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Vassiliev invariants and the cubical knot complex

Ilya Kofman·Xiao-Song Lin
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Abstract

We construct a cubical CW-complex CK(M^3) whose rational cohomology algebra contains Vassiliev invariants of knots in the 3-manifold M^3. We construct \bar{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(\bar{CK}(R^3))=1 and π_2(\bar{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.

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Ilya Kofman, Xiao-Song Lin. 2000-10-02. Vassiliev invariants and the cubical knot complex. https://arxiv.org/abs/math/0010009

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