Mathematical Proceedings of the Cambridge Philosophical Society

Band description of knots and Vassiliev invariants

a1 Department of Mathematics, School of Education, Waseda University, Nishi-Waseda 1-6-1, Shinjuku-ku, Tokyo 169-8050, Japan. e-mail:
a2 Department of Mathematics, Tokyo Gakugei University, Nukuikita 4-1-1, Koganei, Tokyo 184-8501, Japan. e-mail:


In the 1990s, Habiro defined Ck-move of oriented links for each natural number k [5]. A Ck-move is a kind of local move of oriented links, and two oriented knots have the same Vassiliev invariants of order [less-than-or-eq, slant] k−1 if and only if they are transformed into each other by Ck-moves. Thus he has succeeded in deducing a geometric conclusion from an algebraic condition. However, this theorem appears only in his recent paper [6], in which he develops his original clasper theory and obtains the theorem as a consequence of clasper theory. We note that the ‘if’ part of the theorem is also shown in [4], [9], [10] and [16], and in [13] Stanford gives another characterization of knots with the same Vassiliev invariants of order [less-than-or-eq, slant] k−1.

(Received April 5 2000)