Vassiliev invariants and the cubical knot complex

Abstract

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( R 3) by attaching cells to CK( R 3) for every degenerate 1-singular and 2-singular knot, and we show that π 1( CK( R 3))=1 and π 2( CK( R 3))= Z . We give conditions for Vassiliev invariants to be nontrivial in cohomology. In particular, for R 3 we show that v 2 uniquely generates H 2( 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.