A geometric characterization of Vassiliev invariants

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 $\mathbb {S}^1\times \mathbb {S}^2$.