Abstract
Tests of the integrality properties of a scalar operator in topological strings on a resolved conifold background or orientifold of conifold backgrounds have been performed for arborescent knots and some non-arborescent knots. The recent results on polynomials for those knots colored by SU(N ) and SO(N ) adjoint representations [1] are useful to verify Marino’s integrality conjecture up to two boxes in the Young diagram. In this paper, we review the salient aspects of the integrality properties and tabulate explicitly for an arborescent knot and a link. In our knotebook website, we have put these results for over 100 prime knots available in Rolfsen table and some links. The first application of the obtained results, an observation of the Gaussian distribution of the LMOV invariants is also reported.
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A. Mironov and A. Morozov, Universal Racah matrices and adjoint knot polynomials: Arborescent knots, Phys. Lett. B 755 (2016) 47 [arXiv:1511.09077] [INSPIRE].
R. Gopakumar and C. Vafa, On the gauge theory/geometry correspondence, Adv. Theor. Math. Phys. 3 (1999) 1415 [hep-th/9811131] [INSPIRE].
R. Gopakumar and C. Vafa, M theory and topological strings — II, hep-th/9812127 [INSPIRE].
H. Ooguri and C. Vafa, Knot invariants and topological strings, Nucl. Phys. B 577 (2000) 419 [hep-th/9912123] [INSPIRE].
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].
J.W. Alexander, Topological invariants of knots and links, Trans. Am. Math. Soc. 30 (1928) 275.
V.F.R. Jones, Index for subfactors, Invent. Math. 72 (1983) 1 [INSPIRE].
V.F.R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Am. Math. Soc. 12 (1985) 103 [INSPIRE].
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. Fréyd, D. Yetter, J. Hoste, W.B.R. Lickorish, K. Millett and A. Ocneanu, A new polynomial invariant of knots and links, Bull. Am. Math. Soc. 12 (1985) 239 [INSPIRE].
J.H. Przytycki and K.P. Traczyk, Invariants of Conway type, Kobe J. Math. 4 (1987) 115
J.H. Conway, An Enumeration of Knots and Links, and Some of Their Algebraic Properties, in Computational Problems in Abstract Algebra, J. Leech ed., Pergamon Press Ltd., Oxford U.K. (1970), pp. 329-358 [https://doi.org/10.1016/B978-0-08-012975-4.50034-5].
J.M.F. Labastida and M. Mariño, Polynomial invariants for torus knots and topological strings, Commun. Math. Phys. 217 (2001) 423 [hep-th/0004196] [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].
J.M.F. Labastida and M. Mariño, A New point of view in the theory of knot and link invariants, math/0104180 [INSPIRE].
M. Mariño and C. Vafa, Framed knots at large-N , Contemp. Math. 310 (2002) 185 [hep-th/0108064] [INSPIRE].
S. Garoufalidis, P. Kucharski and P. Sulkowski, Knots, BPS states and algebraic curves, Commun. Math. Phys. 346 (2016) 75 [arXiv:1504.06327] [INSPIRE].
P. Kucharski and P. Sulkowski, BPS counting for knots and combinatorics on words, JHEP 11 (2016) 120 [arXiv:1608.06600] [INSPIRE].
W. Luo and S. Zhu, Integrality structures in topological strings I: framed unknot, arXiv:1611.06506 [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].
S. Nawata, P. Ramadevi and Zodinmawia, Colored HOMFLY polynomials from Chern-Simons theory, J. Knot Theory Ramifications 22 (2013) 1350078 [arXiv:1302.5144] [INSPIRE].
Zodinmawia, Knot polynomials from SU(N ) Chern-Simons theory, superpolynomials and super-A-polynomials, Ph.D. Thesis, IIT, Mumbai India (2014).
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].
A. Mironov, A. Morozov and A. Sleptsov, Colored HOMFLY polynomials for the pretzel knots and links, JHEP 07 (2015) 069 [arXiv:1412.8432] [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 that distinguish mutant knots, arXiv:1504.00364 [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. 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 and A. Sleptsov, Colored knot polynomials: HOMFLY in representation [2, 1], Int. J. Mod. Phys. A 30 (2015) 1550169 [arXiv:1508.02870] [INSPIRE].
A. Mironov, A. Morozov, A. Morozov, P. Ramadevi, V.K. Singh and A. Sleptsov, Tabulating knot polynomials for arborescent knots, J. Phys. A 50 (2017) 085201 [arXiv:1601.04199] [INSPIRE].
A. Mironov, A. Morozov, A. Morozov and A. Sleptsov, HOMFLY polynomials in representation [3, 1] for 3-strand braids, JHEP 09 (2016) 134 [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. 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].
M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359 [math/9908171] [INSPIRE].
M. Khovanov, Patterns in knot cohomology I, Exp. Math. 12 (2003) 365 [math/0201306].
M. Khovanov, Categorifications of the colored Jones polynomial, J. Knot Theory Ramifications 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 Theory Ramifications 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].
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, Contemp. Math. 680 (2016) 137 [arXiv:1510.01795] [INSPIRE].
D. Galakhov and G.W. Moore, Comments On The Two-Dimensional Landau-Ginzburg Approach To Link Homology, arXiv:1607.04222 [INSPIRE].
S. Gukov, D. Pei, P. Putrov and C. Vafa, BPS spectra and 3-manifold invariants, arXiv:1701.06567 [INSPIRE].
N.Y. 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 proceedings of Quantum Groups. 8th International Workshop on Mathematical Physics, Clausthal, Germany, 19-26 July 1989, pp. 307-317 [https://doi.org/10.1007/3-540-53503-9 51].
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].
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 proceedings of the 47th International School of Subnuclear Physics: The most unexepted at LHC and the status of high energy frontier (ISSP 2009), Erice, Sicily, Italy, 29 August-7 September 2009 [Subnucl. Ser. 47 (2011) 489] [arXiv:0910.5011] [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. Ramadevi, 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 and T. Sarkar, On link invariants and topological string amplitudes, Nucl. Phys. B 600 (2001) 487 [hep-th/0009188] [INSPIRE].
P. Borhade, P. Ramadevi and T. Sarkar, U(N ) framed links, three manifold invariants and topological strings, Nucl. Phys. B 678 (2004) 656 [hep-th/0306283] [INSPIRE].
P. Ramadevi and Zodinmawia, SU(N ) quantum Racah coefficients and non-torus links, Nucl. Phys. B 870 (2013) 205 [arXiv:1107.3918] [INSPIRE].
P. Ramadevi and Zodinmawia, Reformulated invariants for non-torus knots and links, arXiv:1209.1346 [INSPIRE].
A. Mironov, A. Morozov and A. 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 Scietific Publishins Co. Pte. Ltd. (2013), pp. 101-118 [arXiv:1112.5754] [INSPIRE].
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. 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, On R-matrix approaches to knot invariants, arXiv:1412.8444 [INSPIRE].
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].
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, Int. J. Mod. Phys. A 28 (2013) 1340009 [arXiv:1209.6304] [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 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].
E. Witten, Supersymmetry and Morse theory, J. Diff. Geom. 17 (1982) 661 [INSPIRE].
A. Kapustin and E. Witten, Electric-Magnetic Duality And The Geometric Langlands Program, Commun. Num. Theor. Phys. 1 (2007) 1 [hep-th/0604151] [INSPIRE].
E. Witten, Khovanov Homology And Gauge Theory, in Proceedings of the Freedman Fest, R. Kirby, V. Krushkal and Z. Wang eds., Mathematical Sciences Publishers (2012) [Geom. Topol. Monographs 18 (2012) 291] [arXiv:1108.3103] [INSPIRE].
E. Witten, Two Lectures On The Jones Polynomial And Khovanov Homology, arXiv:1401.6996 [INSPIRE].
E. Witten, Two Lectures on Gauge Theory and Khovanov Homology, arXiv:1603.03854 [INSPIRE].
D.E. Littlewood, The theory of group characters and matrix representations of groups, AMS Chelsea Publishing, AMS, Providence Rhode Island U.S.A. (1958).
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].
A. Mironov, A. Morozov and S. Shakirov, Torus HOMFLY as the Hall-Littlewood Polynomials, J. Phys. A 45 (2012) 355202 [arXiv:1203.0667] [INSPIRE].
J.J. Duistermaat and G. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982) 259 [Addendum ibid. 72 (1983) 153] [INSPIRE].
M. Blau, E. Keski-Vakkuri and A.J. Niemi, Path integrals and geometry of trajectories, Phys. Lett. B 246 (1990) 92 [INSPIRE].
A. Morozov, A.J. Niemi and K. Palo, Supersymmetry and loop space geometry, Phys. Lett. B 271 (1991) 365 [INSPIRE].
A. Hietamaki, A. Morozov, A.J. Niemi and K. Palo, Geometry of N = 1/2 supersymmetry and the Atiyah-Singer index theorem, Phys. Lett. B 263 (1991) 417 [INSPIRE].
A.S. Schwarz and O. Zaboronsky, Supersymmetry and localization, Commun. Math. Phys. 183 (1997) 463 [hep-th/9511112] [INSPIRE].
C. Beasley and E. Witten, Non-Abelian localization for Chern-Simons theory, J. Diff. Geom. 70 (2005) 183 [hep-th/0503126] [INSPIRE].
J. Kallen, Cohomological localization of Chern-Simons theory, JHEP 08 (2011) 008 [arXiv:1104.5353] [INSPIRE].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
A. Alexandrov, A. Mironov and A. Morozov, Instantons and merons in matrix models, Physica D 235 (2007) 126 [hep-th/0608228] [INSPIRE].
A. Alexandrov, A. Mironov and A. Morozov, BGWM as Second Constituent of Complex Matrix Model, JHEP 12 (2009) 053 [arXiv:0906.3305] [INSPIRE].
B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion, Commun. Num. Theor. Phys. 1 (2007) 347 [math-ph/0702045] [INSPIRE].
N. Orantin, Symplectic invariants, Virasoro constraints and Givental decomposition, arXiv:0808.0635 [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. Mironov, A. Morozov and 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].
A. Alexandrov and D. Melnikov, Matrix integral expansion of colored Jones polynomials for figure-eight knot, JETP Lett. 101 (2015) 51 [Pisma Zh. Eksp. Teor. Fiz. 101 (2015) 54] [arXiv:1411.5698] [INSPIRE].
R. Dijkgraaf, H. Fuji and M. Manabe, The Volume Conjecture, Perturbative Knot Invariants and Recursion Relations for Topological Strings, Nucl. Phys. B 849 (2011) 166 [arXiv:1010.4542] [INSPIRE].
A. Mironov, A. Morozov and A. Sleptsov, Genus expansion of HOMFLY polynomials, Theor. Math. Phys. 177 (2013) 1435 [Teor. Mat. Fiz. 177 (2013) 179] [arXiv:1303.1015] [INSPIRE].
A. Mironov, A. Morozov and A. Sleptsov, On genus expansion of knot polynomials and hidden structure of Hurwitz tau-functions, Eur. Phys. J. C 73 (2013) 2492 [arXiv:1304.7499] [INSPIRE].
A. Mironov, A. Morozov, A. Sleptsov and A. Smirnov, On genus expansion of superpolynomials, Nucl. Phys. B 889 (2014) 757 [arXiv:1310.7622] [INSPIRE].
A. Sleptsov, Hidden structures of knot invariants, Int. J. Mod. Phys. A 29 (2014) 1430063 [INSPIRE].
A. Mironov, A. Morozov and S. Natanzon, Complete Set of Cut-and-Join Operators in Hurwitz-Kontsevich Theory, Theor. Math. Phys. 166 (2011) 1 [arXiv:0904.4227] [INSPIRE].
A. Mironov, A. Morozov and S. Natanzon, Algebra of differential operators associated with Young diagrams, J. Geom. Phys. 62 (2012) 148 [arXiv:1012.0433] [INSPIRE].
A. Alexandrov, A. Mironov, A. Morozov and S. Natanzon, Integrability of Hurwitz Partition Functions. I. Summary, J. Phys. A 45 (2012) 045209 [arXiv:1103.4100] [INSPIRE].
A. Alexandrov, A. Mironov, A. Morozov and S. Natanzon, On KP-integrable Hurwitz functions, JHEP 11 (2014) 080 [arXiv:1405.1395] [INSPIRE].
A. Mironov, A. Morozov and S. Natanzon, Integrability properties of Hurwitz partition functions. II. Multiplication of cut-and-join operators and WDVV equations, JHEP 11 (2011) 097 [arXiv:1108.0885] [INSPIRE].
W.Fulton, Young tableaux: with applications to representation theory and geometry, London Mathematical Society (1997).
R. Dijkgraaf, Mirror symmetry and elliptic curves, in The moduli spaces of curves, volume 129, Progress in Mathematics Series, Birkhäuser, Boston U.S.A. (1995), pp. 149-163.
A. Okounkov, Toda equations for Hurwitz numbers, Math. Res. Lett. 7 (2000) 447 [math/0004128] [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].
http://knotebook.org.s3-website-us-west-2.amazonaws.com/knotebook/HOMFLY/ colored.htm.
http://knotebook.org.s3-website-us-west-2.amazonaws.com/knotebook/HOMFLY/ kauffman.htm.
http://knotebook.org.s3-website-us-west-2.amazonaws.com/knotebook/HOMFLY/ universal.htm.
K. Liu and P. Peng, Proof of the Labastida-Mariño-Ooguri-Vafa conjecture, J. Diff. Geom. 85 (2010) 479 [arXiv:0704.1526] [INSPIRE].
K. Liu and P. Peng, On a proof of the Labastida-Mariño-Ooguri-Vafa conjecture, Math. Res. Lett. 17 (2010) 493 [arXiv:1012.2635] [INSPIRE].
M. Mariño, String theory and the Kauffman polynomial, Commun. Math. Phys. 298 (2010) 613 [arXiv:0904.1088] [INSPIRE].
S. Stevan, Chern-Simons Invariants of Torus Links, Annales Henri Poincaré 11 (2010) 1201 [arXiv:1003.2861] [INSPIRE].
C. Paul, P. Borhade and P. Ramadevi, Composite Invariants and Unoriented Topological String Amplitudes, arXiv:1003.5282 [INSPIRE].
C. Paul, P. Borhade and P. Ramadevi, Composite Representation Invariants and Unoriented Topological String Amplitudes, Nucl. Phys. B 841 (2010) 448 [arXiv:1008.3453] [INSPIRE].
S. Nawata, P. Ramadevi and Zodinmawia, Colored Kauffman Homology and Super-A-polynomials, JHEP 01 (2014) 126 [arXiv:1310.2240] [INSPIRE].
A. Caudron, Classification des noeuds et des enlacements, volume 82-4, Publications mathématiques d’Orsay Series, Département de Mathématiques d’Orsay — Université Paris-Sud, Orsay France (1982).
F. Bonahon and L.C. Siebenmann, New geometric splittings of classical knots and the classification and symmetries of arborescent knots, (2016) http://www-bcf.usc.edu/∼fbonahon/Research/Preprints/BonSieb.pdf.
D. Bar-Natan and S. Morrison, Main Page. The Knot Atlas, (2015) http://katlas.org.
M. Aganagic and C. Vafa, Mirror symmetry, D-branes and counting holomorphic discs, hep-th/0012041 [INSPIRE].
S. Sinha and C. Vafa, SO and Sp Chern-Simons at large-N , hep-th/0012136 [INSPIRE].
V. Bouchard, B. Florea and M. Mariño, Counting higher genus curves with crosscaps in Calabi-Yau orientifolds, JHEP 12 (2004) 035 [hep-th/0405083] [INSPIRE].
V. Bouchard, B. Florea and M. Mariño, Topological open string amplitudes on orientifolds, JHEP 02 (2005) 002 [hep-th/0411227] [INSPIRE].
P. Borhade and P. Ramadevi, SO(N ) reformulated link invariants from topological strings, Nucl. Phys. B 727 (2005) 471 [hep-th/0505008] [INSPIRE].
L. Rudolph, A congruence between link polynomials, Math. Proc. Cambridge Philos. Soc. 107 (1990) 319.
H.R. Morton, Integrality of Homfly 1-tangle invariants, Algebr. Geom. Topol. 7 (2007) 327.
H.R. Morton and N.D.A. Ryder, Relations between Kauffman and Homfly satellite invariants, Math. Proc. Cambridge Philos. Soc. 149 (2010) 105 [arXiv:0902.1339].
K. Koike, On the decomposition of tensor products of the representations of the classical groups: by means of the universal characters, Adv. Math. 74 (1989) 57.
A. Mironov, R. Mkrtchyan and A. Morozov, On universal knot polynomials, JHEP 02 (2016) 078 [arXiv:1510.05884] [INSPIRE].
N.M. Dunfield, S. Gukov and J. Rasmussen, The Superpolynomial for knot homologies, Exp. Math. 15 (2006) 129 [math/0505662] [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. Arthamonov, A. Mironov, A. Morozov and A. Morozov, Link polynomial calculus and the AENV conjecture, JHEP 04 (2014) 156 [arXiv:1309.7984] [INSPIRE].
S. Arthamonov, A. Mironov and A. Morozov, Differential hierarchy and additional grading of knot polynomials, Theor. Math. Phys. 179 (2014) 509 [arXiv:1306.5682] [INSPIRE].
S. Gukov, S. Nawata, I. Saberi, M. Stošić and P. Sulkowski, Sequencing BPS Spectra, JHEP 03 (2016) 004 [arXiv:1512.07883] [INSPIRE].
Y. 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].
A. Morozov, Differential expansion and rectangular HOMFLY for the figure eight knot, Nucl. Phys. B 911 (2016) 582 [arXiv:1605.09728] [INSPIRE].
A. Morozov, Factorization of differential expansion for antiparallel double-braid knots, JHEP 09 (2016) 135 [arXiv:1606.06015] [INSPIRE].
I. Tuba and H. Wenzl, Representations of the braid group B 3 and of SL(2, Z), math/9912013.
S. Nawata, P. Ramadevi and Zodinmawia, Multiplicity-free quantum 6j-symbols for \( {U}_q\left(\mathfrak{s}{\mathfrak{l}}_N\right) \), Lett. Math. Phys. 103 (2013) 1389 [arXiv:1302.5143] [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].
P. Vogel, The universal Lie algebra, preprint (1999) http://webusers.imj-prg.fr/∼pierre.vogel/.
M. Kameyama and S. Nawata, Refined large-N duality for knots, arXiv:1703.05408 [INSPIRE].
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Mironov, A., Morozov, A., Morozov, A. et al. Checks of integrality properties in topological strings. J. High Energ. Phys. 2017, 139 (2017). https://doi.org/10.1007/JHEP08(2017)139
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DOI: https://doi.org/10.1007/JHEP08(2017)139