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65
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 52 (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.
Perturbative 3manifolds invariants by cutandpaste topology; GT/9912167
"... We give a purely topological definition of the perturbative quantum invariants of links and 3manifolds associated with ChernSimons field theory. Our definition is as close as possible to one given by Kontsevich. We will also establish some basic properties of these invariants, in particular that t ..."
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Cited by 34 (1 self)
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We give a purely topological definition of the perturbative quantum invariants of links and 3manifolds associated with ChernSimons field theory. Our definition is as close as possible to one given by Kontsevich. We will also establish some basic properties of these invariants, in particular that they are universally finite type with respect to algebraically split surgery and with respect to Torelli surgery. Torelli surgery is a mutual generalization of blink surgery of Garoufalidis and Levine and clasper surgery of Habiro. 1
Claspers and finite type invariants of links
, 2000
"... We introduce the concept of “claspers,” which are surfaces in 3–manifolds with some additional structure on which surgery operations can be performed. Using claspers we define for each positive integer k an equivalence relation on links called “Ck–equivalence,” which is generated by surgery operatio ..."
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Cited by 26 (1 self)
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We introduce the concept of “claspers,” which are surfaces in 3–manifolds with some additional structure on which surgery operations can be performed. Using claspers we define for each positive integer k an equivalence relation on links called “Ck–equivalence,” which is generated by surgery operations of a certain kind called “Ck–moves”. We prove that two knots in the 3–sphere are Ck+1–equivalent if and only if they have equal values of Vassiliev–Goussarov invariants of type k with values in any abelian groups. This result gives a characterization in terms of surgery operations of the informations that can be carried by Vassiliev–Goussarov invariants. In the last section we also describe outlines of some applications of claspers to other fields in 3–dimensional topology.
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 23 (4 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.
The Århus Integral of Rational Homology 3Spheres I: A Highly Non Trivial Flat Connection on S³
, 2002
"... Path integrals do not really exist, but it is very useful to dream that they do and figure out the consequences. Apart from describing much of the physical world as we now know it, these dreams also lead to some highly nontrivial mathematical theorems and theories. We argue that even though nontri ..."
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Cited by 20 (4 self)
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Path integrals do not really exist, but it is very useful to dream that they do and figure out the consequences. Apart from describing much of the physical world as we now know it, these dreams also lead to some highly nontrivial mathematical theorems and theories. We argue that even though nontrivial at connections on S³ do not really exist, it is beneficial to dream that one exists (and, in fact, that it comes from the nonexistent ChernSimons path integral). Dreaming the right way, we are led to a rigorous construction of a universal finitetype invariant of rational homology spheres. We show that this invariant is equal (up to a normalization) to the LMO (LeMurakamiOhtsuki, [LMO]) invariant and that it recovers the Rozansky and Ohtsuki invariants. This is part I of a 4...
The primary approximation to the cohomology of the moduli space of curves and cocycles for the MumfordMoritaMiller classes
, 2001
"... ..."
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 19 (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].
Witten’s Invariants of Rational Homology Spheres at Prime Values of K and Trivial Connection Contribution
, 1995
"... We establish a relation between the coefficients of asymptotic expansion of the trivial connection contribution to Witten’s invariant of rational homology spheres and the invariants that T. Ohtsuki extracted from Witten’s invariant at prime values of K. We also rederive the properties of prime K inv ..."
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Cited by 17 (5 self)
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We establish a relation between the coefficients of asymptotic expansion of the trivial connection contribution to Witten’s invariant of rational homology spheres and the invariants that T. Ohtsuki extracted from Witten’s invariant at prime values of K. We also rederive the properties of prime K invariants discovered by H. Murakami and T. Ohtsuki. We do this by using the bounds on Taylor series expansion of the Jones polynomial of algebraically split links, studied in our previous paper. These bounds are enough to prove that Ohtsuki’s invariants are of finite type. The relation between Ohtsuki’s invariants and trivial connection contribution is verified explicitly for lens spaces and Seifert manifolds.