Band description of knots and Vassiliev invariants
|KOUKI TANIYAMA a1 and AKIRA YASUHARA a2|
a1 Department of Mathematics, School of Education, Waseda University,
Nishi-Waseda 1-6-1, Shinjuku-ku, Tokyo 169-8050, Japan.
a2 Department of Mathematics, Tokyo Gakugei University,
Nukuikita 4-1-1, Koganei, Tokyo 184-8501, Japan.
In the 1990s, Habiro defined Ck-move of oriented links for each natural number k
. 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
, 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 , ,  and , and in  Stanford gives another characterization of knots
with the same Vassiliev invariants of order [less-than-or-eq, slant] k−1.
(Received April 5 2000)