2000/03/03 by Kouki Taniyama, Taniyama, Kouki, Akira Yasuhara +1 · 1 citation
Computer Science · Mathematics · #57M25 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.GT #msc:57M25 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.math/0003021
LaTeX, 16 pages with 22 figures
arxiv created 2000/03/03 · openalex publication_date 2000/03/03 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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-1. In this paper we define Vassiliev invariant of type (k1,...,kl), and show that, for k=k1+...+kl, two oriented knots are transformed into each other by Ck-moves if and only if they have the same Vassiliev invariants of type (k1,...,kl). We introduce a concept ` band description of knots' and give a diagram-oriented proof of this theorem. When k1=...=kl=1, the Vassiliev invariant of type (k1,...,kl) coincides with the Vassiliev invariant of order ≤ l-1 in the usual sense. As a special case, we have Habiro's theorem stated above.