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.