Sequencing BPS spectra

  • Sergei Gukov
  • Satoshi Nawata
  • Ingmar Saberi
  • Marko Stošić
  • Piotr Sułkowski
Open Access
Regular Article - Theoretical Physics

Abstract

This paper provides both a detailed study of color-dependence of link homologies, as realized in physics as certain spaces of BPS states, and a broad study of the behavior of BPS states in general. We consider how the spectrum of BPS states varies as continuous parameters of a theory are perturbed. This question can be posed in a wide variety of physical contexts, and we answer it by proposing that the relationship between unperturbed and perturbed BPS spectra is described by a spectral sequence. These general considerations unify previous applications of spectral sequence techniques to physics, and explain from a physical standpoint the appearance of many spectral sequences relating various link homology theories to one another. We also study structural properties of colored HOMFLY homology for links and evaluate Poincaré polynomials in numerous examples. Among these structural properties is a novel “sliding” property, which can be explained by using (refined) modular S-matrix. This leads to the identification of modular transformations in Chern-Simons theory and 3d \( \mathcal{N}=2 \) theory via the 3d/3d correspondence. Lastly, we introduce the notion of associated varieties as classical limits of recursion relations of colored superpolynomials of links, and study their properties.

Keywords

Differential and Algebraic Geometry Supersymmetry and Duality Topological Field Theories Topological Strings 

References

  1. [1]
    H. Ooguri and C. Vafa, Knot invariants and topological strings, Nucl. Phys. B 577 (2000) 419 [hep-th/9912123] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    S. Gukov, A.S. Schwarz and C. Vafa, Khovanov-Rozansky homology and topological strings, Lett. Math. Phys. 74 (2005) 53 [hep-th/0412243] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    S. Gukov, Gauge theory and knot homologies, Fortsch. Phys. 55 (2007) 473 [arXiv:0706.2369] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    D.E. Diaconescu, V. Shende and C. Vafa, Large-N duality, lagrangian cycles and algebraic knots, Commun. Math. Phys. 319 (2013) 813 [arXiv:1111.6533] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    E. Witten, Fivebranes and knots, arXiv:1101.3216 [INSPIRE].
  6. [6]
    E.S. Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197 (2005) 554 [math/0210213].MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    B. Gornik, Note on Khovanov link cohomology, math/0402266.
  8. [8]
    N.M. Dunfield, S. Gukov and J. Rasmussen, The superpolynomial for knot homologies, Exper. Math.15 (2006) 129 [math/0505662] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    J.A. Rasmussen, Some differentials on Khovanov-Rozansky homology, math/0607544.
  10. [10]
    J.A. Dixon, Calculation of BRS cohomology with spectral sequences, Commun. Math. Phys. 139 (1991) 495 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    P. Bouwknegt, J.G. McCarthy and K. Pilch, BRST analysis of physical states for 2D gravity coupled to c < 1 matter, Commun. Math. Phys. 145 (1992) 541 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    J. de Boer and T. Tjin, Quantization and representation theory of finite W algebras, Commun. Math. Phys. 158 (1993) 485 [hep-th/9211109] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    M. Bertolini, I.V. Melnikov and M.R. Plesser, Hybrid conformal field theories, JHEP 05 (2014) 043 [arXiv:1307.7063] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    K. Wong, Spectral sequences and vacua in N = 2 gauged linear quantum mechanics with potentials, arXiv:1511.05159 [INSPIRE].
  15. [15]
    R. Bott and L. Tu, Differential forms in algebraic topology, Springer Verlag, New York U.S.A. (1982).CrossRefMATHGoogle Scholar
  16. [16]
    G. Kato, The heart of cohomology, Springer Science & Business Media, The Netherlands (2006).MATHGoogle Scholar
  17. [17]
    E. Witten, Dynamical breaking of supersymmetry, Nucl. Phys. B 188 (1981) 513 [INSPIRE].ADSCrossRefMATHGoogle Scholar
  18. [18]
    E. Witten, Supersymmetry and Morse theory, J. Diff. Geom. 17 (1982) 661 [INSPIRE].MathSciNetMATHGoogle Scholar
  19. [19]
    C. Romelsberger, Counting chiral primaries in N = 1, D = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    H. Kim and I. Saberi, Real homotopy theory and supersymmetric quantum mechanics, arXiv:1511.00978 [INSPIRE].
  21. [21]
    E. Witten, On the Landau-Ginzburg description of N = 2 minimal models, Int. J. Mod. Phys. A 9 (1994) 4783 [hep-th/9304026] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of two-dimensional N = 2 gauge theories with rank-one gauge groups, Lett. Math. Phys. 104 (2014) 465 [arXiv:1305.0533] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    A. Gadde and S. Gukov, 2d index and surface operators, JHEP 03 (2014) 080 [arXiv:1305.0266] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    F. Cachazo, M.R. Douglas, N. Seiberg and E. Witten, Chiral rings and anomalies in supersymmetric gauge theory, JHEP 12 (2002) 071 [hep-th/0211170] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, Gauge theories and Macdonald polynomials, Commun. Math. Phys. 319 (2013) 147 [arXiv:1110.3740] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    S. Gukov and M. Stošić, Homological algebra of knots and BPS states, Proc. Symp. Pure Math. 85 (2012) 125 [arXiv:1112.0030] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    E. Gorsky, A. Oblomkov, J. Rasmussen and V. Shende, Torus knots and the rational DAHA, Duke Math. J. 163 (2014) 2709 [arXiv:1207.4523] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    E. Gorsky, S. Gukov and M. Stošić, Quadruply-graded colored homology of knots, arXiv:1304.3481 [INSPIRE].
  30. [30]
    L. Crane and I. Frenkel, Four-dimensional topological field theory, Hopf categories and the canonical bases, J. Math. Phys. 35 (1994) 5136 [hep-th/9405183] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  32. [32]
    N. Yu. Reshetikhin and V.G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Commun. Math. Phys. 127 (1990) 1 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    M. Khovanov and L. Rozansky, Matrix factorizations and link homology, Fund. Math. 199 (2008) 1 [math/0401268].MathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    Y. Yonezawa, Quantum (sl n ,V n ) link invariant and matrix factorizations, Nagoya Math. J. 204 (2011) 69 [arXiv:0906.0220].MathSciNetCrossRefMATHGoogle Scholar
  35. [35]
    B. Webster, Knot invariants and higher representation theory II: the categorification of quantum knot invariants, arXiv:1005.4559.
  36. [36]
    H. Wu, A colored sl(N)-homology for links in S 3, arXiv:0907.0695.
  37. [37]
    B. Cooper and V. Krushkal, Categorification of the Jones-Wenzl projectors, Quant. Topol. 3 (2012) 139 [arXiv:1005.5117].MathSciNetCrossRefMATHGoogle Scholar
  38. [38]
    I. Frenkel, C. Stroppel and J. Sussan, Categorifying fractional Euler characteristics, Jones-Wenzl projector and 3j-symbols, Quant. Topol. 3 (2012) 181 [arXiv:1007.4680].MathSciNetCrossRefMATHGoogle Scholar
  39. [39]
    M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359 [math/9908171] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    M. Khovanov, sl(3) link homology, Algebr. Geom. Topol. 4 (2004) 1045 [math/0304375].MathSciNetCrossRefMATHGoogle Scholar
  41. [41]
    M. Khovanov and L. Rozansky, Matrix factorizations and link homology II, Geom. Topol. 12 (2008) 1387 [math/0505056].MathSciNetCrossRefMATHGoogle Scholar
  42. [42]
    B. Webster and G. Williamson, A geometric construction of colored HOMFLYPT homology, arXiv:0905.0486.
  43. [43]
    S. Gukov and J. Walcher, Matrix factorizations and Kauffman homology, hep-th/0512298 [INSPIRE].
  44. [44]
    S. Nawata, P. Ramadevi and Zodinmawia, Colored Kauffman homology and super-A-polynomials, JHEP 01 (2014) 126 [arXiv:1310.2240] [INSPIRE].
  45. [45]
    H. Awata, S. Gukov, P. Sulkowski and H. Fuji, Volume conjecture: refined and categorified, Adv. Theor. Math. Phys. 16 (2012) 1669 [arXiv:1203.2182] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  46. [46]
    H. Fuji, S. Gukov and P. Sulkowski, Super-A-polynomial for knots and BPS states, Nucl. Phys. B 867 (2013) 506 [arXiv:1205.1515] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  47. [47]
    H. Fuji, S. Gukov, M. Stošić and P. Sułkowski, 3d analogs of Argyres-Douglas theories and knot homologies, JHEP 01 (2013) 175 [arXiv:1209.1416] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    S. Nawata, P. Ramadevi, Zodinmawia and X. Sun, Super-A-polynomials for twist knots, JHEP 11 (2012) 157 [arXiv:1209.1409] [INSPIRE].
  49. [49]
    T. Dimofte, S. Gukov and L. Hollands, Vortex counting and Lagrangian 3-manifolds, Lett. Math. Phys. 98 (2011) 225 [arXiv:1006.0977] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  50. [50]
    M. Aganagic and S. Shakirov, Knot homology and refined Chern-Simons index, Commun. Math. Phys. 333 (2015) 187 [arXiv:1105.5117] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  51. [51]
    R. Gopakumar and C. Vafa, On the gauge theory/geometry correspondence, Adv. Theor. Math. Phys. 3 (1999) 1415 [hep-th/9811131] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  52. [52]
    Y. Terashima and M. Yamazaki, SL(2, R) Chern-Simons, Liouville and gauge theory on duality walls, JHEP 08 (2011) 135 [arXiv:1103.5748] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  53. [53]
    T. Dimofte, D. Gaiotto and S. Gukov, Gauge theories labelled by three-manifolds, Commun. Math. Phys. 325 (2014) 367 [arXiv:1108.4389] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  54. [54]
    T. Dimofte, D. Gaiotto and S. Gukov, 3-manifolds and 3d indices, Adv. Theor. Math. Phys. 17 (2013) 975 [arXiv:1112.5179] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  55. [55]
    T. Dimofte, M. Gabella and A.B. Goncharov, K-decompositions and 3d gauge theories, arXiv:1301.0192 [INSPIRE].
  56. [56]
    J. Yagi, 3d TQFT from 6d SCFT, JHEP 08 (2013) 017 [arXiv:1305.0291] [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    S. Lee and M. Yamazaki, 3d Chern-Simons theory from M5-branes, JHEP 12 (2013) 035 [arXiv:1305.2429] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  58. [58]
    C. Cordova and D.L. Jafferis, Complex Chern-Simons from M5-branes on the squashed three-sphere, arXiv:1305.2891 [INSPIRE].
  59. [59]
    H.-J. Chung, T. Dimofte, S. Gukov and P. Sułkowski, 3d-3d correspondence revisited, arXiv:1405.3663 [INSPIRE].
  60. [60]
    J.A. Harvey and G.W. Moore, Algebras, BPS states and strings, Nucl. Phys. B 463 (1996) 315 [hep-th/9510182] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  61. [61]
    I. Cherednik, Jones polynomials of torus knots via DAHA, arXiv:1111.6195 [INSPIRE].
  62. [62]
    S. Nawata and A. Oblomkov, Lectures on knot homology, arXiv:1510.01795 [INSPIRE].
  63. [63]
    A. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  64. [64]
    N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, arXiv:0908.4052 [INSPIRE].
  65. [65]
    S.H. Katz, A. Klemm and C. Vafa, Geometric engineering of quantum field theories, Nucl. Phys. B 497 (1997) 173 [hep-th/9609239] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  66. [66]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  67. [67]
    E. Frenkel, S. Gukov and J. Teschner, Surface operators and separation of variables, JHEP 01 (2016) 179 [arXiv:1506.07508] [INSPIRE].CrossRefADSGoogle Scholar
  68. [68]
    C.H. Taubes, Lagrangians for the Gopakumar-Vafa conjecture, Adv. Theor. Math. Phys. 5 (2001) 139 [math/0201219] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  69. [69]
    M. Khovanov and L. Rozansky, Virtual crossings, convolutions and a categorification of the SO(2N) Kauffman polynomial, math/0701333.
  70. [70]
    S. Chun, S. Gukov and D. Roggenkamp, Junctions of surface operators and categorification of quantum groups, arXiv:1507.06318 [INSPIRE].
  71. [71]
    I. Brunner and D. Roggenkamp, B-type defects in Landau-Ginzburg models, JHEP 08 (2007) 093 [arXiv:0707.0922] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  72. [72]
    H. Murakami, T. Ohtsuki and S. Yamada, Homfly polynomial via an invariant of colored planar graphs, Enseign. Math. 44 (1998) 325.MathSciNetMATHGoogle Scholar
  73. [73]
    J. Walcher, Stability of Landau-Ginzburg branes, J. Math. Phys. 46 (2005) 082305 [hep-th/0412274] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  74. [74]
    I. Brunner, D. Roggenkamp and S. Rossi, Defect perturbations in Landau-Ginzburg models, JHEP 03 (2010) 015 [arXiv:0909.0696] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  75. [75]
    P. Seidel and R.P. Thomas, Braid group actions on derived categories of coherent sheaves, Duke Math. 108 (2001) 37 [math/0001043] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  76. [76]
    L. Lewark and A. Lobb, New quantum obstructions to slicenes, arXiv:1501.07138.
  77. [77]
    D. Bar-Natan, On Khovanov’s categorification of the Jones polynomial, Algebr. Geom. Topol. 2 (2002) 337 [math/0201043].MathSciNetCrossRefMATHGoogle Scholar
  78. [78]
    K. Habiro, A unified Witten-Reshetikhin-Turaev invariant for integral homology spheres, Invent. Math. 171 (2007) 1 [math/0605314].ADSMathSciNetCrossRefMATHGoogle Scholar
  79. [79]
    G.E. Andrews, Multiple series Rogers-Ramanujan type identities, Pacific J. Math. 114 (1984) 267.MathSciNetCrossRefMATHGoogle Scholar
  80. [80]
    A. Beilinson and J. Bernstein, Localization of g-modules, Comptes Rendus Acad. Sci. Ser. I Math. 292 (1981) 15.MathSciNetMATHGoogle Scholar
  81. [81]
    D.A. Vogan, The method of coadjoint orbits for real reductive groups, in Representation theory of Lie groups 8, Park City UT U.S.A. (1998), pg. 179.Google Scholar
  82. [82]
    D. Cooper, M. Culler, H. Gillet, D. Long and P. Shalen, Plane curves associated to character varieties of 3-manifolds, Invent. Math. 118 (1994) 47.ADSMathSciNetCrossRefMATHGoogle Scholar
  83. [83]
    R.M. Kashaev, The hyperbolic volume of knots from quantum dilogarithm, Lett. Math. Phys. 39 (1997) 269 [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  84. [84]
    H. Murakami and J. Murakami, The colored Jones polynomials and the simplicial volume of a knot, Acta Math. 186 (2001) 85 [math/9905075].MathSciNetCrossRefMATHGoogle Scholar
  85. [85]
    S. Gukov, Three-dimensional quantum gravity, Chern-Simons theory and the A polynomial, Commun. Math. Phys. 255 (2005) 577 [hep-th/0306165] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  86. [86]
    S. Garoufalidis and T. Le, The colored Jones function is q-holonomic, Geom. Topol. 9 (2004) 1253 [math/0309214].MathSciNetCrossRefMATHGoogle Scholar
  87. [87]
    S. Garoufalidis, On the characteristic and deformation varieties of a knot, Geom. Topol. Monogr. 7 (2004) 291 [math/0306230].MathSciNetCrossRefMATHGoogle Scholar
  88. [88]
    L. Ng, Combinatorial knot contact homology and transverse knots, Adv. Math. 227 (2011) 2189 [arXiv:1010.0451].MathSciNetCrossRefMATHGoogle Scholar
  89. [89]
    L. Ng, Framed knot contact homology, Duke Math. J. 141 (2008) 365 [math/0407071].MathSciNetCrossRefMATHGoogle Scholar
  90. [90]
    M. Aganagic and C. Vafa, Large-N duality, mirror symmetry and a Q-deformed A-polynomial for knots, arXiv:1204.4709 [INSPIRE].
  91. [91]
    M. Aganagic, T. Ekholm, L. Ng and C. Vafa, Topological strings, D-model and knot contact homology, Adv. Theor. Math. Phys. 18 (2014) 827 [arXiv:1304.5778] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  92. [92]
    S. Arthamonov, A. Mironov, A. Morozov and A. Morozov, Link polynomial calculus and the AENV conjecture, JHEP 04 (2014) 156 [arXiv:1309.7984] [INSPIRE].ADSCrossRefGoogle Scholar
  93. [93]
    H. Fuji and P. Sulkowski, Super-A-polynomial, arXiv:1303.3709 [INSPIRE].
  94. [94]
    S. Gukov and P. Sulkowski, A-polynomial, B-model and quantization, JHEP 02 (2012) 070 [arXiv:1108.0002] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  95. [95]
    H.R. Morton and P.R. Cromwell, Distinguishing mutants by knot polynomials, J. Knot Theor. 5 (1996) 225.MathSciNetCrossRefMATHGoogle Scholar
  96. [96]
    S.M. Wehrli, Khovanov homology and Conway mutation, math/0301312.
  97. [97]
    E. Witten, SL(2, Z) action on three-dimensional conformal field theories with Abelian symmetry, hep-th/0307041 [INSPIRE].
  98. [98]
    A. Gadde, S. Gukov and P. Putrov, Walls, lines and spectral dualities in 3d gauge theories, JHEP 05 (2014) 047 [arXiv:1302.0015] [INSPIRE].ADSCrossRefGoogle Scholar
  99. [99]
    J.A. Rasmussen, Khovanov homology and the slice genus, Invent. Math. 182 (2010) 419 [math/0402131].ADSMathSciNetCrossRefMATHGoogle Scholar
  100. [100]
    S. Nawata, P. Ramadevi and Zodinmawia, Colored HOMFLY polynomials from Chern-Simons theory, J. Knot Theor. 22 (2013) 1350078 [arXiv:1302.5144] [INSPIRE].
  101. [101]
    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].ADSMathSciNetCrossRefGoogle Scholar
  102. [102]
    M. Rosso and V. Jones, On the invariants of torus knots derived from quantum groups, J. Knot Theor. 2 (1993) 97.MathSciNetCrossRefMATHGoogle Scholar
  103. [103]
    Y.-Z. Huang and L. Kong, Modular invariance for conformal full field algebras, Trans. Amer. Math. Soc. 362 (2010) 3027 [math/0609570] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  104. [104]
    A. Brini, B. Eynard and M. Mariño, Torus knots and mirror symmetry, Annales Henri Poincaré 13 (2012) 1873 [arXiv:1105.2012] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  105. [105]
    K. Kawagoe, On the formulae for the colored HOMFLY polynomials, arXiv:1210.7574 [INSPIRE].
  106. [106]
    K. Habiro, On the colored Jones polynomials of some simple links, Surikaisekikenkyusho Kokyuroku 1172 (2000) 34.MathSciNetMATHGoogle Scholar
  107. [107]
    K. Habiro, On the quantum sl 2 invariants of knots and integral homology spheres, Geom. Topol. Monogr. 4 (2002) 55 [math/0211044].MathSciNetCrossRefMATHGoogle Scholar
  108. [108]
    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].ADSCrossRefGoogle Scholar
  109. [109]
    K. Bringmann, K. Hikami and J. Lovejoy, On the modularity of the inified WRT invariants of certain Seifert manifold, Adv. Appl. Math. 46 (2011) 86.MathSciNetCrossRefMATHGoogle Scholar
  110. [110]
    E. Gorsky and A. Negut, Refined knot invariants and Hilbert schemes, J. Math. Pure. Appl. 104 (2015) 403 [arXiv:1304.3328] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  111. [111]
    I.G. Macdonald, Symmetric functions and Hall polynomials, second ed., Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York U.S.A. (1995).Google Scholar
  112. [112]
    S. Shakirov, Colored knot amplitudes and Hall-Littlewood polynomials, arXiv:1308.3838 [INSPIRE].
  113. [113]
    A. Iqbal and C. Kozcaz, Refined Hopf link revisited, JHEP 04 (2012) 046 [arXiv:1111.0525] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  114. [114]
    P. Wedrich, Categorified sl(N) invariants of colored rational tangles, arXiv:1404.2736.
  115. [115]
    J. Batson and C. Seed, A link splitting spectral sequence in Khovanov homology, Duke Math. J. 164 (2015) 801 [arXiv:1303.6240].MathSciNetCrossRefMATHGoogle Scholar
  116. [116]
    B. Cooper, private communication.Google Scholar
  117. [117]
    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., Singapore (2011), pg. 101 [arXiv:1112.5754].
  118. [118]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  119. [119]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  120. [120]
    H. Queffelec and D. Rose, Sutured annular Khovanov-Rozansky homology, arXiv:1506.08188.
  121. [121]
    R. Gelca, On the relation between the A-polynomial and the Jones polynomial, Proc. Amer. Math. Soc. 130 (2002) 1235 [math/0004158].MathSciNetCrossRefMATHGoogle Scholar
  122. [122]
    S. Garoufalidis, The colored HOMFLY polynomial is q-holonomic, arXiv:1211.6388 [INSPIRE].
  123. [123]
    J. Gu, H. Jockers, A. Klemm and M. Soroush, Knot invariants from topological recursion on augmentation varieties, Commun. Math. Phys. 336 (2015) 987 [arXiv:1401.5095] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  124. [124]
    T. Dimofte and S. Gukov, Quantum field theory and the volume conjecture, Contemp. Math. 541 (2011) 41 [arXiv:1003.4808] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  125. [125]
    L. Ng AugmentationVarietiesforLinks.nb, http://www.math.duke.edu/~ng/.
  126. [126]
    J.H. Conway, An enumeration of knots and links, and some of their algebraic properties, Comput. Probl. Abstr. Alg. (1970) 329.Google Scholar
  127. [127]
    M. Khovanov, Patterns in knot cohomology, I, Exper. Math. 12 (2003) 365 [math/0201306].MathSciNetCrossRefMATHGoogle Scholar
  128. [128]
    H. Jockers, A. Klemm and M. Soroush, Torus knots and the topological vertex, Lett. Math. Phys. 104 (2014) 953 [arXiv:1212.0321] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  129. [129]
    J.M.F. Labastida and M. Mariño, A new point of view in the theory of knot and link invariants, math/0104180 [INSPIRE].
  130. [130]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  131. [131]
    J.M.F. Labastida, M. Mariño and C. Vafa, Knots, links and branes at large-N, JHEP 11 (2000) 007 [hep-th/0010102] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  132. [132]
    S. Garoufalidis, P. Kucharski and P. Sulkowski, Knots, BPS states and algebraic curves, arXiv:1504.06327 [INSPIRE].
  133. [133]
    P. Paule, The concept of Bailey chains, http://www.emis.de/journals/SLC/opapers/s18paule.pdf.
  134. [134]
    N. Carqueville and D. Murfet, Computing Khovanov-Rozansky homology and defect fusion, Algebr. Geom. Topol. 14 (2014) 489 [arXiv:1108.1081] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  135. [135]
  136. [136]
    M. Stošić, Khovanov homology of torus links, Topol. Appl. 153 (2009) 533 [math/0606656].MathSciNetMATHGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Sergei Gukov
    • 1
    • 2
  • Satoshi Nawata
    • 1
    • 3
  • Ingmar Saberi
    • 1
  • Marko Stošić
    • 4
    • 5
  • Piotr Sułkowski
    • 1
    • 6
  1. 1.Walter Burke Institute for Theoretical PhysicsCalifornia Institute of TechnologyPasadenaU.S.A.
  2. 2.Max-Planck-Institut für MathematikBonnGermany
  3. 3.Centre for Quantum Geometry of Moduli SpacesUniversity of AarhusAarhusDenmark
  4. 4.CAMGSD, Departamento de Matemática, Instituto Superior TécnicoLisbonPortugal
  5. 5.Mathematical Institute SANUBelgradeSerbia
  6. 6.Faculty of PhysicsUniversity of WarsawWarsawPoland

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