Geometry & Topology

Calculus of clovers and finite type invariants of 3–manifolds

Stavros Garoufalidis, Mikhail Goussarov, and Michael Polyak

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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}

3–manifolds Y–graphs finite type invariants clovers


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.

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  • T D Cochran, P M Melvin, Finite type invariants of 3–manifolds, Rice University and Bryn Mawr College preprint, November 1997, available from arxiv:math.GT/9805026
  • S Garoufalidis, On finite type 3–manifold invariants I, J. Knot Theory and its Ramifications, 5 (1996) 441–462
  • S Garoufalidis, J Levine, Finite type 3–manifold invariants, the mapping class group and blinks, J. Diff. Geom. 47 (1997) 257–320
  • M Goussarov, Knotted graphs and a geometrical technique of n–equivalence, St. Petersbourg Math. Journal, to appear, POMI preprint 1995 in Russian, available at$\sim$drorbn/Goussarov,
  • M Goussarov, Finite type invariants and $n$–equivalence of $3$–manifolds, Comp. Rend. Ac. Sci. Paris, 329 Série I (1999) 517–522
  • K Habiro, Claspers and finite type invariants of links, Geometry and Topology, 4 (2000) 1–83
  • D Johnson, An abelian quotient of the mapping class group, Math. Ann. 249 (1980) 225–242
  • D Johnson, A survey of the Torelli group, from “Low-dimensional topology (San Francisco, Calif. 1981)”, Contemporary Math. 20 (1983) 163–179
  • R Kirby, A calculus for framed links in $S^3$, Invent. Math. 45 (1978) 35–56
  • S Matveev, Generalized surgeries of three-dimensional manifolds and representations of homology spheres, Mat. Zametki, 42 (1987) no. 2, 268–278; English translation in Math. Notices Acad. Sci. USSR, 42:2
  • S Matveev, M Polyak, A geometrical presentation of the surface mapping class group and surgery, Comm. Math. Phys. 160 (1994) 537–556
  • H Murakami, Y Nakanishi, On a certain move generating link-homology, Math. Ann. 284 (1989) 75–89
  • T Ohtsuki, Finite type invariants of integral homology 3–spheres, J. Knot Theory and its Ramifications, 5 (1996) 101–115