Claspers and finite type invariants of links

TL;DR

This paper introduces claspers, a new tool in 3-manifold topology, to define link equivalences and connect them with Vassiliev invariants, providing a surgery-based characterization of these invariants.

Contribution

The paper introduces claspers and C_k-equivalence, establishing a surgery-based characterization of Vassiliev invariants for links.

Findings

C_{k+1}-equivalence characterized by Vassiliev invariants of type k

Claspers provide a new method for link surgery operations

Applications of claspers extend to other areas in 3D topology

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.