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32
An invariant of integral homology 3spheres which is universal for all finite type invariants, preprint
, 1996
"... Abstract. In [LMO] a 3manifold 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 3valent graphs. Here we show that for homology 3spheres the invariant Ω is universal for all finite ..."
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Cited by 58 (4 self)
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Abstract. In [LMO] a 3manifold 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 3valent graphs. Here we show that for homology 3spheres 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.
Finite Type 3Manifold Invariants, The Mapping Class Group And Blinks
, 1996
"... The goal of the present paper is to find higher genus surgery formulas for the set of finite type invariants of integral homology 3spheres, 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 ..."
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Cited by 29 (5 self)
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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 3spheres, 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 3spheres (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.
On finite type 3manifold invariants III: manifold weight systems, Topology, in press. M.N. Gusarov, On nequivalence of knots and invariants of finite degrees, Topology of manifolds and varieties, edited by
, 1994
"... Abstract. The present paper is a continuation of [Oh2] and [GL] devoted to the study of finite type invariants of integral homology 3spheres. We introduce the notion of manifold weight systems, and show that type m invariants of integral homology 3spheres are determined (modulo invariants of type ..."
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Cited by 24 (10 self)
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Abstract. The present paper is a continuation of [Oh2] and [GL] devoted to the study of finite type invariants of integral homology 3spheres. We introduce the notion of manifold weight systems, and show that type m invariants of integral homology 3spheres 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, cocommutative 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
The Alexander polynomial and finite type 3manifold invariants
 Math. Ann
"... Abstract. Using elementary counting methods, we calculate the universal perturbative invariant (also known as the LMO invariant) of a 3manifold M, satisfying H1(M, Z) = Z, in terms of the Alexander polynomial of M. We show that +1 surgery on a knot in the 3sphere induces an injective map from fin ..."
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Cited by 22 (7 self)
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Abstract. Using elementary counting methods, we calculate the universal perturbative invariant (also known as the LMO invariant) of a 3manifold M, satisfying H1(M, Z) = Z, in terms of the Alexander polynomial of M. We show that +1 surgery on a knot in the 3sphere induces an injective map from finite type invariants of integral homology 3spheres 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 3spheres, lie in the algebra of AlexanderConway weight systems, thus answering the questions raised in [Ga].
The SU(3) Casson Invariant For Integral Homology 3Spheres
, 1998
"... We derive a gauge theoretic invariant of integral homology 3spheres 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, p ..."
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Cited by 21 (9 self)
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We derive a gauge theoretic invariant of integral homology 3spheres 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 3spheres.
Integration of singular braid invariants and graph cohomology
, 1995
"... Abstract. We prove necessary and sucient conditions for an arbitrary invariant of braids with m double points to be the \mth derivative " of a braid invariant. We show that the \primary obstruction to integration " is the only obstruction. This gives a slight generalization of the existen ..."
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Cited by 15 (0 self)
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Abstract. We prove necessary and sucient conditions for an arbitrary invariant of braids with m double points to be the \mth derivative " of a braid invariant. We show that the \primary obstruction to integration " is the only obstruction. This gives a slight generalization of the existence theorem for Vassiliev invariants of braids. We give a direct proof by induction on m which works for invariants with values in any abelian group. We nd that to prove our theorem, we must show that every relation among fourterm relations satises a certain geometric condition. To nd the relations among relations we show that H1 of a variant of Kontsevich’s graph complex vanishes. We discuss related open questions for invariants of links and other things. 1.
On Ohtsuki's Invariants of Integral Homology 3Spheres
, 1999
"... We provide some more explicit formulae to facilitate the computation of Ohtsuki's rational invariants,k, ~ of integral homology 3spheres extracted from ReshetikhinTuraev SU(2) quantum invariants. Several interesting consequences will follow from our computation of A2. One of them says that ..."
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Cited by 10 (3 self)
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We provide some more explicit formulae to facilitate the computation of Ohtsuki's rational invariants,k, ~ of integral homology 3spheres extracted from ReshetikhinTuraev 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 3spheres obtained from surgery on knots by using the Jones polynomial.
3manifold invariants from cosets
"... 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 3manifold invariants. ..."
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Cited by 8 (4 self)
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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 3manifold 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 3manifold 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 3manifold invariants of the coset may be finer than the products of the 3manifold 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.