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Calculus of clovers and finite type invariants of 3–manifolds

Stavros Garoufalidis, Mikhail Goussarov and Michael Polyak

Geometry & Topology 5 (2001) 75–108

arXiv: math.GT/0005192


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.

3–manifolds, Y–graphs, finite type invariants, clovers
Mathematical Subject Classification 2000
Primary: 57N10, 57M27
Secondary: 57M25
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Received: 19 October 2000
Accepted: 28 January 2001
Published: 10 February 2001
Proposed: Robion Kirby
Seconded: Joan Birman, Cameron Gordon
Stavros Garoufalidis
School of Mathematics
Georgia Institute of Technology
GA 30332-0160
Mikhail Goussarov
Michael Polyak
School of Mathematics
Tel-Aviv University
69978 Tel-Aviv