#### 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.

Notes on file formats
Kazuo Habiro

Graduate School of Mathematical Sciences, University of Tokyo

3-8-1 Komaba Meguro-ku, Tokyo 153, Japan

Email: habiro@ms.u-tokyo.ac.jp

GT home page

## Archival Version

**These pages are not updated anymore.
They reflect the state of
.
For the current production of this journal, please refer to
http://msp.warwick.ac.uk/.
**