Vassiliev Polynomial

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

Birman and Lin (1993) subsequently found a way to translate this scheme into a set of rules and list of potential startingpoints, which makes analysis of Vassiliev polynomials much simpler. Bar-Natan (1995) and Birman and Lin (1993) provedthat 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 Serieswhose 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.

• van der Grinten Projection
• Vandermonde Determinant
• Vandermonde Identity
• Vandermonde Matrix
• Vandermonde's Sum
• Vandermonde Theorem
• van der Pol Equation
• van der Waerden Number
• van der Waerden's Theorem
• Vandiver's Criteria
• Vanishing Point
• van Kampen's Theorem
• van Wijngaarden-Deker-Brent Method
• Varga's Constant
• Variance
• Difference of Successes
• Difference Operator
• Difference Quotient
• Difference Set
• Difference Table
• Different
• Differentiable
• Differentiable Manifold
• Differential
• Differential Calculus