Geometry & Topology, Vol. 4 (2000) Paper no. 1, pages 1--83.

Claspers and finite type invariants of links

Kazuo Habiro

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.

Keywords. Vassiliev-Goussarov invariant, clasper, link, string link

AMS subject classification. Primary: 57M25. Secondary: 57M05, 18D10.

DOI: 10.2140/gt.2000.4.1

E-print: arXiv:math.GT/0001185

Submitted to GT on 30 October 1999. (Revised 27 January 2000.) Paper accepted 14 January 2000. Paper published 28 January 2000.

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Kazuo Habiro
Graduate School of Mathematical Sciences, University of Tokyo
3-8-1 Komaba Meguro-ku, Tokyo 153, Japan

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