On a theorem of Goussarov

Abstract

In this paper, the easier methods of my thesis are applied to give a simple proof of a theorem of Goussarov. The theorem relates two possible notions of finite type equivalence of knots, links or string links, showing that the resulting filtrations are the same up to a degree shift by a factor of two. This is then applied to the situation of rooted claspers to show that rooted claspers of sufficiently high degree must preserve type k invariants. As a consequence, grope cobordisms of sufficiently high class must preserve type k invariants. This result is applied in math.GT/0012118, to show Theorem 2 of that paper.