiv:1207.0279] [INSPIRE].
A. Anokhina, A. Mironov, A. Morozov and A. Morozov, Knot polynomials in the first non-symmetric representation, Nucl. Phys. B 882 (2014) 171 [arXiv:1211.6375] [INSPIRE].
A. Anokhina, A. Mironov, A. Morozov and A. Morozov, Colored HOMFLY polynomials as multiple sums over paths or standard Young tableaux, Adv. High Energy Phys. 2013 (2013) 931830 [arXiv:1304.1486] [INSPIRE].
A. Anokhina, On R-matrix approaches to knot invariants, arXiv:1412.8444 [INSPIRE].
S. Gukov and M. Stošić, Homological algebra of knots and BPS states, Proc. Symp. Pure Math. 85 (2012) 125 [arXiv:1112.0030] [INSPIRE].
S. Nawata, P. Ramadevi, Zodinmawia and X. Sun, Super-A-polynomials for twist knots, JHEP 11 (2012) 157 [arXiv:1209.1409] [INSPIRE].
S. Nawata, P. Ramadevi and Zodinmawia, Multiplicity-free quantum 6j-symbols for U q (\( \mathfrak{s}{\mathfrak{l}}_N \)), Lett. Math. Phys. 103 (2013) 1389 [arXiv:1302.5143] [INSPIRE].
S. Nawata, P. Ramadevi and Zodinmawia, Colored HOMFLY polynomials from Chern-Simons theory, J. Knot Theor. 22 (2013) 1350078 [arXiv:1302.5144] [INSPIRE].
S. Nawata, P. Ramadevi and Zodinmawia, Colored Kauffman homology and super-A-polynomials, JHEP 01 (2014) 126 [arXiv:1310.2240] [INSPIRE].
Zodinmawia, Knot polynomials from SU(N) Chern-Simons theory, superpolynomials and super-A-polynomials, Ph.D. thesis, IIT, Mumbai India (2014).
A. Anokhina and A. Morozov, Cabling procedure for the colored HOMFLY polynomials, Theor. Math. Phys. 178 (2014) 1 [Teor. Mat. Fiz. 178 (2014) 3] [arXiv:1307.2216] [INSPIRE].
S. Nawata, P. Ramadevi and Zodinmawia, Colored Kauffman homology and super-A-polynomials, JHEP 01 (2014) 126 [arXiv:1310.2240] [INSPIRE].
S. Gukov, S. Nawata, I. Saberi, M. Stošić and P. Sulkowski, Sequencing BPS spectra, JHEP 03 (2016) 004 [arXiv:1512.07883] [INSPIRE].
D. Galakhov, D. Melnikov, A. Mironov, A. Morozov and A. Sleptsov, Colored knot polynomials for arbitrary pretzel knots and links, Phys. Lett. B 743 (2015) 71 [arXiv:1412.2616] [INSPIRE].
D. Galakhov, D. Melnikov, A. Mironov and A. Morozov, Knot invariants from Virasoro related representation and pretzel knots, Nucl. Phys. B 899 (2015) 194 [arXiv:1502.02621] [INSPIRE].
S. Nawata, P. Ramadevi and V.K. Singh, Colored HOMFLY polynomials can distinguish mutant knots, arXiv:1504.00364 [INSPIRE].
S. Garoufalidis, A.D. Lauda and T.T.Q. Lê, The colored HOMFLYPT function is q-holonomic, arXiv:1604.08502 [INSPIRE].
M. Rosso and V.F.R. Jones, On the invariants of torus knots derived from quantum groups, J. Knot Theor. 2 (1993) 97.
X.-S. Lin and H. Zheng, On the Hecke algebras and the colored HOMFLY polynomial, Trans. Amer. Math. Soc. 362 (2010) 1 [math/0601267].
S. Stevan, Chern-Simons invariants of torus links, Annales Henri Poincaré 11 (2010) 1201 [arXiv:1003.2861] [INSPIRE].
M. Tierz, Soft matrix models and Chern-Simons partition functions, Mod. Phys. Lett. A 19 (2004) 1365 [hep-th/0212128] [INSPIRE].
A. Brini, B. Eynard and M. Mariño, Torus knots and mirror symmetry, Annales Henri Poincaré 13 (2012) 1873 [arXiv:1105.2012] [INSPIRE].
A. Aleksandrov, A.D. Mironov, A. Morozov and A.A. Morozov, Towards matrix model representation of HOMFLY polynomials, JETP Lett. 100 (2014) 271 [Pisma Zh. Eksp. Teor. Fiz. 100 (2014) 297] [arXiv:1407.3754] [INSPIRE].
M. Aganagic and S. Shakirov, Knot homology and refined Chern-Simons index, Commun. Math. Phys. 333 (2015) 187 [arXiv:1105.5117] [INSPIRE].
M. Aganagic and S. Shakirov, Refined Chern-Simons theory and knot homology, Proc. Symp. Pure Math. 85 (2012) 3 [arXiv:1202.2489] [INSPIRE].
M. Aganagic and S. Shakirov, Refined Chern-Simons theory and topological string, arXiv:1210.2733 [INSPIRE].
A. Mironov, A. Morozov, S. Shakirov and A. Sleptsov, Interplay between MacDonald and Hall-Littlewood expansions of extended torus superpolynomials, JHEP 05 (2012) 070 [arXiv:1201.3339] [INSPIRE].
P. Dunin-Barkowski, A. Mironov, A. Morozov, A. Sleptsov and A. Smirnov, Superpolynomials for toric knots from evolution induced by cut-and-join operators, JHEP 03 (2013) 021 [arXiv:1106.4305] [INSPIRE].
I. Cherednik, Jones polynomials of torus knots via DAHA, arXiv:1111.6195 [INSPIRE].
E. Gorsky and A. Negut, Refined knot invariants and Hilbert schemes, J. Math. Pure. Appl. 104 (2015) 403 [arXiv:1304.3328] [INSPIRE].
I. Cherednik and I. Danilenko, DAHA and iterated torus knots, Algebr. Geom. Topol. 16 (2016) 843 [arXiv:1408.4348].
S. Arthamonov and S. Shakirov, Refined Chern-Simons theory in genus two, arXiv:1504.02620 [INSPIRE].
E. Gorsky, S. Gukov and M. Stošić, Quadruply-graded colored homology of knots, arXiv:1304.3481 [INSPIRE].
A. Mironov, A. Morozov, A. Morozov, P. Ramadevi and V.K. Singh, Colored HOMFLY polynomials of knots presented as double fat diagrams, JHEP 07 (2015) 109 [arXiv:1504.00371] [INSPIRE].
A. Caudron, Classification des noeuds et des enlacements (in French), Publ. Math. Orsay 82-4, University of Paris XI, Orsay France (1982).
F. Bonahon and L.C. Siebenmann, New geometric splittings of classical knots and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Preprints/BonSieb.pdf, (2010).
H. Itoyama, A. Mironov, A. Morozov and A. Morozov, Character expansion for HOMFLY polynomials. III. All 3-strand braids in the first symmetric representation, Int. J. Mod. Phys. A 27 (2012) 1250099 [arXiv:1204.4785] [INSPIRE].
S. Nawata, P. Ramadevi, Zodinmawia and X. Sun, Super-A-polynomials for twist knots, JHEP 11 (2012) 157 [arXiv:1209.1409] [INSPIRE].
H. Fuji, S. Gukov, M. Stošić and P. Sulkowski, 3d analogs of Argyres-Douglas theories and knot homologies, JHEP 01 (2013) 175 [arXiv:1209.1416] [INSPIRE].
A. Mironov, A. Morozov and A. Sleptsov, Colored HOMFLY polynomials for the pretzel knots and links, JHEP 07 (2015) 069 [arXiv:1412.8432] [INSPIRE].
J. Gu and H. Jockers, A note on colored HOMFLY polynomials for hyperbolic knots from WZW models, Commun. Math. Phys. 338 (2015) 393 [arXiv:1407.5643] [INSPIRE].
A. Mironov, A. Morozov, A. Morozov and A. Sleptsov, Racah matrices and hidden integrability in evolution of knots, Phys. Lett. B 760 (2016) 45 [arXiv:1605.04881] [INSPIRE].
A. Mironov, A. Morozov, A. Morozov and A. Sleptsov, HOMFLY polynomials in representation [3, 1] for 3-strand braids, arXiv:1605.02313 [INSPIRE].
A. Mironov, A. Morozov, A. Morozov and A. Sleptsov, Quantum Racah matrices and 3-strand braids in irreps R with |R| = 4, JETP Lett. 104 (2016) 56 [Pisma Zh. Eksp. Teor. Fiz. 104 (2016) 52] [arXiv:1605.03098] [INSPIRE].
A. Morozov, Differential expansion and rectangular HOMFLY for the figure eight knot, Nucl. Phys. B 911 (2016) 582 [arXiv:1605.09728] [INSPIRE].
H. Itoyama, A. Mironov, A. Morozov and A. Morozov, HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations, JHEP 07 (2012) 131 [arXiv:1203.5978] [INSPIRE].
A. Mironov, A. Morozov and A. Morozov, Evolution method and “differential hierarchy” of colored knot polynomials, AIP Conf. Proc. 1562 (2013) 123 [arXiv:1306.3197] [INSPIRE].
S.B. Arthamonov, A. Mironov and A. Morozov, Differential hierarchy and additional grading of knot polynomials, Theor. Math. Phys. 179 (2014) 509 [Teor. Mat. Fiz. 179 (2014) 147] [arXiv:1306.5682] [INSPIRE].
S. Arthamonov, A. Mironov, A. Morozov and A. Morozov, Link polynomial calculus and the AENV conjecture, JHEP 04 (2014) 156 [arXiv:1309.7984] [INSPIRE].
Ya. Kononov and A. Morozov, On the defect and stability of differential expansion, JETP Lett. 101 (2015) 831 [Pisma Zh. Eksp. Teor. Fiz. 101 (2015) 931] [arXiv:1504.07146] [INSPIRE].
N.M. Dunfield, S. Gukov and J. Rasmussen, The superpolynomial for knot homologies, Exper. Math. 15 (2006) 129 [math/0505662] [INSPIRE].
M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359 [math/9908171] [INSPIRE].
M. Khovanov, Patterns in knot cohomology I, Exper. Math. 12 (2003) 365 [math/0201306].
M. Khovanov, Categorifications of the colored Jones polynomial, J. Knot Theor. 14 (2005) 111 [math/0302060].
M. Khovanov, sl(3) link homology, Algebr. Geom. Topol. 4 (2004) 1045 [math/0304375].
M. Khovanov, Triply-graded link homology and Hochschild homology of Soergel bimodules, Int. J. Math. 18 (2007) 869 [math/0510265].
M. Khovanov, Link homology and categorification, math/0605339.
M. Khovanov, Categorifications from planar diagrammatics, arXiv:1008.5084.
D. Bar-Natan, On Khovanov’s categorification of the Jones polynomial, Algebr. Geom. Topol. 2 (2002) 337 [math/0201043].
D. Bar-Natan, Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005) 1443 [math/0410495].
D. Bar-Natan, Fast Khovanov homology computations, J. Knot Theor. 16 (2007) 243 [math/0606318].
M. Khovanov and L. Rozansky, Matrix factorizations and link homology, Fund. Math. 199 (2008) 191 [math/0401268].
M. Khovanov and L. Rozansky, Matrix factorizations and link homology II, Geom. Topol. 12 (2008) 1387 [math/0505056].
M. Khovanov and L. Rozansky, Virtual crossings, convolutions and a categorification of the SO(2N) Kauffman polynomial, math/0701333.
N. Carqueville and D. Murfet, Computing Khovanov-Rozansky homology and defect fusion, Algebr. Geom. Topol. 14 (2014) 489 [arXiv:1108.1081] [INSPIRE].
V. Dolotin and A. Morozov, Introduction to Khovanov homologies. I. Unreduced Jones superpolynomial, JHEP 01 (2013) 065 [arXiv:1208.4994] [INSPIRE].
V. Dolotin and A. Morozov, Introduction to Khovanov homologies. II. Reduced Jones superpolynomials, J. Phys. Conf. Ser. 411 (2013) 012013 [arXiv:1209.5109] [INSPIRE].
V. Dolotin and A. Morozov, Introduction to Khovanov homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants, Nucl. Phys. B 878 (2014) 12 [arXiv:1308.5759] [INSPIRE].
E. Witten, Two lectures on the Jones polynomial and Khovanov homology, arXiv:1401.6996 [INSPIRE].
A. Anokhina and A. Morozov, Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary T-deformation of HOMFLY, JHEP 07 (2014) 063 [arXiv:1403.8087] [INSPIRE].
S. Nawata and A. Oblomkov, Lectures on knot homology, arXiv:1510.01795 [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1606.06015
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Morozov, A. Factorization of differential expansion for antiparallel double-braid knots. J. High Energ. Phys. 2016, 135 (2016). https://doi.org/10.1007/JHEP09(2016)135
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2016)135