Skip to main content
SpringerLink
Log in

Rectangular superpolynomials for the figure-eight knot 41

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We rewrite the recently proposed differential expansion formula for HOMFLY polynomials of the knot 41 in an arbitrary rectangular representation R = [rs] as a sum over all Young subdiagrams λ of R with surprisingly simple coefficients of the Z factors. Intriguingly, these coefficients are constructed from the quantum dimensions of symmetric representations of the groups SL(r) and SL(s) and restrict the summation to diagrams with no more than s rows and r columns. Moreover, the β-deformation to Macdonald dimensions yields polynomials with positive integer coefficients, which are plausible candidates for the role of superpolynomials for rectangular representations. Both the polynomiality and the positivity of the coefficients are nonobvious, nevertheless true. This generalizes the previously known formulas for symmetric representations to arbitrary rectangular representations. The differential expansion allows introducing additional gradings. For the trefoil knot 31, to which our results for the knot 41 are immediately extended, we obtain the so-called fourth grading of hyperpolynomials. The property of factorization in roots of unity is preserved even in the five-graded case.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S.-S. Chern and J. Simons, “Characteristic forms and geometric invariants,” Ann. Math., 99, 48–69 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  2. E. Witten, “Quantum field theory and the Jones polynomial,” Commun. Math. Phys., 121, 351–399 (1989).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. J. W. Alexander, “Topological invariants of knots and links,” Trans. Amer. Math. Soc., 30, 275–306 (1928)

    Article  MathSciNet  MATH  Google Scholar 

  4. J. H. Conway, “An enumeration of knots and links, and some of their algebraic properties,” in: Computational Problems in Abstract Algebra (Sci. Res. Council Atlas Computer Lab., Oxford, 29 August–2 September 1967, J. Leech, ed.), Pergamon, Oxford (1970), pp. 329–358

    Google Scholar 

  5. V. F. R. Jones, “Index for subfactors,” Invent. Math., 72, 1–25 (1983); “A polynomial invariant for knots via von Neumann algebras,” Bull. Amer. Math. Soc., n.s., 12, 103–111 (1985); “Hecke algebra representations of braid groups and link polynomials,” Ann. Math., 126, 335–388 (1987)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. L. H. Kauffman, “State models and the Jones polynomial,” Topology, 26, 395–407 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  7. P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett, and A. Ocneanu, “A new polynomial invariant of knots and links,” Bull. Amer. Math. Soc., n.s., 12, 239–246 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  8. J. H. Przytycki and K. P. Traczyk, “Invariants of links of Conway type,” Kobe J. Math., 4, 115–139 (1987)

    MathSciNet  MATH  Google Scholar 

  9. A. Morozov and A. Smirnov, “Towards the proof of AGT relations with the help of the generalized Jack polynomials,” Lett. Math. Phys., 104, 585–612 (2014); arXiv:1307.2576v2 [hep-th] (2013).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. R. Gopakumar and C. Vafa, “Topological gravity as large N topological gauge theory,” Adv. Theor. Math. Phys., 2, 413–442 (1998); arXiv:hep-th/9802016v2 (1998); “M-Theory and Topological Strings–I,” arXiv:hepth/9809187v1 (1998); “M-Theory and topological strings-II,” arXiv:hep-th/9812127v1 (1998); “On the gauge theory/geometry correspondence,” Adv. Theor. Math. Phys., 3, 1415–1443 (1999); arXiv:hep-th/9811131v1 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. H. Ooguri and C. Vafa, “Knot invariants and topological strings,” Nucl. Phys. B, 577, 419–438 (2000); arXiv:hep-th/9912123v3 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. S. Gukov, A. Schwarz, and C. Vafa, “Khovanov–Rozansky homology and topological strings,” Lett. Math. Phys., 74, 53–74 (2005); arXiv:hep-th/0412243v3 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. M. Dedushenko and E. Witten, “Some details on the Gopakumar–Vafa and Ooguri–Vafa formulas,” Adv. Theor. Math. Phys., 20, 1–133 (2016); arXiv:1411.7108v2 [hep-th] (2014).

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Khovanov, “A categorification of the Jones polynomial,” Duke Math. J., 101, 359–426 (2000); arXiv: math/9908171v2 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Khovanov and L. Rozansky, “Matrix factorizations and link homology,” Fund. Math., 199, 1–91 (2008); arXiv:math/0401268v2 (2004); “Virtual crossings, convolutions, and a categorification of the SO(2N) Kauffman polynomial,” J. Gökova Geom. Topol., 1, 116–214 (2007); arXiv:math/0701333v1 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. N. Carqueville and D. Murfet, “Computing Khovanov–Rozansky homology and defect fusion,” Algebr. Geom. Topol., 14, 489–537 (2014); arXiv:1108.1081v3 [math.QA] (2011).

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Yu. Morozov, “Are there p-adic knot invariants?” Theor. Math. Phys., 187, 447–454 (2016); arXiv: 1509.04928v2 [hep-th] (2015).

    Article  MathSciNet  MATH  Google Scholar 

  18. E. Gorsky, S. Gukov, and M. Stosic, “Quadruply-graded colored homology of knots,” arXiv:1304.3481v1 [math.QA] (2013).

  19. S. B. Arthamonov, A. D. Mironov, and A. Yu. Morozov, “Differential hierarchy and additional grading of knot polynomials,” Theor. Math. Phys., 179, 509–542 (2014); arXiv:1306.5682v1 [hep-th] (2013).

    Article  MathSciNet  MATH  Google Scholar 

  20. H. Nakajima, “More lectures on Hilbert schemes of points on surfaces,” in: Development of Moduli Theory–Kyoto 2013 (Adv. Stud. Pure Math., Vol. 69, O. Fujino, S. Kondô, A. Moriwaki, M.-H. Saito, and K. Yoshioka, eds.), Math. Soc. Japan, Tokyo (2016), pp. 173–205

    Google Scholar 

  21. A. Okounkov, “Lectures on K-theoretic computations in enumerative geometry,” arXiv:1512.07363v2 [math.AG] (2015)

  22. N. Nekrasov and A. Okounkov, “Membranes and sheaves,” Algebr. Geom., 3, 320–369 (2016); arXiv:1404.2323v1 [math.AG] (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. E. Carlsson, N. Nekrasov, and A. Okounkov, “Five dimensional gauge theories and vertex operators,” Moscow Math. J., 14, 39–61 (2014); arXiv:1308.2465v1 [math.RT] (2013)

    MathSciNet  MATH  Google Scholar 

  24. E. Carlsson and A. Okounkov, “Exts and vertex operators,” Duke Math. J., 161, 1797–1815 (2012); arXiv:0801.2565v2 [math.AG] (2008).

    Article  MathSciNet  MATH  Google Scholar 

  25. J. Ding and K. Iohara, “Generalization and deformation of Drinfeld quantum affine algebras,” Lett. Math. Phys., 41, 181–193 (1997); arXiv:q-alg/9608002v2 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. K. Miki, “A (q, y) analog of the W1+8 algebra,” J. Math. Phys., 48, 123520 (2007).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. B. Feigin, E. Feigin, M. Jimbo, T. Miwa, and E. Mukhin, “Quantum continuous gl8: Semi-infinite construction of representations,” Kyoto J. Math., 51, 337–364 (2011); arXiv:1002.3100v1 [math.QA] (2010); “Quantum continuous gl8: Tensor products of Fock modules and Wn characters,” Kyoto J. Math., 51, 365–392 (2011); arXiv:1002.3113v1 [math.QA] (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Mironov, A. Morozov, Sh. Shakirov, and A. Smirnov, “Proving AGT conjecture as HS duality: Extension to five dimensions,” Nucl. Phys. B, 855, 128–151 (2012); arXiv:1105.0948v1 [hep-th] (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. H. Awata, B. Feigin, and J. Shiraishi, “Quantum algebraic approach to refined topological vertex,” JHEP, 1203, 041 (2012); arXiv:1112.6074v1 [hepth] (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. D. Maulik and A. Okounkov, “Quantum groups and quantum cohomology,” arXiv:1211.1287v2 [math.AG] (2012)

  31. N. Nekrasov, V. Pestun, and S. Shatashvili, “Quantum geometry and quiver gauge theories,” arXiv:1312.6689v1 [hep-th] (2013)

  32. M. Bershtein and O. Foda, “AGT, Burge pairs and minimal models,” JHEP, 1406, 177 (2014); arXiv:1404.7075v3 [hep-th] (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. V. Belavin, O. Foda, and R. Santachiara, “AGT, N-Burge partitions, and WN minimal models,” JHEP, 1510, 073 (2015); arXiv:1507.03540v2 [hep-th] (2015)

    Article  ADS  Google Scholar 

  34. A. Morozov and Y. Zenkevich, “Decomposing Nekrasov decomposition,” JHEP, 1602, 098 (2016); arXiv:1510.01896v1 [hep-th] (2015)

    Article  ADS  MathSciNet  Google Scholar 

  35. J.-E. Bourgine, Y. Matsuo, and H. Zhang, “Holomorphic field realization of SHc and quantum geometry of quiver gauge theories,” JHEP, 1604, 167 (2016); arXiv:1512.02492v3 [hep-th] (2015)

    ADS  Google Scholar 

  36. N. Nekrasov, “BPS/CFT correspondence: Non-perturbative Dyson–Schwinger equations and qq-characters,” JHEP, 1603, 181 (2016); arXiv:1512.05388v2 [hep-th] (2015); “BPS/CFT correspondence II: Instantons at crossroads, moduli, and compactness theorem,” arXiv:1608.07272v1 [hep-th] (2016)

    Article  ADS  Google Scholar 

  37. T. Kimura and V. Pestun, “Quiver W-algebras,” arXiv:1512.08533v3 [hep-th] (2015); “Quiver elliptic W-algebras,” arXiv:1608.04651v2 [hep-th] (2016)

  38. A. Mironov, A. Morozov, and Y. Zenkevich, “Spectral duality in elliptic systems, six-dimensional gauge theories, and topological strings,” JHEP, 1605, 121 (2016); arXiv:1603.00304v1 [hep-th] (2016); “Ding–Iohara–Miki symmetry of network matrix models,” Phys. Lett. B, 762, 196–208 (2016); arXiv:1603.05467v4 [hep-th] (2016)

    Article  ADS  MathSciNet  Google Scholar 

  39. H. Awata, H. Kanno, T. Matsumoto, A. Mironov, A. Morozov, An. Morozov, Yu. Ohkubo, and Y. Zenkevich, “Explicit examples of DIM constraints for network matrix models,” JHEP, 1607, 103 (2016); arXiv:1604.08366v3 [hep-th] (2016); “Toric Calabi–Yau threefolds as quantum integrable systems: R-matrix and RTT relations,” JHEP, 1610, 047 (2016); arXiv:1608.05351v2 [hep-th] (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. A. Nedelin, F. Nieri, and M. Zabzine, “q-Virasoro modular double and 3d partition functions,” Commun. Math. Phys., 353, 1059–1102 (2017); arXiv:1605.07029v3 [hep-th] (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. J.-E. Bourgine, M. Fukuda, Y. Matsuo, H. Zhang, and R.-D. Zhu, “Coherent states in quantum W1+8c algebra and qq-character for 5d Super Yang–Mills,” Prog. Theor. Exp. Phys., 2016, 123B05; arXiv:1606.08020v3 [hep-th] (2016).

    Google Scholar 

  42. N. M. Dunfield, S. Gukov, and J. Rasmussen, “The superpolynomial for knot homologies,” Experiment. Math., 15, 129–159 (2006); arXiv:math/0505662v2 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  43. M. Aganagic and Sh. Shakirov, “Knot homology from refined Chern–Simons theory,” arXiv:1105.5117v2 [hepth] (2011); “Refined Chern–Simons theory and knot homology,” in: String-Math 2011 (Proc. Symp. Pure Math., Vol. 85, J. Block, J. Distler, R. Donagi, and E. Sharpe, eds.), Amer. Math. Soc., Providence, R. I. (2012), pp. 3–31; arXiv:1202.2489v1 [hep-th] (2012); “Refined Chern–Simons theory and topological string,” arXiv:1210.2733v1 [hep-th] (2012).

  44. P. Dunin-Barkowski, A. Mironov, A. Morozov, A. Sleptsov, and A. Smirnov, “Superpolynomials for torus knots from evolution induced by cut-and-join operators,” JHEP, 1303, 021 (2013); arXiv:1106.4305v4 [hep-th] (2011).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. I. Cherednik, “Jones polynomials of torus knots via DAHA,” Int. Math. Res. Not., 2013, 5366–5425 (2013); arXiv:1111.6195v10 [math.QA] (2011).

    Article  MathSciNet  MATH  Google Scholar 

  46. A. Mironov, A. Morozov, Sh. Shakirov, and A. Sleptsov, “Interplay between MacDonald and Hall–Littlewood expansions of extended torus superpolynomials,” JHEP, 1205, 70 (2012); arXiv:1201.3339v2 [hep-th] (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. A. Mironov, A. Morozov, and Sh. Shakirov, “Torus HOMFLY as the Hall–Littlewood polynomials,” J. Phys. A: Math. Theor., 45, 355202 (2012); arXiv:1203.0667v1 [hep-th] (2012).

    Article  MATH  Google Scholar 

  48. A. Yu. Morozov, “Challenges of ß-deformation,” Theor. Math. Phys., 173, 1417–1437 (2012); arXiv:1201.4595v2 [hep-th] (2012).

    Article  MathSciNet  MATH  Google Scholar 

  49. H. Itoyama, A. Mironov, A. Morozov, and An. Morozov, “HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations,” JHEP, 1207, 131 (2012); arXiv:1203.5978v5 [hep-th] (2012).

    Article  ADS  MathSciNet  Google Scholar 

  50. H. Fuji, S. Gukov, and P. Sulkovski, “Super-A-polynomial for knots and BPS states,” Nucl. Phys. B, 867, 506–546 (2013); arXiv:1205.1515v2 [hep-th] (2012).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  51. Ant. Morozov, “Special colored superpolynomials and their representation-dependence,” JHEP, 1212, 116 (2012); arXiv:1208.3544v1 [hep-th] (2012).

    Article  ADS  MathSciNet  Google Scholar 

  52. S. Nawata, P. Ramadevi, Zodinmawia, and X. Sun, “Super-A-polynomials for twist knots,” JHEP, 1211, 157 (2012); arXiv:1209.1409v4 [hep-th] (2012).

    Article  ADS  MathSciNet  Google Scholar 

  53. H. Fuji and P. Sulkowski, “Super-A-polynomial,” in: String-Math 2012 (Proc. Symp. Pure Math., Vol. 90, R. Donagi, S. Katz, A. Klemm, and D. R. Morrison, eds.), Amer. Math. Soc., Providence, R. I. (2015), pp. 277–304; arXiv:1303.3709v1 [math.AG] (2013).

    Google Scholar 

  54. S. Nawata, P. Ramadevi, and Zodinmawia, “Colored Kauffman homology and super-A-polynomials,” JHEPs, 1401, 126 (2014); arXiv:1310.2240v4 [hep-th] (2013).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. A. Mironov, A. Morozov, A. Sleptsov, and A. Smirnov, “On genus expansion of superpolynomials,” Nucl. Phys. B, 889, 757–777 (2014); arXiv:1310.7622v2 [hep-th] (2013).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  56. Yu. Berest and P. Samuelson, “Double affine Hecke algebras and generalized Jones polynomials,” Compositio Mathematica, 152, 1333–1384 (2016); arXiv:1402.6032v2 [math.QA] (2014).

    Article  MathSciNet  MATH  Google Scholar 

  57. I. Cherednik and I. Danilenko, “DAHA and iterated torus knots,” Algebr. Geom. Topol., 16, 843–898 (2016); arXiv:1408.4348v2 [math.QA] (2014).

    Article  MathSciNet  MATH  Google Scholar 

  58. S. Arthamonov and Sh. Shakirov, “Refined Chern–Simons theory in genus two,” arXiv:1504.02620v3 [hep-th] (2015).

  59. E. Witten, “Two lectures on gauge theory and Khovanov homology,” arXiv:1603.03854v2 [math.GT] (2016).

  60. S. Gukov, S. Nawata, I. Saberi, M. Stosic, and P. Sulkowski, “Sequencing BPS spectra,” JHEP, 1603, 004 (2016); arXiv:1512.07883v3 [hep-th] (2015).

    Article  ADS  MathSciNet  Google Scholar 

  61. S. Nawata and A. Oblomkov, “Lectures on knot homology,” in: Physics and Mathematics of Link Homology (Contemp. Math., Vol. 680, S. Gukov, M. Khovanov, and J. Walcher, eds.), Amer. Math. Soc., Providence, R. I. (2016), pp. 137–177; arXiv:1510.01795v4 [math-ph] (2015).

    Chapter  Google Scholar 

  62. D. Bar-Natan, “On Khovanov’s categorification of the Jones polynomial,” Algebr. Geom. Topol., 2, 337–370 (2002); arXiv:math/0201043v3 [math.QA] (2002)

    Article  MathSciNet  MATH  Google Scholar 

  63. 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, 12–81 (2014); arXiv:1308.5759v2 [hep-th] (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  64. A. Morozov, And. Morozov, and A. Popolitov, “On matrix-model approach to simplified Khovanov–Rozansky calculus,” Phys. Lett. B, 749, 309–325 (2015); arXiv:1506.07516v2 [hep-th] (2015)

    Article  ADS  MATH  Google Scholar 

  65. A. Yu. Morozov, A. A. Morozov, and A. V. Popolitov, “Matrix model and dimensions at hypercube vertices,” Theor. Math. Phys., 192, 1039–1079 (2017).

    Article  MathSciNet  Google Scholar 

  66. A. Anokhina and A. Morozov, “Towards R-matrix construction of Khovanov–Rozansky polynomials: I. Primary T-deformation of HOMFLY,” JHEP, 1407, 063 (2014); arXiv:1403.8087v2 [hep-th] (2014).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  67. N. Yu. Reshetikhin and V. G. Turaev, “Ribbon graphs and their invariants derived from quantum groups,” Commun. Math. Phys., 127, 1–26 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  68. E. Guadagnini, M. Martellini, and M. Mintchev, “Chern–Simons field theory and quantum groups,” in: Quantum Groups (Lect. Notes Phys., Vol. 370, H. D. Doebner and J. D. Hennig, eds.), World Scientific, Singapore (1990), pp. 307–317; “Chern–Simons holonomies and the appearance of quantum groups,” Phys. Lett. B, 235, 275–281 (1990)

    Google Scholar 

  69. V. G. Turaev and O. Ya. Viro, “State sum invariants of 3-manifolds and quantum 6j-symbols,” Topology, 31, 865–902 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  70. 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, 284–313 (2010); arXiv:1001.2003v2 [hep-th] (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  71. A. Smirnov, “Notes on Chern–Simons theory in the temporal gauge,” arXiv:0910.5011v1 [hep-th] (2009).

  72. R. K. Kaul and T. R. Govindarajan, “Three-dimensional Chern–Simons theory as a theory of knots and links,” Nucl. Phys. B, 380, 293–333 (1992); arXiv:hep-th/9111063v1 (1991); “Three-dimensional Chern–Simons theory as a theory of knots and links: II. Multicoloured links,” Nucl. Phys. B, 393, 392–412 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  73. P. Rama Devi, T. R. Govindarajan, and R. K. Kaul, “Three-dimensional Chern–Simons theory as a theory of knots and links: III. Compact semi-simple group,” Nucl. Phys. B, 402, 548–566 (1993); arXiv:hep-th/9212110v1 (1992); “Knot invariants from rational conformal field theories,” Nucl. Phys. B, 422, 291–306 (1994); arXiv:hepth/9312215v1 (1993); “Representations of composite braids and invariants for mutant knots and links in Chern–Simons field theories,” Modern Phys. Lett. A, 10, 1635–1658 (1995); arXiv:hep-th/9412084v1 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  74. P. Ramadevi and T. Sarkar, “On link invariants and topological string amplitudes,” Nucl. Phys. B, 600, 487–511 (2001); arXiv:hep-th/0009188v4 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  75. Zodinmawia and P. Ramadevi, “SU(N) quantum Racah coefficients and non-torus links,” Nucl. Phys. B, 870, 205–242 (2013); arXiv:1107.3918v7 [hep-th] (2011); “Reformulated invariants for non-torus knots and links,” arXiv:1209.1346v1 [hep-th] (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  76. S. Nawata, P. Ramadevi, and Zodinmawia, “Colored HOMFLY polynomials from Chern–Simons theory,” J. Knot Theory Ramifications, 22, 1350078 (2013); arXiv:1302.51444 [hep-th] (2013).

  77. A. Mironov, A. Morozov, and And. 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, Singapore (2013), pp. 101–118; arXiv:1112.5754v1 [hep-th] (2011); “Character expansion for HOMFLY polynomials: II. Fundamental representation. Up to five strands in braid,” JHEP, 1203, 034 (2012); arXiv:1112.2654v2 [math.QA] (2011)

    Google Scholar 

  78. S. Nawata, P. Ramadevi, and V. K. Singh, “Colored HOMFLY polynomials that distinguish mutant knots,” arXiv:1504.00364v2 [math.GT] (2015)

  79. A. Mironov, A. Morozov, An. Morozov, P. Ramadevi, V. Kumar Singh, and A. Sleptsov, “Colored HOMFLY polynomials of knots presented as double fat diagrams,” JHEP, 1507, 109 (2015); arXiv:1504.00371v3 [hep-th] (2015); “Tabulating knot polynomials for arborescent knots,” J. Phys. A: Math. Theor., 50, 085201 (2017); arXiv:1601.04199v2 [hep-th] (2016)

    Article  ADS  MathSciNet  Google Scholar 

  80. A. Mironov and A. Morozov, “Towards effective topological field theory for knots,” Nucl. Phys. B, 899, 395–413 (2015); arXiv:1506.00339v2 [hep-th] (2015).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  81. L. H. Kauffman, “State models and the Jones polynomial,” Topology, 26, 395–407 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  82. L. H. Kauffman and P. Vogel, “Link polynomials and a graphical calculus,” J. Knot Theory Ramifications, 1, 59–104 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  83. P. Vogel, “The universal Lie algebra,” http://www.math.jussieu.fr/~vogel/A299.ps.gz (1999); “Algebraic structures on modules of diagrams,” J. Pure Appl. Algebra, 215, 1292–1339 (2011)

    Article  MathSciNet  Google Scholar 

  84. P. Deligne, “La série,” C. R. Acad. Sci. Paris. Sér. I, 322, 321–326 (1996)

    MATH  Google Scholar 

  85. A. Cohen and R. de Man, “Computational evidence for Deligne’s conjecture regarding exceptional Lie groups,” C. R. Acad. Sci. Paris. Sér. I, 322, 427–432 (1996)

    MathSciNet  MATH  Google Scholar 

  86. J. M. Landsberg and L. Manivel, “Series of Lie groups,” Michigan Math. J., 52, 453–479 (2004); arXiv:math.AG/0203241v2 (2002); “Triality, exceptional Lie algebras, and Deligne dimension formulas,” Adv. Math., 171, 59–85 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  87. A. Mironov, R. Mkrtchyan, and A. Morozov, “On universal knot polynomials,” JHEP, 1602, 078 (2016); arXiv:1510.05884v2 [hep-th] (2015); “Universal Racah matrices and adjoint knot polynomials: Arborescent knots,” Phys. Lett. B, 755, 47–57 (2016); arXiv:1511.09077v1 [hep-th] (2015).

    Article  ADS  MathSciNet  Google Scholar 

  88. D. Galakhov, D. Melnikov, A. Mironov, A. Morozov, and A. Sleptsov, “Colored knot polynomials for arbitrary pretzel knots and links,” Phys. Lett. B, 743, 71–74 (2015); arXiv:1412.2616v1 [hep-th] (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  89. A. Mironov, A. Morozov, and A. Sleptsov, “Colored HOMFLY polynomials for the pretzel knots and links,” JHEP, 1507, 069 (2015); arXiv:1412.8432v2 [hep-th] (2014).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  90. A. Mironov, A. Morozov, and And. Morozov, “Evolution method and ‘differential hierarchy’ of colored knot polynomials,” in: Nonlinear and Modern Mathematical Physics (AIP Conf. Proc., Vol. 1562, W.-X. Ma and D. Kaup, eds.), AIP, Melville, N. Y. (2013), pp. 123–155; arXiv:1306.3197v1 [hep-th] (2013).

    Google Scholar 

  91. S. Arthamonov, A. Mironov, A. Morozov, and An. Morozov, “Link polynomial calculus and the AENV conjecture,” JHEP, 1704, 156 (2017); arXiv:1309.7984v3 [hep-th] (2013).

    ADS  Google Scholar 

  92. Ya. Kononov and A. Morozov, “On the defect and stability of differential expansion,” JETP Lett., 101, 831–834 (2015); arXiv:1504.07146v3 [hep-th] (2015).

    Article  ADS  Google Scholar 

  93. A. Morozov, “Differential expansion and rectangular HOMFLY for the figure eight knot,” Nucl. Phys., 911, 582–605 (2015); arXiv:1605.09728v3 [hep-th] (2016).

    Article  ADS  MATH  Google Scholar 

  94. A. S. Anokhina and A. A. Morozov, “Cabling procedure for the colored HOMFLY polynomials,” Theor. Math. Phys., 178, 1–58 (2014); arXiv:1307.2216v2 [hep-th] (2013).

    Article  MathSciNet  MATH  Google Scholar 

  95. D. Galakhov and G. W. Moore, “Comments on the two-dimensional Landau–Ginzburg approach to link homology,” arXiv:1607.04222v1 [hep-th] (2016).

  96. A. Anokhina, A. Mironov, A. Morozov, and An. Morozov, “Knot polynomials in the first non-symmetric representation,” Nucl. Phys. B, 882, 171–194 (2014); arXiv:1211.6375v1 [hep-th] (2012).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  97. A. Mironov, A. Morozov, An. Morozov, and A. Sleptsov, “Colored knot polynomials: HOMFLY in representation [2, 1],” Internat. J. Modern Phys. A, 30, 1550169 (2015); arXiv:1508.02870v1 [hep-th] (2015).

    Article  ADS  MATH  Google Scholar 

  98. A. Mironov, A. Morozov, An. Morozov, and A. Sleptsov, “HOMFLY polynomials in representation [3, 1] for 3-strand braids,” JHEP, 1609, 134 (2016); arXiv:1605.02313v1 [hep-th] (2016).

    Article  ADS  MathSciNet  Google Scholar 

  99. K. Liu and P. Peng, “Proof of the Labastida–Mari˜no–Ooguri–Vafa conjecture,” J. Differential Geom., 85, 479–525 (2010); arXiv:0704.1526v3 [math.QA] (2007)

    Article  MathSciNet  MATH  Google Scholar 

  100. S. Zhu, “Colored HOMFLY polynomial via skein theory,” JHEP, 10, 229 (2013); arXiv:1206.5886v1 [math.GT] (2012).

    Article  ADS  Google Scholar 

  101. A. Morozov, “Factorization of differential expansion for antiparallel double-braid knots,” JHEP, 1609, 135 (2016); arXiv:1606.06015v7 [hep-th] (2016).

    Article  ADS  Google Scholar 

  102. P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, New York (1994).

    Book  MATH  Google Scholar 

  103. Ya. Kononov and A. Morozov, “Factorization of colored knot polynomials at roots of unity,” Phys. Lett. B, 747, 500–510 (2015); arXiv:1505.06170v1 [hep-th] (2015).

    Article  ADS  MATH  Google Scholar 

  104. A. Morozov, “The first-order deviation of superpolynomial in an arbitrary representation from the special polynomial,” JETP Lett., 97, 171–172 (2013); arXiv:1211.4596v2 [hep-th] (2012).

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. National Research University “Higher School of Economics,”, Moscow, Russia

    Ya. A. Kononov

  2. Landau Institute for Theoretical Physics, RAS, Chernogolovka, Moscow Oblast, Russia

    Ya. A. Kononov

  3. Institute for Theoretical and Experimental Physics, Moscow, Russia

    A. Yu. Morozov

  4. National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Moscow, Russia

    A. Yu. Morozov

  5. Institute for Information Transmission Problems, RAS, Moscow, Russia

    A. Yu. Morozov

Authors
  1. Ya. A. Kononov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Yu. Morozov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ya. A. Kononov.

Additional information

This research is supported by the Russian Foundation for Basic Research (Grant Nos. 15-51-52031-HHC a, 15-52-50041-YaF, 16-51-53034-GFEN, and 16-51-45029-Ind).

The research of Ya. A. Kononov is supported in part by the Russian Foundation for Basic Research (Grant Nos. 16-01-00291 and 16-31-00484-mol_a) and the Simons Foundation.

The research of A. Yu. Morozov is supported in part by the Russian Foundation for Basic Research (Grant Nos. 16-02-01021 and 15-31-20832-mol_a_ved).

Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 193, No. 2, pp. 256–275, November, 2017.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kononov, Y.A., Morozov, A.Y. Rectangular superpolynomials for the figure-eight knot 41. Theor Math Phys 193, 1630–1646 (2017). https://doi.org/10.1134/S0040577917110058

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0040577917110058

Keywords

Search

Navigation