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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.
Garoufalidis, Stavros 1 ; Goussarov, Mikhail 2 ; Polyak, Michael 3
@article{GT_2001_5_1_a2,
author = {Garoufalidis, Stavros and Goussarov, Mikhail and Polyak, Michael},
title = {Calculus of clovers and finite type invariants of 3{\textendash}manifolds},
journal = {Geometry & topology},
pages = {75--108},
publisher = {mathdoc},
volume = {5},
number = {1},
year = {2001},
doi = {10.2140/gt.2001.5.75},
url = {http://geodesic.mathdoc.fr/articles/10.2140/gt.2001.5.75/}
}
TY - JOUR AU - Garoufalidis, Stavros AU - Goussarov, Mikhail AU - Polyak, Michael TI - Calculus of clovers and finite type invariants of 3–manifolds JO - Geometry & topology PY - 2001 SP - 75 EP - 108 VL - 5 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/gt.2001.5.75/ DO - 10.2140/gt.2001.5.75 ID - GT_2001_5_1_a2 ER -
%0 Journal Article %A Garoufalidis, Stavros %A Goussarov, Mikhail %A Polyak, Michael %T Calculus of clovers and finite type invariants of 3–manifolds %J Geometry & topology %D 2001 %P 75-108 %V 5 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/gt.2001.5.75/ %R 10.2140/gt.2001.5.75 %F GT_2001_5_1_a2
Garoufalidis, Stavros; Goussarov, Mikhail; Polyak, Michael. Calculus of clovers and finite type invariants of 3–manifolds. Geometry & topology, Tome 5 (2001) no. 1, pp. 75-108. doi : 10.2140/gt.2001.5.75. http://geodesic.mathdoc.fr/articles/10.2140/gt.2001.5.75/
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