Volume 4, issue 1 (2000)

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Claspers and finite type invariants of links

Kazuo Habiro

Geometry & Topology 4 (2000) 1–83

arXiv: math.GT/0001185


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 “Ck–equivalence,” which is generated by surgery operations of a certain kind called “Ck–moves”. We prove that two knots in the 3–sphere are Ck+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.

Vassiliev–Goussarov invariant, clasper, link, string link
Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 57M05, 18D10
Forward citations
Received: 30 October 1999
Revised: 27 January 2000
Accepted: 14 January 1999
Published: 28 January 2000
Proposed: Frances Kirwan
Seconded: Joan Birman, Robion Kirby
Kazuo Habiro
Graduate School of Mathematical Sciences
University of Tokyo
3-8-1 Komaba Meguro-ku
Tokyo 153