Transactions of the American Mathematical Society 355 (2003), 4825-4846 Preprint version available at A GEOMETRIC CHARACTERIZATION OF VASSILIEV INVARIANTS MICHAEL EISERMANN 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 ≤ m if and only if it is a polynomial of degree ≤ m on every geometric sequence of knots. Here a sequence Kz with z ? Z is called geometric if the knots Kz coincide outside a ball B, inside of which they satisfy Kz ?B = ?z for all z and some pure braid ? . 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 S1 ? S2 that can be distinguished by Z/2-invariants of finite type but not by rational invariants of finite type. In order to obtain such torsion invariants we construct over Z a universal Vassiliev invariant of degree 1 for knots in S1?S2. Introduction and statement of results A Vassiliev invariant is a map v : K ? A from the set of knots K to an abelian group A such that v satisfies a certain finiteness condition (see 1). Vassiliev invariants are commonly interpreted as polynomials on the set of knots [2, 3, 21].
- coefficients c?? ?
- vassiliev invariant
- zbn ?
- singular knots
- knot invariant
- trivial torsion
- torsion invariants
- zd ?
- geometric characterization