Vassiliev invariants and the cubical knot complex

We construct a cubical CW-complex CK(M) whose rational cohomology algebra contains Vassiliev invariants of knots in the 3-manifold M. We construct CK(R3) by attaching cells to CK(R3) for every degenerate 1-singular and 2-singular knot, and we show that π1(CK(R3))=1 and π2(CK(R3))=Z. We give conditions for Vassiliev invariants to be nontrivial in cohomology. In particular, for R3 we show that v2 uniquely generates H2(CK,D), where D is the subcomplex of degenerate singular knots. More generally, we show that any Vassiliev invariant coming from the Conway polynomial is nontrivial in cohomology. The cup product in H∗(CK) provides a new graded commutative algebra of Vassiliev invariants evaluated on ordered singular knots. We show how the cup product arises naturally from a cocommutative differential graded Hopf algebra of ordered chord diagrams.