arXiv subjects
Publications and source records attributed to Ka Yi Ng.
Dror Bar-Natan and Ofer Ron have questioned the accuracy of the statement on page 453, line 2 minus of my paper "Groups of Ribbon Knots", Topology Vol 37, No. 2, 441-458, that "It follows from the inductive proof that $\cal{R}_n$ is of index 2." See their article posted on arXiv: math.GT/0309282. We give here full details of how to carry out the inductive step.
Given a positive integer $n$, we say that two knots are $V_n$-equivalent if they have the same Vassiliev invariants of order $\le n$. We showed that the $V_n$-equivalence classes of ribbon knots form a group, the operation being induced by the connected sums of knots. The group is free abelian and its rank is equal to the dimension of the space of primitive Vassiliev invariants of order $\le n$. The $n$-equivalence classes (following Gusarov's notion) of ribbon knots also constitute a group. As a corollary, any non-ribbon knot whose Arf invariant istrivial cannot be distinguished from ribbon knots by finitely many Vassiliev invariants. Furthermore, all additive knot cobordism invariants are not of finite type. As a bi-product, we prove that the number of independent Vassiliev invariants of order $n$ is bounded above by $(n-2)!/2$ for $n>5$.