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In 1993 K. Habiro defined $C_k$-move of oriented links and around 1994 heproved that two oriented knots are transformed into each other by $C_k$-movesif and only if they have the same Vassiliev invariants of order $\leq k-1$. Inthis paper we define Vassiliev invariant of type $(k_1,...,k_l)$, and showthat, for $k=k_1+...+k_l$, two oriented knots are transformed into each otherby $C_k$-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 adiagram-oriented proof of this theorem. When $k_1=...=k_l=1$, the Vassilievinvariant of type $(k_1,...,k_l)$ coincides with the Vassiliev invariant oforder $\leq l-1$ in the usual sense. As a special case, we have Habiro'stheorem stated above.

Band description of knots and Vassiliev invariants