Due to the technical nature of their original definition through the Vassiliev spectral sequence, the study of finite-type knot invariants has generally proceeded from the notion of Vassiliev derivative of a knot invariant. We initiate a more concrete, detailed and geometric analysis of the classical Vassiliev spectral sequence by constructing an inverse system of "unstable" Vassiliev spectral sequences on the spaces of plumbers' knots. We define an unstable analog of the Vassiliev derivative for invariants of plumbers' knots and use this to show that the classical sequence is contained in the inverse limit of the unstable ones. Time permitting, we will use the geometry of the spaces of plumbers' knots to investigate the combinatorics of the unstable sequences. |