IMPORTANT NOTICE

The AMS website will be down for maintenance on May 23 between 6:00am - 8:00am EDT. For questions please contact AMS Customer Service at cust-serv@ams.org or (800) 321-4267 (U.S. & Canada), (401) 455-4000 (Worldwide).

 

Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Vanishing of a certain kind
of Vassiliev invariants of 2-knots


Author: Seiichi Kamada
Journal: Proc. Amer. Math. Soc. 127 (1999), 3421-3426
MSC (1991): Primary 57Q45, 57M25
DOI: https://doi.org/10.1090/S0002-9939-99-04924-2
Published electronically: May 27, 1999
MathSciNet review: 1610788
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In knot theory, Vassiliev's 1-knot invariants are defined in a combinatorial way as finite type invariants. By a natural generalization of the combinatorial definition, one has a certain family of 2-knot invariants, which should be called finite type 2-knot invariants. They form a subspace of the whole space of ``Vassiliev 2-knot invariants''. In this paper we prove that it is 1-dimensional.


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

  • [1] J. S. Birman and X.-S. Lin, Knot polynomials and Vassiliev's invariant, Invent. Math. 11 (1993), 225-270. MR 94d:57010
  • [2] J. Boyle, Classifying 1-handles attached to knotted surfaces, Trans. Amer. Math. Soc. 306 (1988), 475-487. MR 89f:57032
  • [3] A. J. Casson, Three lectures on new-infinite constructions in 4-dimensional manifolds, A la Recherche de la Topologie Perdue, Progress in Math., 62, Birkhäuser, 1986, pp. 201-214. CMP 19:16
  • [4] M. W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242-276. MR 22:9980
  • [5] F. Hosokawa and A. Kawauchi, Proposals for unknotted surfaces in four-spaces, Osaka J. Math. 16 (1979), 233-248. MR 81c:57018
  • [6] S. Kamada, Unknotting operations for 1- and 2-dimensional links, The Proceedings of the fifth Korea-Japan School of Knots and Links, held in Taejon, Korea, 1997, to appear.
  • [7] R. Kirby, The topology of $4$-manifolds, Lecture Notes in Mathematics, 1374, Springer-Verlag, 1989. MR 90j:57012
  • [8] P. Kirk, Link maps in the four sphere, Differential Topology, Lect. Notes Math., 1350, Springer-Verlag, 1988, pp. 31-43. MR 90e:57043
  • [9] G.-S. Li, An invariant of link homotopy in dimension four, Topology 36 (1997), 881-897. MR 98g:57038
  • [10] S. Smale, A classification of immersions of the two-sphere, Trans. Amer. Math. Soc. 90 (1959), 281-290. MR 21:2984
  • [11] -, The classification of immersions of spheres in Euclidean spaces, Ann. Math. 69 (1959), 327-344.
  • [12] V. A. Vassiliev, Cohomology of knot space, Theory of Singularities and Its Applications, Advances in Soviet Math. vol. 1, Amer. Math. Soc., 1990. MR 92a:57016

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 57Q45, 57M25

Retrieve articles in all journals with MSC (1991): 57Q45, 57M25


Additional Information

Seiichi Kamada
Affiliation: Department of Mathematics, Osaka City University, Sumiyoshi-ku, Osaka 558, Japan
Email: kamada@sci.osaka-cu.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-99-04924-2
Keywords: 2-knot, Vassiliev invariant, finite type invariant, finger move, regular homotopy
Received by editor(s): May 22, 1997
Received by editor(s) in revised form: February 1, 1998
Published electronically: May 27, 1999
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society