Vassiliev invariants and the cubical knot complex

详细信息    查看全文

• 作者：Kofman ; Ilya ; Lin ; Xiao-Song
• 关键词：Primary 57M27 ; Secondary 55U15 ; Vassiliev invariants ; Cubical CW-complex ; Hopf algebra
• 刊名：Topology
• 出版年：2003
• 出版时间：January, 2003
• 年：2003
• 卷：42
• 期：1
• 页码：83-101
• 全文大小：369 K

文摘

We construct a cubical CW-complex CK(M) whose rational cohomology algebra contains Vassiliev invariants of knots in the 3-manifold M. We construct (R³) by attaching cells to CK(R³) for every degenerate 1-singular and 2-singular knot, and we show that π₁((R³))=1 and π₂((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.