#### Geometry & Topology, Vol. 5 (2001)
Paper no. 3, pages 75--108.

## Calculus of clovers and finite type invariants of 3-manifolds

### Stavros Garoufalidis, Mikhail Goussarov, Michael Polyak

**Abstract**.
A clover is a framed trivalent graph with some additional structure,
embedded in a 3-manifold. We define surgery on clovers, generalizing
surgery on Y-graphs used earlier by the second author to define a new
theory of finite-type invariants of 3-manifolds. We give a systematic
exposition of a topological calculus of clovers and use it to deduce
some important results about the corresponding theory of finite type
invariants. In particular, we give a description of the weight systems
in terms of uni-trivalent graphs modulo the AS and IHX relations,
reminiscent of the similar results for links. We then compare several
definitions of finite type invariants of homology spheres (based on
surgery on Y-graphs, blinks, algebraically split links, and boundary
links) and prove in a self-contained way their equivalence.
**Keywords**.
3-manifolds, Y-graphs, finite type invariants, clovers

**AMS subject classification**.
Primary: 57N10, 57M27.
Secondary: 57M25.

**DOI:** 10.2140/gt.2001.5.75

**E-print:** `arXiv:math.GT/0005192`

Submitted to GT on 19 October 2000.
Paper accepted 28 January 2001.
Paper published 10 February 2001.

Notes on file formats
Stavros Garoufalidis, Mikhail Goussarov, Michael Polyak

SG: School of Mathematics, Georgia Institute of Technology

Atlanta, GA 30332-0160 USA

MP: School of Mathematics, Tel-Aviv University

69978 Tel-Aviv, Israel

Email: stavros@math.gatech.edu, polyak@math.tau.ac.il

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