Vassiliev invariants and the cubical knot complex
| dc.creator | Kofman, Ilya | |
| dc.creator | Lin, Xiao-Song | |
| dc.date | 2000-10-02 | |
| dc.date.accessioned | 2026-07-07T04:37:46Z | |
| dc.date.available | 2026-07-07T04:37:46Z | |
| dc.description | 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. | |
| dc.description | 41 pages, 15 figures | |
| dc.identifier | https://arxiv.org/abs/math/0010009 | |
| dc.identifier | http://arxiv.org/abs/math/0010009 | |
| dc.identifier | Topology 42 (2003), 83-101. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/60025 | |
| dc.subject | Geometric Topology | |
| dc.title | Vassiliev invariants and the cubical knot complex | |
| dc.type | text |