A geometric characterization of Vassiliev invariants

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