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Band description of knots and Vassiliev invariants
by
Kouki Taniyama
Tokyo Woman's Christian University
Coauthors: Akira Yasuhara (Tokyo Gakugei University and GWU)
In 1993 K. Habiro defined Ck-move of oriented links and around 1994 he proved that two oriented knots are transformed into each other by Ck-moves if and only if they have the same Vassiliev invariants of order < k. However this deep theorem appears only in his recent paper that develops his original clasper theory. In this talk we define Vassiliev invariant of type (k1, ..., km). When k1= ... = km=1 the invariant coincides with Vassiliev invariant of order < m in the usual sense. Let k=k1+ ... +km. We show that two oriented knots are transformed into each other by Ck-moves if and only if they have the same Vassiliev invariants of type (k1, ..., km). As a corollary we have Habiro's Theorem. Our proof is based on a concept which we call band description of knots. Our proof is elementary and completely self-contained.
Date received: January 17, 2000
Copyright © 2000 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # caea-09.