Vassiliev (1990) introduced a radically new way of looking at Knots by considering a multidimensional space in which each point represents a possible 3-D knot configuration. If two Knots are equivalent, a path then exists in this space from one to the other. The paths can be associated with polynomial invariants.

Birman and Lin (1993) subsequently found a way to translate this scheme into a set of rules and list of potential starting points, which makes analysis of Vassiliev polynomials much simpler. Bar-Natan (1995) and Birman and Lin (1993) proved that Jones Polynomials and several related expressions are directly connected (Peterson 1992). In fact, substituting the Power series for as the variable in the Jones Polynomial yields a Power Series whose Coefficients are Vassiliev polynomials (Birman and Lin 1993). Bar-Natan (1995) also discovered a link with Feynman diagrams (Peterson 1992).

**References**

Bar-Natan, D. ``On the Vassiliev Knot Invariants.'' *Topology* **34**, 423-472, 1995.

Birman, J. S. ``New Points of View in Knot Theory.'' *Bull. Amer. Math. Soc.* **28**, 253-287, 1993.

Birman, J. S. and Lin, X.-S. ``Knot Polynomials and Vassiliev's Invariants.'' *Invent. Math.* **111**, 225-270, 1993.

Peterson, I. ``Knotty Views: Tying Together Different Ways of Looking at Knots.'' *Sci. News* **141**, 186-187, 1992.

Praslov, V. V. and Sossinsky, A. B.
*Knots, Links, Braids and 3-Manifolds: An Introduction to the New Invariants in Low-Dimensional Topology.*
Providence, RI: Amer. Math. Soc., 1996.

Stoimenow, A. ``Degree-3 Vassiliev Invariants.'' http://www.informatik.hu-berlin.de/~stoimeno/vas3.html.

Vassiliev, V. A. ``Cohomology of Knot Spaces.'' In *Theory of Singularities and Its Applications* (Ed. V. I. Arnold).
Providence, RI: Amer. Math. Soc., pp. 23-69, 1990.

Vassiliev, V. A. *Complements of Discriminants of Smooth Maps: Topology and Applications.* Providence, RI: Amer. Math. Soc., 1992.

© 1996-9

1999-05-26