Geometry & Topology

Calculus of clovers and finite type invariants of 3–manifolds

Stavros Garoufalidis, Mikhail Goussarov, and Michael Polyak

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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.

Article information

Source
Geom. Topol., Volume 5, Number 1 (2001), 75-108.

Dates
Received: 19 October 2000
Accepted: 28 January 2001
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513882984

Digital Object Identifier
doi:10.2140/gt.2001.5.75

Mathematical Reviews number (MathSciNet)
MR1812435

Zentralblatt MATH identifier
1066.57015

Subjects
Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx] 57M27: Invariants of knots and 3-manifolds
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

Keywords
3–manifolds Y–graphs finite type invariants clovers

Citation

Garoufalidis, Stavros; Goussarov, Mikhail; Polyak, Michael. Calculus of clovers and finite type invariants of 3–manifolds. Geom. Topol. 5 (2001), no. 1, 75--108. doi:10.2140/gt.2001.5.75. https://projecteuclid.org/euclid.gt/1513882984


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