Abstract. In [LMO] a 3-manifold invariant Ω(M) is constructed using a modification of the Kontsevich integral and the Kirby calculus. The invariant Ω takes values in a graded Hopf algebra of Feynman 3-valent graphs. Here we show that for homology 3-spheres the invariant Ω is universal for all finite type invariants, i.e. Ωn is an invariant order 3n which dominates all other invariants of the same order. Some corollaries are discussed. 1.

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The goal of the present paper is to find higher genus surgery formulas for the set of finite type invariants of integral homology 3-spheres, and to develop a theory of finite type invariants which will be applied in a subsequent publication [GL3] in the study of subgroups of the mapping class group. The main result is to show that six filtrations on the vector space generated by oriented integral homology 3-spheres (three coming from surgery on special classes of links and three coming from subgroups of the mapping class group) are equal. En route we introduce the notion of blink (a special case of a link) and of a new subgroup of the mapping class group.

Abstract. The present paper is a continuation of [Oh2] and [GL] devoted to the study of finite type invariants of integral homology 3-spheres. We introduce the notion of manifold weight systems, and show that type m invariants of integral homology 3-spheres are determined (modulo invariants of type m − 1) by their associated manifold weight systems. In particular we deduce a vanishing theorem for finite type invariants. We show that the space of manifold weight systems forms a commutative, co-commutative Hopf algebra and that the map from finite type invariants to manifold weight systems is an algebra map. We conclude with better bounds for the graded space of finite type invariants of integral homology

. Using elementary counting methods, we calculate the universal perturbative invariant (also known as the LMO invariant) of a 3-manifold M , satisfying H 1 (M; Z) = Z, in terms of the Alexander polynomial of M . We show that +1 surgery on a knot in the 3-sphere induces an injective map from finite type invariants of integral homology 3-spheres to finite type invariants of knots. We also show that weight systems of degree 2m on knots, obtained by applying finite type 3m invariants of integral homology 3-spheres, lie in the algebra of Alexander-Conway weight systems, thus answering the questions raised in [Ga]. Contents 1. Introduction 1 1.1. History 1 1.2. Statement of the results 2 1.3. Acknowledgment 3 2. Preliminaries 4 2.1. Preliminaries on Chinese characters 4 2.2. The Alexander-Conway polynomial and its weight system 5 2.3. Preliminaries on the LMO invariant 7 3. Proofs 8 References 10 1. Introduction 1.1. History. In their fundamental paper, T.T.Q. Le, J. Murakami and T. Ohtsu...

. We derive a gauge theoretic invariant of integral homology 3-spheres which counts gauge orbits of irreducible, perturbed flat SU(3) connections with sign given by spectral flow. To compensate for the dependence of this sum on perturbations, the invariant includes contributions from the reducible, perturbed flat orbits. Our formula for the correction term generalizes that given by Walker in his extension of Casson's SU(2) invariant to rational homology 3-spheres. 1. Introduction Since its introduction in 1985, Casson's invariant [3, 1] has been the focus of intense study. For example, it has been shown that it extends as a Q-valued invariant of oriented 3-manifolds which retains most of the important properties of the original invariant (for details, see [25, 14] and the references contained therein). Its relevance to gauge theory was recognized by C. Taubes, who related it to the Euler characteristic for the instanton homology groups defined by A. Floer [24, 6]. Because Casson's inv...

We are now embarrassingly rich in knot and 3-manifold invariants. We have to organize these invariants systematically and find out ways to make use of them. The theory of finite type knot invariants, or Vassiliev invariants, has been very successful in accomplishing the first

We provide some more explicit formulae to facilitate the computation of Ohtsuki's rational invariants,k, ~ of integral homology 3-spheres extracted from Reshetikhin-Turaev SU(2) quantum invariants. Several interesting consequences will follow from our computation of A2. One of them says that A2 is always an integer divisible by 3. It seems interesting to compare this result with the fact shown by Murakami that)u is 6 times the Casson invariant. Other consequences include some general criteria for distinguishing homology 3-spheres obtained from surgery on knots by using the Jones polynomial.

Abstract. We construct unitary modular categories for a general class of coset conformal field theories based on our previous study of these theories in the algebraic quantum field theory framework using subfactor theory. We also consider the calculations of the corresponding 3-manifold invariants. It is shown that under certain index conditions the link invaraints colored by the representations of coset factorize into the products of the the link invaraints colored by the representations of the two groups in the coset. But the 3-manifold invariants do not behave so simply in general due to the nontrivial branching and selection rules of the coset. Examples in the parafermion cosets and diagonal cosets show that 3-manifold invariants of the coset may be finer than the products of the 3-manifold invariants associated with the two groups in the coset, and these two invariants do not seem to be simply related in some cases, for an example, in the cases when there are issues of “fixed point resolutions”. In the later case our framework provides a mathematical understanding of the underlying unitary modular categories which has not been obtained by other methods.

. Let Z(M) be the 3-manifold invariant of Le, Murakami and Ohtsuki. We show that Z(M) = 1 + o(n), where o(n) denotes terms of degree n, if M is a homology 3-sphere obtained from S 3 by surgery on an n-component Brunnian link whose Milnor -invariants of length 2n vanish. We prove a realization theorem which is a partial converse to the above theorem. Using the Milnor filtration on links, we define a new bifiltration on the Q vector space with basis the set of oriented diffeomorphism classes of homology 3-spheres. This includes the Milnor level 2 filtration defined by Ohtsuki. We show that the Milnor level 2 and level 3 filtrations coincide after reindexing. October 23, 1998 1. Introduction The field of finite type 3-manifold invariants (also known as "perturbative quantum invariants") has developed quite rapidly over the past two years. A universal invariant of 3-manifolds, taking values in an algebra of Feynman diagrams, was introduced by Le, Murakami and Ohtsuki [LMO...