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.
Similar content being viewed by others
References
D. Bar-Natan,On the Vassiliev knot invariants, Topology34 (1995), 423–472.
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.
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.
S. V. Chmutov and A. N. Varchenko,Remarks on the Vassilliev knot invariants coming from sl 2, Topology36 (1997), 153–178.
V. Hinich and A. Vaintrob,Cyclic operads and algebra of chord diagrams, Selecta Mathematica, New Series8 (2002), 237–282, arXiv:math.QA/0005197.
C. Kassel,Quantum Groups, GTM155, Springer-Verlag, Heidelberg, 1994.
R. Kirby and P. Melvin,Dedekind sums, μ-invariants and the signature cocycle, Mathematische Annalen299 (1994), 231–267.
M. Kontsevich,Deformation quantization of Poisson manifolds, Publications Mathématiques de l’Institut des Hautes Études Scientifiques, preprint, September 1997, arXiv:q-alg/9709040.
R. Lawrence and L. Rozansky,Witten-Reshetikhin-Turaev invariants of Seifert manifolds, Communications in Mathematical Physics205 (1999), 287–314.
T. Q. T. Le,The Lê-Murakami-Ohtsuki invariant, SUNY at Buffalo preprint, June 1999.
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.
T. Q. T. Le and J. Murakami,Parallel version of the universal Vassiliev-Kontsevich invariant, Journal of Pure and Applied Algebra121 (1997), 271–291.
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.
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.
C. Lescop,Global surgery formula for the Casson-Walker invariant, Annals of Mathematics Studies140, Princeton University Press, Princeton, 1996.
N. D. Mermin,Is the moon there when nobody looks? Reality and the quantum theory, Physics Today39 (1985), 38–47.
T. Mochizuki,On the morphism of Duflo-Kirillov type, Journal of Geometry and Physics41 (2002), 73–113.
J. M. Montesinos,Classical Tessellations and Three-manifolds, Springer-Verlag, Berlin, 1985.
D. Rolfsen,Knots and Links, Mathematics Lecture Series7, Publish or Perish, Wilmington, 1976.
P. Scott,The geometries of 3-manifolds, The Bulletin of the London Mathematical Society15 (1983), 401–487.
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.
K. Walker,An extension of Casson’s invariant, Annals of Mathematics Studies126, Princeton University Press, Princeton, 1992.
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.
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.
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.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bar-Natan, D., Lawrence, R. A rational surgery formula for the LMO invariant. Isr. J. Math. 140, 29–60 (2004). https://doi.org/10.1007/BF02786626
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02786626