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Vassiliev Polynomial

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 $e^x$ 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).


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.''

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 Eric W. Weisstein