Advertisement

Israel Journal of Mathematics

, Volume 140, Issue 1, pp 29–60 | Cite as

A rational surgery formula for the LMO invariant

  • Dror Bar-Natan
  • Ruth Lawrence
Article

Abstract

We write a formula for the LMO invariant of a rational homology sphere presented as a rational surgery on a link inS 3. Our main tool is a careful use of the Århus integral and the (now proven) “Wheels” and “Wheeling” conjectures of B-N, Garoufalidis, Rozansky and Thurston. As steps, side benefits and asides we give explicit formulas for the values of the Kontsevich integral on the Hopf link and on Hopf chains, and for the LMO invariant of lens spaces and Seifert fibered spaces. We find that the LMO invariant does not separate lens spaces, is far from separating general Seifert fibered spaces, but does separate Seifert fibered spaces which are integral homology spheres.

Keywords

Lens Space Continue Fraction Expansion Rational Homology Univalent Vertex Link Relation 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [B-N]
    D. Bar-Natan,On the Vassiliev knot invariants, Topology34 (1995), 423–472.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [BGRT]
    D. Bar-Natan, S. Garoufalidis, L. Rozansky and D. P. Thurston,Wheels, wheeling, and the Kontsevich integral of the unknot, Israel Journal of Mathematics119 (2000), 217–237.zbMATHMathSciNetGoogle Scholar
  3. [BLT]
    D. Bar-Natan, T. Q. T. Le, and D. P. Thurston,Two applications of elementary knot theory to Lie algebras and Vassiliev invariants, Geometry and Topology7 (2003), 1–31.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [CV]
    S. V. Chmutov and A. N. Varchenko,Remarks on the Vassilliev knot invariants coming from sl 2, Topology36 (1997), 153–178.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [HV]
    V. Hinich and A. Vaintrob,Cyclic operads and algebra of chord diagrams, Selecta Mathematica, New Series8 (2002), 237–282, arXiv:math.QA/0005197.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [Kas]
    C. Kassel,Quantum Groups, GTM155, Springer-Verlag, Heidelberg, 1994.Google Scholar
  7. [KM]
    R. Kirby and P. Melvin,Dedekind sums, μ-invariants and the signature cocycle, Mathematische Annalen299 (1994), 231–267.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [Ko]
    M. Kontsevich,Deformation quantization of Poisson manifolds, Publications Mathématiques de l’Institut des Hautes Études Scientifiques, preprint, September 1997, arXiv:q-alg/9709040.Google Scholar
  9. [LR]
    R. Lawrence and L. Rozansky,Witten-Reshetikhin-Turaev invariants of Seifert manifolds, Communications in Mathematical Physics205 (1999), 287–314.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [Le]
    T. Q. T. Le,The Lê-Murakami-Ohtsuki invariant, SUNY at Buffalo preprint, June 1999.Google Scholar
  11. [LM1]
    T. Q. T. Le and J. Murakami,The universal Vassiliev-Kontsevich invariant for framed oriented links, Compositio Mathematica102 (1996), 41–64, arXiv:hep-th/9401016.zbMATHMathSciNetGoogle Scholar
  12. [LM2]
    T. Q. T. Le and J. Murakami,Parallel version of the universal Vassiliev-Kontsevich invariant, Journal of Pure and Applied Algebra121 (1997), 271–291.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [LMMO]
    T. Q. T. Le, J. Murakami, H. Murakami and T. Ohtsuki,A three-manifold invariant derived from the universal Vassiliev-Kontsevich invariant, Proceedings of the Japan Academy, Series A71 (1995), 125–127.zbMATHMathSciNetCrossRefGoogle Scholar
  14. [LMO]
    T. Q. T. Le, J. Murakami and T. Ohtsuki,On a universal perturbative invariant of 3-manifolds, Topology37 (1998), 539–574, arXiv:q-alg/9512002.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [Les]
    C. Lescop,Global surgery formula for the Casson-Walker invariant, Annals of Mathematics Studies140, Princeton University Press, Princeton, 1996.zbMATHGoogle Scholar
  16. [Me]
    N. D. Mermin,Is the moon there when nobody looks? Reality and the quantum theory, Physics Today39 (1985), 38–47.Google Scholar
  17. [Moc]
    T. Mochizuki,On the morphism of Duflo-Kirillov type, Journal of Geometry and Physics41 (2002), 73–113.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [Mon]
    J. M. Montesinos,Classical Tessellations and Three-manifolds, Springer-Verlag, Berlin, 1985.Google Scholar
  19. [Ro]
    D. Rolfsen,Knots and Links, Mathematics Lecture Series7, Publish or Perish, Wilmington, 1976.zbMATHGoogle Scholar
  20. [Sc]
    P. Scott,The geometries of 3-manifolds, The Bulletin of the London Mathematical Society15 (1983), 401–487.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [Th]
    D. P. Thurston,Wheeling: A Diagrammatic Analogue of the Duflo Isomorphism, Ph.D. thesis, University of California at Berkeley, May 2000, arXiv:math.QA/0006083.Google Scholar
  22. [Wa]
    K. Walker,An extension of Casson’s invariant, Annals of Mathematics Studies126, Princeton University Press, Princeton, 1992.zbMATHGoogle Scholar
  23. [Å-I]
    D. Bar-Natan, S. Garoufalidis, L. Rozansky and D. P. Thurston,The Århus integral of rational homology 3-spheres I: A highly non-trivial flat connection on S 3, Selecta Mathematica, New Series8 (2002), 315–339, arXiv:q-alg/9706004.zbMATHCrossRefMathSciNetGoogle Scholar
  24. [Å-II]
    D. Bar-Natan, S. Garoufalidis, L. Rozansky and D. P. Thurston,The Århus integral of rational homology 3-spheres II: Invariance and Universality, Selecta Mathematica, New Series88 (2002), 341–371, arXiv:math.QA/9801049.CrossRefMathSciNetGoogle Scholar
  25. [Å-III]
    D. Bar-Natan, S. Garoufalidis, L. Rozansky and D. P. Thurston,The Århus integral of rational homology 3-spheres III: The relation with the Le-Murakami-Ohtsuki invariant, Selecta Mathematica, to appear, arXiv:math.QA/9808013.Google Scholar

Copyright information

© The Hebrew University Magnes Press 2004

Authors and Affiliations

  • Dror Bar-Natan
    • 1
  • Ruth Lawrence
    • 1
  1. 1.Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael

Personalised recommendations