Calculus of clovers and finite type invariants of 3–manifolds

By Stavros Garoufalidis, M. Goussarov, Michael Polyak. 2000

115 citations recorded by the source corpus.

A clover is a framed trivalent graph with some additional structure, embedded in a 3{manifold. We dene surgery on clovers, generalizing surgery on Y{graphs used earlier by the second author to dene a new theory of nite-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 nite 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 denitions of nite 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3a175388644a95c1df4dcb5d2710f3b51a743637

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Grope cobordism of classical knots (2000, unknown): “This is in [12], but their Borromean rings are oriented oppositely, so the gure should not look identical! One can also apply the technique of Proposition 6 to obtain this picture.”
Problems on invariants of knots and 3-manifolds (2004, methodology): “Let FY d (RM) [137] denote the submodule of RM spanned by [M;G] such that M is an integral homology 3-sphere and G is a collection of d disjoint Y-graphs (see Figure 11) in M , where [M;G] is de ned similarly as [M;L] (see [137]).”
Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant (2008, background): “Fundamental in their works is the notion of Ydequivalence for 3-manifolds [12, 15, 11, 9].”
Homology cylinders: an enlargement of the mapping class group (2000, methodology): “Now consider the corresponding β: T Hg → V . We claim that β(FY 2 (Hg)) = 0 and so we get an induced β: GY 1 (Hg) → V . As is pointed out in [H] and [ GGP ], surgery on a clasper of degree n is the same as cutting and pasting along some imbedded surface using an element of the n-th lower central ser”