$C_k$-moves on spatial theta-curves and Vassiliev invariants

Details $C_k$-moves on spatial theta-curves and Vassiliev invariants $C_k$-moves on spatial theta-curves and Vassiliev invariants

2001-04-18

arxiv.org/abs/math/0104177v1

Mathematics

Geometric Topology

LaTeX, 15 pages with 12 figures

Scientific paper

The $C_k$-equivalence is an equivalence relation generated by $C_k$-moves defined by Habiro. Habiro showed that the set of $C_k$-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 $\leq k-1$. We see that the set of $C_k$-equivalence classes of the spatial $\theta$-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 $\leq k-1$. However the group is not necessarily abelian. In fact, we show that it is nonabelian for $k\geq 12$. As an easy consequence, we have the set of $C_k$-equivalence classes of $m$-string links, which forms a group under the composition, is nonabelian for $k\geq 12$ and $m\geq 2$.

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