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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A geometric characterization of Vassiliev invariants


Author: Michael Eisermann
Journal: Trans. Amer. Math. Soc. 355 (2003), 4825-4846
MSC (2000): Primary 57M27, 57M25, 20F36
Published electronically: July 24, 2003
MathSciNet review: 1997586
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Abstract: It is a well-known paradigm to consider Vassiliev invariants as polynomials on the set of knots. We prove the following characterization: a rational knot invariant is a Vassiliev invariant of degree $\le m$ if and only if it is a polynomial of degree $\le m$ on every geometric sequence of knots. Here a sequence $K_z$ with $z\in\mathbb{Z} $ is called geometric if the knots $K_z$ coincide outside a ball $B$, inside of which they satisfy $K_z \cap B = \tau^z$ for all $z$ and some pure braid $\tau$. As an application we show that the torsion in the braid group over the sphere induces torsion at the level of Vassiliev invariants: there exist knots in $\mathbb{S} ^1\times\mathbb{S} ^2$that can be distinguished by $\mathbb{Z} {/}{2}$-invariants of finite type but not by rational invariants of finite type. In order to obtain such torsion invariants we construct over $\mathbb{Z} $ a universal Vassiliev invariant of degree $1$ for knots in $ \mathbb{S} ^1\times\mathbb{S} ^2$.


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Additional Information

Michael Eisermann
Affiliation: UMPA, École Normale Supérieure de Lyon, 46 allée d’Italie, 69364 Lyon, France
Address at time of publication: Institut Fourier, Université Grenoble I, France
Email: Michael.Eisermann@umpa.ens-lyon.fr, Michael.Eisermann@ujf-grenoble.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03117-9
PII: S 0002-9947(03)03117-9
Keywords: Vassiliev invariant, invariant of finite type, twist sequence, geometric sequence of knots, torsion in the braid group over the sphere, Dirac twist, Dirac's spin trick
Received by editor(s): March 5, 2001
Received by editor(s) in revised form: May 20, 2002
Published electronically: July 24, 2003
Article copyright: © Copyright 2003 American Mathematical Society