Vassiliev invariants and the cubical knot complex

الأصل · EN

We construct a cubical CW-complex CK(M³) whose rational cohomology algebra contains Vassiliev invariants of knots in the 3-manifold M³. 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.