Summary
A fundamental relationship is established between Jones' knot invariants and Vassiliev's knot invariants. Since Vassiliev's knot invariants have a firm grounding in classical topology, one obtains as a result a first step in understanding the Jones polynomial by topological methods.
Similar content being viewed by others
References
[B] Birman, J.S., New points of view in knot and link theory, Bull AMS (to appear)
[BN] Bar-Natan, D.: Perturbative Chern-Simons theory, Phd thesis, Princeton University, 1990
[B-T] Birman, J.S. Kanenobu, T.: Jones' braid-plat formula and a new surgery triple Proc. of Am. Math. Soc. 1988102, 687–695
[FHLMOY] Freyd, P. Yetter, D. Hoste, J. Lickorish, W. Millet, K. Ocneanu, A.: A new polynomial invariant of knots and links Bull Am. Math. Soc.12, 1985
[J1] Jones, V.F.R.: A polynomial invariant for knots via von Neumann algebras. Bull of AMS12, 103–111 1985
[J2] Jones, V.F.R.: Hecke algebra representations of braid groups and link polynomials. Ann. Math.126, 335–388 1987
[J3] Jones, V.F.R.: On knot invariants related to some statistical mechanical models. Pac. J. Math.137, 311–334 1989
[K1] Kauffman, L.: An invariant of regular isotopy. Trans Am. Math. Soc318, 417–471 1990
[K2] Kauffman, L.: Knots and Physics, Singapore: World Scientific Press, 1992
[L] Lin, XS.: Vertex models, quantum groups and Vassiliev's knot invariants. Columbia University, (preprint), 1992
[S] Stanford, T.: Finite-type invariants of knots, links and graphs. Columbia University, (preprint), 1992
[R] Reshetiken, N.: Quantized universal enveloping algebras, the Yang-Baxter equation, and invariants of links. Leningrad, (preprint), 1988
[V] Vassiliev, V.A.: Cohomology of knot spaces. In: Arnold V.I. (ed.): Theory of singularities and its applications. (Adv. Sov. Math., vol. 1) Providence, RI. Am. Math. Soc. 1990
[Y] Yamada, S.: An invariant of spacial graphs. J. Graph. Theory13, 537–551 1989
Author information
Authors and Affiliations
Additional information
Oblatum 20-V-1991 & 10-VI-1992
Research supported in part by NSF Grant DMS-88-055627.
Research supported in part by NSF Grant DMS-90-04017.
Rights and permissions
About this article
Cite this article
Birman, J.S., Lin, XS. Knot polynomials and Vassiliev's invariants. Invent Math 111, 225–270 (1993). https://doi.org/10.1007/BF01231287
Issue Date:
DOI: https://doi.org/10.1007/BF01231287