On Vassiliev Invariants of Degrees 2 and 3 for Torus Knots

Abstract

We consider the $\mathbf{R}$-valued Vassiliev invariants of degrees 2 and 3 normalized by the conditions that they take values 0 on the unknot and 1 on the trefoil. We give certain answers for a problem due to N. Okuda about these two invariants. Moreover, we prove a conjecture due to Simon Willerton concerning the degree-3 Vassiliev invariant in the case of torus knots.