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Transactions of the American Mathematical Society

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On Vassiliev knot invariants induced
from finite type 3-manifold invariants


Authors: Matt Greenwood and Xiao-Song Lin
Journal: Trans. Amer. Math. Soc. 351 (1999), 3659-3672
MSC (1991): Primary 57M25
DOI: https://doi.org/10.1090/S0002-9947-99-02139-X
Published electronically: May 3, 1999
MathSciNet review: 1467465
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the knot invariant induced by a $\mathbb{Z}$-homology 3-sphere invariant of order $\leq k$ in Ohtsuki's sense, where $k\geq 4$, is of order $\leq k-2$. The method developed in our computation shows that there is no $\mathbb{Z}$-homology 3-sphere invariant of order 5.


References [Enhancements On Off] (What's this?)

  • [BN1] D. Bar-Natan, On the Vassiliev knot invariants, Topology, 34(1995), pp. 3659-3672. MR 97d:57004
  • [B-L] J. Birman and X.-S. Lin, Knot polynomials and Vassiliev's invariants. Invent. Math., 111(1993), pp. 225-270. MR 94d:57010
  • [G] S. Garoufalidis, On finite type 3-Manifold Invariants I, Jour. of Knot Theory and Its Ramifications, 5(1996), pp. 441-461. MR 97j:57019
  • [O1] T. Ohtsuki, A polynomial invariant of integral homology 3-spheres, Math. Proc. Camb. Phil. Soc., 117(1995), pp. 3659-3672. MR 95i:57021
  • [O2] T. Ohtsuki, Finite type invariants of integral homology 3-spheres, Jour. of Knot Theory and Its Ramifications, 5(1996), pp. 3659-3672. MR 97i:57019
  • [Gu] M. Gusarov, On n-equivalence of knots and invariants of finite degree, Advances in Soviet Math. AMS, 18(1994), pp. 3659-3672. MR 96i:57005
  • [K] M. Kontsevich, Vassiliev's knot invariant, Adv. Sov. Math., 16(1993), part 2, pp. 137-150. MR 94k:57014
  • [L] J. Levine, Surgery on links and the $\bar\mu$-invariants, Topology, 26(1987), pp. 3659-3672. MR 88d:57005
  • [Li] X.-S. Lin, Finite type invariants of integral homology 3-spheres: A survey, Knot Theory, Banach Center Publication, vol. 42, Inst. of Math., Polish Acad. of Sci., Warszawa, 1998, pp. 205-220.
  • [R1] L. Rozansky, The trivial connection contribution to Witten's invariant and finite type invariants of rational homology spheres, preprint UMTG-182-95, q-alg/9503011
  • [R2] L. Rozansky, Witten's invariants of rational homology spheres at prime values of $K$ and trivial connection contribution, preprint UMTG-183-95, q-alg/9504015.
  • [V] V. Vassiliev, Cohomology of knot spaces, Theory of Singularities and Its Application (ed. V. Arnold), Amer. Math. Soc., 1990. MR 92a:57016

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Additional Information

Matt Greenwood
Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
Email: matt@math.columbia.edu

Xiao-Song Lin
Affiliation: Department of Mathematics, University of California, Riverside, California 92521
Email: xl@math.ucr.edu

DOI: https://doi.org/10.1090/S0002-9947-99-02139-X
Received by editor(s): June 29, 1995
Received by editor(s) in revised form: May 2, 1997
Published electronically: May 3, 1999
Additional Notes: The second author is supported in part by NSF
Article copyright: © Copyright 1999 American Mathematical Society

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