Band description of knots and Vassiliev invariants

Details Band description of knots and Vassiliev invariants Band description of knots and Vassiliev invariants

2000-03-03

arxiv.org/abs/math/0003021v1

Mathematics

Geometric Topology

LaTeX, 16 pages with 22 figures

Scientific paper

In 1993 K. Habiro defined $C_k$-move of oriented links and around 1994 he proved that two oriented knots are transformed into each other by $C_k$-moves if and only if they have the same Vassiliev invariants of order $\leq k-1$. In this paper we define Vassiliev invariant of type $(k_1,...,k_l)$, and show that, for $k=k_1+...+k_l$, two oriented knots are transformed into each other by $C_k$-moves if and only if they have the same Vassiliev invariants of type $(k_1,...,k_l)$. We introduce a concept ` band description of knots' and give a diagram-oriented proof of this theorem. When $k_1=...=k_l=1$, the Vassiliev invariant of type $(k_1,...,k_l)$ coincides with the Vassiliev invariant of order $\leq l-1$ in the usual sense. As a special case, we have Habiro's theorem stated above.

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