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A unified Witten–Reshetikhin–Turaev invariant for integral homology spheres

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Abstract

We construct an invariant J M of integral homology spheres M with values in a completion \(\widehat{\mathbb{Z}[q]}\) of the polynomial ring ℤ[q] such that the evaluation at each root of unity ζ gives the the SU(2) Witten–Reshetikhin–Turaev invariant τζ(M) of M at ζ. Thus J M unifies all the SU(2) Witten–Reshetikhin–Turaev invariants of M. It also follows that τζ(M) as a function on ζ behaves like an “analytic function” defined on the set of roots of unity.

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References

  1. Akutsu, Y., Deguchi, T., Ohtsuki, T.: Invariants of colored links. J. Knot Theory Ramifications 1(2), 161–184 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bar-Natan, D., Garoufalidis, S.: On the Melvin–Morton–Rozansky conjecture. Invent. Math. 125(1), 103–133 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. Beliakova, A., Blanchet, C., Le, T.: Laplace transform and universal sl(2) invariants. Preprint, math.QA/0509394

  4. Cochran, T.D., Melvin, P.: Finite type invariants of 3-manifolds. Invent. Math. 140, 45–100 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Crane, L., Yetter, D.: On algebraic structures implicit in topological quantum field theories. J. Knot Theory Ramifications 8(2), 125–163 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. Fenn, R., Rourke, C.: On Kirby’s calculus of links. Topology 18(1), 1–15 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  7. Garoufalidis, S., Lê, T.T.Q.: Is the Jones polynomial of a knot really a polynomial? J. Knot Theory Ramifications 15, 983–1000 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Garoufalidis, S., Levine, J.: Tree-level invariants of three-manifolds, Massey products and the Johnson homomorphism. In: Graphs and Patterns in Mathematics and Theoretical Physics. Proc. Sympos. Pure Math., vol. 3, pp. 173–203. Am. Math. Soc., Providence, RI (2005)

    Google Scholar 

  9. Gilmer, P.M.: Integrality for TQFTs. Duke Math. J. 125(2), 389–413 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Gilmer, P.M., Masbaum, G.: Integral lattices in TQFT. To appear in Ann. Sci. Éc. Norm. Supér., IV. Sér.

  11. Gilmer, P.M., Masbaum, G., van Wamelen, P.: Integral bases for TQFT modules and unimodular representations of mapping class groups. Comment. Math. Helv. 79(2), 260–284 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Goussarov, M.: Finite type invariants and n-equivalence of 3-manifolds. C. R. Acad. Sci. Paris, Sèr. I 329(6), 517–522 (1999)

    MATH  MathSciNet  Google Scholar 

  13. Gouvêa, F.Q.: p-adic numbers. An introduction, 2nd edn. Universitext. Springer, Berlin (1997)

  14. Habegger, N.: Milnor, Johnson, and tree level perturbative invariants. Preprint

  15. Habiro, K.: Claspers and finite type invariants of links. Geom. Topol. 4, 1–83 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Habiro, K.: On the colored Jones polynomials of some simple links. Recent progress towards the volume conjecture (Kyoto, 2000). RIMS Kokyuroku 1172, 34–43 (2000)

    MATH  MathSciNet  Google Scholar 

  17. Habiro, K.: On the quantum sl 2 invariants of knots and integral homology spheres. In: Invariants of Knots and 3-manifolds (Kyoto 2001). Geom. Topol. Monogr., vol. 4, pp. 55–68 (2002)

  18. Habiro, K.: Cyclotomic completions of polynomial rings. Publ. Res. Inst. Math. Sci. 40, 1127–1146 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  19. Habiro, K.: An integral form of the quantized enveloping algebra of sl 2 and its completions. J. Pure Appl. Algebra 211, 265–292 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. Habiro, K.: Bottom tangles and universal invariants. Algebr. Geom. Topol. 6, 1113–1214 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  21. Habiro, K.: Refined Kirby calculus for integral homology spheres. Geom. Topol. 10, 1285–1317 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. Habiro, K.: Spanning surfaces and the Jones polynomial. In preparation

  23. Habiro, K., Le, T.T.Q.: In preparation

  24. Hikami, K.: Quantum invariant, modular form, and lattice points. Int. Math. Res. Not. 2005(3), 121–154 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  25. Hikami, K.: On the quantum invariant for the Brieskorn homology spheres. Int. J. Math. 16(6), 661–685 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  26. Hikami, K.: Mock (false) theta functions as quantum invariants. Regul. Chaotic Dyn. 10(4), 509–530 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hikami, K.: Quantum invariants, modular forms, and lattice points II. J. Math. Phys. 47, 102301, 32pp. (2006)

    Google Scholar 

  28. Hoste, J.: A formula for Casson’s invariant. Trans. Am. Math. Soc. 297(2), 547–562 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  29. Huynh, V., Le, T.T.Q.: On the colored Jones polynomial and the Kashaev invariant. Fundam. Prikl. Mat. 11, 57–78 (2005)

    MathSciNet  Google Scholar 

  30. Jones, V.F.R.: A new knot polynomial and von Neumann algebras. Notices Am. Math. Soc. 33(2), 219–225 (1986)

    Google Scholar 

  31. Jones, V.F.R.: Hecke algebra representations of braid groups and link polynomials. Ann. Math. (2) 126, 335–388 (1987)

    Article  Google Scholar 

  32. Kashaev, R.M.: A link invariant from quantum dilogarithm. Modern Phys. Lett. A 10, 1409–1418 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  33. Kauffman, L.H.: Gauss codes, quantum groups and ribbon Hopf algebras. Rev. Math. Phys. 5(4), 735–773 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  34. Kauffman, L., Radford, D.E.: Oriented quantum algebras, categories and invariants of knots and links. J. Knot Theory Ramifications 10(7), 1047–1084 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  35. Kerler, T.: Genealogy of non-perturbative quantum invariants of 3–manifolds: The surgical family. In: Geometry and Physics (Aarhus, 1995). Lect. Notes Pure Appl. Math., vol. 184, pp. 503–547. Dekker, New York (1997)

    Google Scholar 

  36. Kerler, T.: Bridged links and tangle presentations of cobordism categories. Adv. Math. 141(2), 207–281 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  37. Kirby, R.: A calculus for framed links in S 3. Invent. Math. 45(1), 35–56 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  38. Kirby, R., Melvin, P.: The 3-manifold invariants of Witten and Reshetikhin–Turaev for sl(2,C). Invent. Math. 105(3), 473–545 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  39. Kirby, R., Melvin, P., Zhang, X.: Quantum invariants at the sixth root of unity. Commun. Math. Phys. 151(3), 607–617 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  40. Koblitz, N.: p-Adic Numbers, p-Adic Analysis, and Zeta-Functions, 2nd edn. Grad. Texts Math., vol. 58. Springer, New York (1984)

    Google Scholar 

  41. Kricker, A., Spence, B.: Ohtsuki’s invariants are of finite type. J. Knot Theory Ramifications 6(4), 583–597 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  42. Lawrence, R.J.: A universal link invariant using quantum groups. In: Differential Geometric Methods in Theoretical Physics (Chester, 1988), pp. 55–63. World Sci. Publishing, Teaneck, NJ (1989)

    Google Scholar 

  43. Lawrence, R.J.: A universal link invariant. In: The Interface of Mathematics and Particle Physics (Oxford, 1988). Inst. Math. Appl. Conf. Ser. New Ser., vol. 24, pp. 151–156. Oxford University Press, New York (1990)

    Google Scholar 

  44. Lawrence, R.J.: Asymptotic expansions of Witten–Reshetikhin–Turaev invariants for some simple 3-manifolds. J. Math. Phys. 36, 6106–6129 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  45. Lawrence, R.J.: Witten–Reshetikhin–Turaev invariants of 3-manifolds as holomorphic functions. In: Geometry and Physics (Aarhus, 1995). Lect. Notes Pure Appl. Math., vol. 184, pp. 363–377. Dekker, New York (1997)

    Google Scholar 

  46. Lawrence, R., Rozansky, L.: Witten–Reshetikhin–Turaev invariants of Seifert manifolds. Commun. Math. Phys. 205(2), 287–314 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  47. Lawrence, R., Zagier, D.: Modular forms and quantum invariants of 3-manifolds. Asian J. Math. 3, 93–107 (1999)

    MATH  MathSciNet  Google Scholar 

  48. Le, T.T.Q.: An invariant of integral homology 3-spheres which is universal for all finite type invariants. In: Solitons, Geometry, and Topology: On the Crossroad. Trans. Am. Math. Soc., vol. 179, pp. 75–100. Am. Math. Soc., Providence, RI (1997)

    Google Scholar 

  49. Le, T.T.Q.: On perturbative PSU(n) invariants of rational homology 3-spheres. Topology 39(4), 813–849 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  50. Le, T.T.Q.: Quantum invariants of 3-manifolds: integrality, splitting, and perturbative expansion. Topol. Appl. 127(1–2), 125–152 (2003)

    Article  MATH  Google Scholar 

  51. Le, T.T.Q.: Strong integrality of quantum invariants of 3-manifolds. To appear in Trans. Am. Math. Soc.

  52. Le, T.T.Q., Murakami, J., Ohtsuki, T.: On a universal perturbative invariant of 3-manifolds. Topology 37(3), 539–574 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  53. Lee, H.C.: Tangles, links, and twisted quantum groups. In: Physics, Geometry and Topology (Banff, 1989). NATO ASI Ser., Ser. B, Phys., vol. 238, pp. 623–655. Plenum, New York (1990)

    Google Scholar 

  54. Lin, X., Wang, Z.: On Ohtsuki’s invariants of integral homology 3-spheres. Acta Math. Sin. 15(3), 293–316 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  55. Lin, X.-S., Wang, Z.: Fermat limit and congruence of Ohtsuki invariants. In: Proceedings of the Kirbyfest (Berkeley, CA, 1998). Geom. Topol. Monogr., vol. 2, pp. 321–333. Geom. Topol. Publ., Coventry (1999)

    Chapter  Google Scholar 

  56. Majid, S.: Algebras and Hopf algebras in braided categories. In: Advances in Hopf Algebras. Lect. Notes Pure Appl. Math., vol. 158, pp. 55–105. Dekker, New York (1994)

    Google Scholar 

  57. Majid, S.: Foundations of quantum group theory. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  58. Masbaum, G.: Skein-theoretical derivation of some formulas of Habiro. Algebr. Geom. Topol. 3, 537–556 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  59. Masbaum, G., Roberts, J.D.: A simple proof of integrality of quantum invariants at prime roots of unity. Math. Proc. Camb. Philos. Soc. 121(3), 443–454 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  60. Masbaum, G., Wenzl, H.: Integral modular categories and integrality of quantum invariants at roots of unity of prime order. J. Reine Angew. Math. 505, 209–235 (1998)

    MATH  MathSciNet  Google Scholar 

  61. Melvin, P.M., Morton, H.R.: The coloured Jones function. Commun. Math. Phys. 169, 501–520 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  62. Murakami, H.: Quantum SU(2)-invariants dominate Casson’s SU(2)-invariant. Math. Proc. Camb. Philos. Soc. 115(2), 253–281 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  63. Murakami, H.: Quantum SO(3)-invariants dominate the SU(2)-invariant of Casson and Walker. Math. Proc. Camb. Philos. Soc. 117(2), 237–249 (1995)

    MATH  MathSciNet  Google Scholar 

  64. Murakami, H., Murakami, J.: The coloured Jones polynomials and the simplicial volume of a knot. Acta Math. 186(1), 85–104 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  65. Murakami, J.: A state model for the multi-variable Alexander polynomial. Pac. J. Math. 157, 109–135 (1993)

    MATH  MathSciNet  Google Scholar 

  66. Ohtsuki, T.: Colored ribbon Hopf algebras and universal invariants of framed links. J. Knot Theory Ramifications 2(2), 211–232 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  67. Ohtsuki, T.: A polynomial invariant of integral homology 3-spheres. Math. Proc. Camb. Philos. Soc. 117(1), 83–112 (1995)

    MATH  MathSciNet  Google Scholar 

  68. Ohtsuki, T.: A polynomial invariant of rational homology 3-spheres. Invent. Math. 123(2), 241–257 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  69. Ohtsuki, T.: Finite type invariants of integral homology 3-spheres. J. Knot Theory Ramifications 5(1), 101–115 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  70. Ohtsuki, T.: The perturbative SO(3) invariant of rational homology 3-spheres recovers from the universal perturbative invariant. Topology 39, 1103–1135 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  71. Ohtsuki, T. (ed.): Problems on invariants of knots and 3-manifolds. In: Invariants of Knots and 3-Manifolds (Kyoto 2001). Geom. Topol. Monogr., vol. 4, pp. 377–572. Geom. Topol. Publ., Coventry (2002)

  72. Reshetikhin, N.Y.: Quasitriangle Hopf algebras and invariants of tangles. Algebr. Anal. 1(2), 169–188 (1989) (Russian) (English transl.: Leningr. Math. J. 1(2), 491–513 (1990))

    MathSciNet  Google Scholar 

  73. Reshetikhin, N., Turaev, V.G.: Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103(3), 547–597 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  74. Rozansky, L.: The universal R-matrix, Burau representation, and the Melvin–Morton expansion of the colored Jones polynomial. Adv. Math. 134, 1–31 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  75. Rozansky, L.: On p-adic properties of the Witten–Reshetikhin–Turaev invariant. In: Primes and Knots. Contemp. Math., vol. 416, pp. 213–236. Am. Math. Soc., Providence, RI (2006)

    Google Scholar 

  76. Takata, T., Yokota, Y.: The PSU(N) invariants of 3-manifolds are algebraic integers. J. Knot Theory Ramifications 8(4), 521–532 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  77. Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121(3), 351–399 (1989)

    Article  MATH  MathSciNet  Google Scholar 

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  1. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Kazuo Habiro

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Habiro, K. A unified Witten–Reshetikhin–Turaev invariant for integral homology spheres . Invent. math. 171, 1–81 (2008). https://doi.org/10.1007/s00222-007-0071-0

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