Knot polynomials and Vassiliev’s invariants. (English) Zbl 0812.57011

The main result of the paper offers an interpretation of the Jones’ polynomial (and further polynomials created in its wake) in terms of more classical concepts of topology. V. A. Vassiliev introduced in ‘Theory of singularities and its applications’ [Adv. Sov. Math. 1, 23-69 (1990; Zbl 0727.57008)] (a virtually identical version is Chapter V of [V. A. Vassiliev, Complements of discriminants of smooth maps: topology and applications (1992; Zbl 0762.55001)]) new invariants of knots based on a spectral sequence associated to the space of smooth maps from the circle to the 3-sphere. The authors prove that after a change of variables which transforms the knot polynomials into power series the coefficients of these series are Vassiliev type invariants. In order to prove this they give an axiomatic description of Vassiliev invariants and then show that the coefficients of the knot power series satisfy the axioms. This axiomatic approach has proved extremely useful, making Vassiliev invariants much more accessible.
Reviewer: E.Vogt (Berlin)


57M25 Knots and links in the \(3\)-sphere (MSC2010)
55T99 Spectral sequences in algebraic topology
58D10 Spaces of embeddings and immersions
57R40 Embeddings in differential topology
57R45 Singularities of differentiable mappings in differential topology
Full Text: DOI EuDML


[1] [B] Birman, J.S., New points of view in knot and link theory, Bull AMS (to appear) · Zbl 0785.57001
[2] [BN] Bar-Natan, D.: Perturbative Chern-Simons theory, Phd thesis, Princeton University, 1990
[3] [B-T] Birman, J.S. Kanenobu, T.: Jones’ braid-plat formula and a new surgery triple Proc. of Am. Math. Soc. 1988102, 687-695 · Zbl 0643.57007
[4] [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 · Zbl 0572.57002
[5] [J1] Jones, V.F.R.: A polynomial invariant for knots via von Neumann algebras. Bull of AMS12, 103-111 1985 · Zbl 0564.57006
[6] [J2] Jones, V.F.R.: Hecke algebra representations of braid groups and link polynomials. Ann. Math.126, 335-388 1987 · Zbl 0631.57005
[7] [J3] Jones, V.F.R.: On knot invariants related to some statistical mechanical models. Pac. J. Math.137, 311-334 1989 · Zbl 0695.46029
[8] [K1] Kauffman, L.: An invariant of regular isotopy. Trans Am. Math. Soc318, 417-471 1990 · Zbl 0763.57004
[9] [K2] Kauffman, L.: Knots and Physics, Singapore: World Scientific Press, 1992
[10] [L] Lin, XS.: Vertex models, quantum groups and Vassiliev’s knot invariants. Columbia University, (preprint), 1992
[11] [S] Stanford, T.: Finite-type invariants of knots, links and graphs. Columbia University, (preprint), 1992 · Zbl 0863.57005
[12] [R] Reshetiken, N.: Quantized universal enveloping algebras, the Yang-Baxter equation, and invariants of links. Leningrad, (preprint), 1988
[13] [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
[14] [Y] Yamada, S.: An invariant of spacial graphs. J. Graph. Theory13, 537-551 1989 · Zbl 0682.57003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.