Claspers and finite type invariants of links

Abstract

We introduce the concept of `claspers,' which are surfaces in 3-manifolds with some additional structure on which surgery operations can be performed. Using claspers we define for each positive integer k an equivalence relation on links called `C_k-equivalence,' which is generated by surgery operations of a certain kind called `C_k-moves'. We prove that two knots in the 3-sphere are C_{k+1}-equivalent if and only if they have equal values of Vassiliev-Goussarov invariants of type k with values in any abelian groups. This result gives a characterization in terms of surgery operations of the informations that can be carried by Vassiliev-Goussarov invariants. In the last section we also describe outlines of some applications of claspers to other fields in 3-dimensional topology.