Groups of ribbon knots

Abstract

We prove that for each positive integer n, the V n -equivalence classes of ribbon knot types form a subgroup R n , of index two, of the free abelian group V n constructed by the author and Stanford. As a corollary, any non-ribbon knot whose Arf invariant is trivial cannot be distinguished from ribbon knots by finitely many independent Vassiliev invariants. Furthermore, except the Arf invariant, all non-trivial additive knot cobordism invariants are not of finite type. We prove a few more consequences about the relationship between knot cobordism and V n -equivalence of knots. As a by-product, we prove that the number of independent Vassiliev invariants of order n is bounded above by (n − 2)! 2 if n > 5, improving the previously known upper bound of ( n − 1)!.