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Annales Henri Poincaré

, Volume 16, Issue 8, pp 1937–1967 | Cite as

Torus Knots in Lens Spaces and Topological Strings

  • Sebastien StevanEmail author
Article

Abstract

We study the invariant of knots in lens spaces defined from quantum Chern–Simons theory. By means of the knot operator formalism, we derive a generalization of the Rosso-Jones formula for torus knots in L(p,1). In the second part of the paper, we propose a B-model topological string theory description of torus knots in L(2,1).

Keywords

Topological String Spectral Curve Algebraic Curve Bergman Kernel Lens Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Aganagic M., Klemm A., Vafa C.: Disk instantons, mirror symmetry and the duality Web. Z. Naturforsch. 57, 1–28 (2002). arXiv:hep-th/0105045 MathSciNetADSCrossRefzbMATHGoogle Scholar
  2. 2.
    Aganagic, M., Vafa, C.: Mirror symmetry, D-branes and counting holomorphic discs. (2000). arXiv:hep-th/0012041.
  3. 3.
    Aganagic, M., Vafa, C.: Large N duality, mirror symmetry, and a Q-deformed A-polynomial for Knots. (2012). arXiv:1204.4709
  4. 4.
    Aganagic, M., et al.:Matrix model as a mirror of Chern–Simons theory. J. High Energy Phys. 2, 10 (2004). arXiv:hep-th/0211098
  5. 5.
    Akemann G.: Higher genus correlators for the hermitian matrix model with multiple cuts. Nucl. Phys. B 482, 403–430 (1996) arXiv:hep-th/9606004 MathSciNetADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Auckly D., Koshkin S.: Introduction to the Gopakumar-Vafa Large N Duality. Geom. Top. Monogr. 8, 195–456 (2006). arXiv:math.GT/0701568 MathSciNetGoogle Scholar
  7. 7.
    Beasley C., Witten E.: Non-abelian localization for Chern–Simons theory. J. Diff. Geom. 70, 183–323 (2005). arXiv:hep-th/0503126 MathSciNetzbMATHGoogle Scholar
  8. 8.
    Berge J.: The knots in D 2 × S 1 which have nontrivial Dehn surgeries that yield D 2 × S 1. Topol. Appl. 38, 1–19 (1991)MathSciNetADSCrossRefzbMATHGoogle Scholar
  9. 9.
    Bonahon F.: Difféotopies des espaces lenticulaires. Topology 22(3), 305–314 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bouchard, V., et al.: Remodeling the B-model. Commun. Math. Phys. 287, 117–178 (2009). arXiv:0709.1453 [hep-th]
  11. 11.
    Brini A., Eynard B., Mariño M.: Torus knots and mirror symmetry. Ann. Henri Poincar é13(8), 1873–1910 (2012). arXiv:1105.2012 ADSCrossRefzbMATHGoogle Scholar
  12. 12.
    Brini, A., et al.: Chern–Simons theory on L(p,q) lens spaces and Gopakumar-Vafa duality. J. Geom. Phys. 60, 417–429 (2010). arXiv:0809.1610 [math-ph]
  13. 13.
    Cattabriga, A., Manfredi, E., Mulazzani, M.: On knots and links in lens spaces. (2012). arXiv:1209.6532 [math.GT]
  14. 14.
    Chern S.-S., Simons J.H.: Characteristic forms and geometric invariants. Ann. Math. 99(1), 48–69 (1974)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Cornwell, C.: A polynomial invariant for links in lens spaces. J. Knot Theory Ramif. 21, 1250060 (2012). arXiv:1002.1543 [math.GT]
  16. 16.
    Diaconescu, D.-E., Shende, V., Vafa, C.: Large N duality, lagrangian cycles, and algebraic knots. (2011). arXiv:1111.6533 [hep-th]
  17. 17.
    Drobotukhina J.: An analogue of the Jones polynomial for links in \({\mathbb{R}P^3}\) and a generalization of the Kauffman-Murasugi Theorem. Leningrad Math. J. 2(3), 613–630 (1991)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Eynard B., Orantin N.: Invariants of algebraic curves and topological expansion. Commun. Number Theory Phys. 1(2), 347–452 (2007). arXiv:math-ph/0702045 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Freyd, P., et al. A new polynomial invariant of knots and links. Bull. Am. Math. Soc. 12, 239–246 (1985). euclid.bams/1183552531
  20. 20.
    Gabai D.: Surgery on knots in solid tori. Topology 28(1), 1–6 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gabai D.: 1-Bridge braids in solid tori. Topol. Appl. 37, 221–235 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gang, D.: Chern–Simons theory on L(p,q) lens spaces and localization. (2009). arXiv:0912.4664 [hep-th]
  23. 23.
    Geiges, H., Onaran, S.: Legendrian rational unknots in lens spaces. (2013). arXiv:1302.3792 [math.SG]
  24. 24.
    Gepner D., Witten E.: String theory on group manifolds. Nucl. Phys. B 278(3), 493–549 (1986)MathSciNetADSCrossRefGoogle Scholar
  25. 25.
    Gopakumar R., Vafa C.: On the gauge theory/geometry correspondence. Adv. Theor. Math. Phys. 3, 1415–1443 (1999). arxiv:hep-th/9811131 MathSciNetzbMATHGoogle Scholar
  26. 26.
    Halmagyi, N., Yasnov, V.: The spectral curve of the lens space matrix model. J. High Energy Phys. 11, 104 (2009). arXiv:hep-th/0311117
  27. 27.
    Hansen S.K., Takata T.: Reshetikhin-Turaev invariants of Seifert 3-manifolds for classical simple Lie algebras. J. Knot Theory Ramif. 13(5), 617–668 (2004). arXiv:math/0209403 MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Hori K., et al.: Mirror symmetry. Am. Math. Soc. (2003)Google Scholar
  29. 29.
    Hoste J., Przytycki J.H.: The (2, ∞)-skein module of lens spaces; a generalization of the Jones polynomial. J. Knot Theory Ramif. 2(3), 321–333 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Huynh V.Q., Lê T.T.Q.: Twisted alexander polynomial of links in the projective space. J. Knot Theory Ramif. 17(4), 411–438 (2008). arXiv:0706.2017 [math.GT]CrossRefzbMATHGoogle Scholar
  31. 31.
    Isidro J.M., Labastida J.M.F., Ramallo A.V.: Polynomials for torus links from Chern–Simons gauge theories. Nucl. Phys. B 398, 187–236 (1993). arXiv:hep-th/9210124 MathSciNetADSCrossRefGoogle Scholar
  32. 32.
    Jockers, H., Klemm, A., Soroush, M.: Torus knots and the topological vertex. (2012). arXiv:1212.0321 [hep-th]
  33. 33.
    Jones V.F.R.: On knot invariants related to some statistical mechanical models. Pacific J. Math. 137(2), 311–334 (1989). euclid.pjm/1102650387 MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Kac V.G., Peterson D.H.: Infinite-dimensional lie algebras, theta functions and modular forms. Adv. Math. 53, 125–264 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Kalfagianni, E.: An intrinsic approach to invariants of framed links in 3-manifolds. Q. Top. 2, 71–96 (2011) arXiv:1001.0174 [math.GT]
  36. 36.
    Kalfagianni E., Lin X.-S.: The HOMFLY polynomial for links inrational homology 3-spheres. Topology 38(1), 95–115 (1999). arXiv:q-alg/9509010 MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Labastida J.M.F., Llatas P.M., Ramallo A.V.: Knot operators in Chern–Simons gauge theory. Nucl. Phys. B 348, 651–692 (1991)MathSciNetADSCrossRefGoogle Scholar
  38. 38.
    Labastida J.M.F., Mariño M.: Polynomial invariants for torus knots and topological strings. Commun. Math. Phys. 217, 423–449 (2001). arXiv:hep-th/0004196 ADSCrossRefzbMATHGoogle Scholar
  39. 39.
    Labastida, J.M.F., Mariño, M., Vafa, C.: Knots, links and branes at large N. J. High Energy Phys. 11, 7 (2000). arXiv:hep-th/0010102
  40. 40.
    Labastida J.M.F., Ramallo A.V.: Operator formalism for Chern–Simons theories. Phys. Lett. B 227, 92–102 (1989)MathSciNetADSCrossRefGoogle Scholar
  41. 41.
    Lin X.-S.: Representations of knot groups and twisted Alexander polynomials. Acta Math. Sin. 17(3), 361–380 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Lin X.-S., Zheng H.: On the Hecke algebras and the colored HOMFLY polynomial. Trans. Am. Math. Soc. 362, 1–18 (2010). arXiv:math/0601267 MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Mariño M.: Chern–Simons theory, matrix integrals, and perturbative three-manifold invariants. Commun. Math. Phys. 253, 25–49 (2005). arXiv:hep-th/0207096 ADSCrossRefzbMATHGoogle Scholar
  44. 44.
    Marino, M.: Open string amplitudes and large order behavior in topological string theory. J. High Energy Phys. 03 60 (2008). arXiv:hep-th/0612127
  45. 45.
    Mariño M.: String theory and the Kauffman polynomial. Commun. Math. Phys. 298, 613–643 (2010). arXiv:0904.1088 [hep-th]ADSCrossRefzbMATHGoogle Scholar
  46. 46.
    Marino, M., Pasquetti, S., Putrov, P.: Large N duality beyond the genus expansion. J. High Energy Phys. 07, 74 (2010). arXiv:0911.4692 [hep-th]
  47. 47.
    Morton H.R.: Invariants of links and 3-manifolds from skein theory and from quantum groups. In: Bozhüyük, M.E. (eds) Topics in knot theory, number 399 in NATO Adv. Sci. Inst. Ser., pp. 107–156. Kluwer Acad. Publ, Dordrecht (1993)Google Scholar
  48. 48.
    Morton H.R.: Skein theory and the murphy operators. J. Knot Theory Ramif. 11, 475–492 (2002). arXiv:math/0102098 MathSciNetADSCrossRefzbMATHGoogle Scholar
  49. 49.
    Morton H.R., Manchón P.M.G.: Geometrical relations and plethysms in the Homfly skein of the annulus. J. Lond. Math. Soc. 78, 305–328 (2008). arXiv:0707.2851 [math.GT]MathSciNetADSCrossRefzbMATHGoogle Scholar
  50. 50.
    Onaran, S.C.: Legendrian knots in lens spaces. (2010). arXiv:1012.3047 [math.GT]
  51. 51.
    Ooguri H., Vafa C.: Knot invariants and topological strings. Nucl. Phys. B 577, 419–438 (2000). arXiv:hep-th/9912123 MathSciNetADSCrossRefzbMATHGoogle Scholar
  52. 52.
    Prasolov, V.V., Sossinsky, A.B.: Knots, links, braids and 3-manifolds. Translations of Mathematical Monographs, vol. 154, pp. 191–194. American Mathematical Society (1997)Google Scholar
  53. 53.
    Przytycki J.H.: Skein modules of 3-manifolds. Bull. Polish Acad. Sci. 39, 91–100 (1991). arXiv:math/0611797 MathSciNetzbMATHGoogle Scholar
  54. 54.
    Rosso M., Jones V.: On the invariants of torus knots derived from quantum groups. J. Knot Theory Ramif. 2(1), 97–112 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Rozansky L.: A contribution of the trivial connection to jones polynomial and Witten’s invariant of 3d manifolds I. Commun. Math. Phys. 175(2), 275–296 (1996). euclid.cmp/1104275925 MathSciNetADSCrossRefzbMATHGoogle Scholar
  56. 56.
    Stevan S.: Chern–Simons invariants of torus links. Ann. Henri Poincaré 11((7), 1201–1224 (2010). arXiv:1003.2861 [hep-th]MathSciNetADSCrossRefzbMATHGoogle Scholar
  57. 57.
    ’t Hooft G.: A planar diagram theory for strong interactions. Nucl. Phys. B 72, 461–473 (1974)MathSciNetADSCrossRefGoogle Scholar
  58. 58.
    Taubes C.H.: Lagrangians for the Gopakumar-Vafa conjecture. Geom. Top. Monogr. 8, 73–95 (2006). arXiv:math/0201219 MathSciNetCrossRefGoogle Scholar
  59. 59.
    Turaev V.G.: Conway and Kauffman modules of a solid torus. J. Math. Sci. 52(1), 2799–2805 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Witten E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351–399 (1989). euclid.cmp/1104178138 ADSCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Section de mathématiquesUniversité de GenèveGenève 4Switzerland

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