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Vassiliev invariants and the Hopf algebra of chord diagrams

Published online by Cambridge University Press:  24 October 2008

Simon Willerton
Affiliation:
Department of Mathematics and Statistics, University of Edinburgh, King's Buildings, Edinburgh EH9 3JZ

Abstract

This paper is closely related to Bar-Natan's work, and fills in some of the gaps in [1]. Following his analogy of the extension of knot invariants to knots with double points to the notion of multivariate calculus on polynomials, we introduce a new notation which facilitates the formulation of a Leibniz type formula for the product of two Vassiliev invariants. This leads us to see how Bar-Natan's co-product of chord diagrams corresponds to multiplication of Vassiliev invariants. We also include a proof that the multiplication in is a consequence of Bar-Natan's 4T relation.

The last part of this paper consists of a proof that the space of weight systems is a sub-Hopf algebra of the space *, by means of the canonical projection.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

[1]Bar-Natan, D.. On the Vassiliev knot invariants. Topology (to appear).Google Scholar
[2]Bar-Natan, D.. Talk given at the Isaac Newton Institute, Cambridge, November 1992.Google Scholar
[3]Milnor, J. and Moore, J.. On the structure of Hopf algebras. Annals of Math. 81 (1965), 211264.CrossRefGoogle Scholar
[4]Vassiliev, V. A.. Complements of discriminants of smooth maps: topology and applications, Trans. of Math. Mono. 98 (Amer. Math. Soc., 1992).CrossRefGoogle Scholar
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