2001/04/18 by Akira Yasuhara, Yasuhara, Akira
Computer Science · Mathematics · #Computational Geometry and Mesh Generation #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math.GT #msc:57M25 #msc:57M27
paper · pdf · doi:10.48550/arxiv.math/0104177
LaTeX, 15 pages with 12 figures
arxiv created 2001/04/18 · arxiv updated 2009/11/30
The Ck-equivalence is an equivalence relation generated by Ck-moves defined by Habiro. Habiro showed that the set of Ck-equivalence classes of the knots forms an abelian group under the connected sum and it can be classified by the additive Vassiliev invariant of order ≤ k-1. We see that the set of Ck-equivalence classes of the spatial θ-curves forms a group under the vertex connected sum and that if the group is abelian, then it can be classified by the additive Vassiliev invariant of order ≤ k-1. However the group is not necessarily abelian. In fact, we show that it is nonabelian for k≥ 12. As an easy consequence, we have the set of Ck-equivalence classes of m-string links, which forms a group under the composition, is nonabelian for k≥ 12 and m≥ 2.