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14
A rational noncommutative invariant of boundary links
 GEOM. AND TOPOLOGY
, 2003
"... In 1999, Rozansky conjectured the existence of a rational presentation of the Kontsevich integral of a knot. Roughly speaking, this rational presentation of the Kontsevich integral would sum formal power series into rational functions with prescribed denominators. Rozansky’s conjecture was soon pr ..."
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In 1999, Rozansky conjectured the existence of a rational presentation of the Kontsevich integral of a knot. Roughly speaking, this rational presentation of the Kontsevich integral would sum formal power series into rational functions with prescribed denominators. Rozansky’s conjecture was soon proven by the first author. We begin our paper by reviewing Rozansky’s conjecture and the main ideas that lead to its proof. The natural question of extending this conjecture to links leads to the class of boundary links, and a proof of Rozansky’s conjecture in this case. A subtle issue is the fact that a “hair ” map which replaces beads by the exponential of hair is not 11. This raises the question of whether a rational invariant of boundary links exists in an appropriate space of trivalent graphs whose edges are decorated by rational functions in noncommuting variables. A main result of the paper is to construct such an invariant, using the socalled surgery view of bounadry links and after developing a formal diagrammatic Gaussian integration. Since our invariant is one of many rational forms of the Kontsevich integral, one may ask if our invariant is in some sense canonical. We prove that this is indeed the case, by axiomatically characterizing our invariant as a universal finite type invariant of boundary links with respect to the null move. Finally, we discuss relations between our rational invariant and homology surgery, and give some applications to low dimensional topology.
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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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].
A universal U(1)RCC invariant of links and rationality conjecture, arXiv:math.GT/0201139
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Finite type invariants of cyclic branched covers
, 2002
"... Given a knot in an integer homology sphere, one can construct a family of closed 3manifolds (parametrized by the positive integers), namely the cyclic branched coverings of the knot. In this paper we give a formula for the the CassonWalker invariants of these 3manifolds in terms of residues of a ..."
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Cited by 5 (1 self)
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Given a knot in an integer homology sphere, one can construct a family of closed 3manifolds (parametrized by the positive integers), namely the cyclic branched coverings of the knot. In this paper we give a formula for the the CassonWalker invariants of these 3manifolds in terms of residues of a rational function (which measures the 2loop part of the Kontsevich integral of a knot) and the signature function of the knot. Our main result actually computes the LMO invariant of cyclic branched covers in terms of a rational invariant of the knot and its signature function.
On the Kontsevich integral of Brunnian links
, 2006
"... The purpose of the paper is twofold. First, we give a short proof using the Kontsevich integral for the fact that the restriction of an invariant of degree 2n to (n+1)– component Brunnian links can be expressed as a quadratic form on the Milnor ¯µ linkhomotopy invariants of length n + 1. Second, we ..."
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The purpose of the paper is twofold. First, we give a short proof using the Kontsevich integral for the fact that the restriction of an invariant of degree 2n to (n+1)– component Brunnian links can be expressed as a quadratic form on the Milnor ¯µ linkhomotopy invariants of length n + 1. Second, we describe the structure of the Brunnian part of the degree–2n graded quotient of the Goussarov–Vassiliev filtration for (n+1)–component links.
VassilievKontsevich invariants and Parseval’s theorem
, 2009
"... We use an example to provide evidence for the statement: the VassilievKontsevich invariants kn of a knot (or braid) k can be redefined so that k = P∞ 0 kn. This constructs a knot from its VassilievKontsevich invariants, like a power series expansion. The example is pure braids on two strands P2 ∼ ..."
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We use an example to provide evidence for the statement: the VassilievKontsevich invariants kn of a knot (or braid) k can be redefined so that k = P∞ 0 kn. This constructs a knot from its VassilievKontsevich invariants, like a power series expansion. The example is pure braids on two strands P2 ∼ = Z, which leads to solving eτ = q for τ a Laurent series in q. We set τ = P∞ 1 (−1)n+1(qn − q−n)/n and use Parseval’s theorem for Fourier series to prove eτ = q. Finally we describe some problems, particularly a Plancherel theorem for braid groups, whose solution would take us towards a proof of k = P∞ 0 kn.
RATIONALITY: FROM LIE ALGEBRAS TO LIE GROUPS
, 2002
"... Abstract. On the level of Lie algebras, the Kontsevich integral of a knot (a graphvalued invariant) becomes the colored Jones function (a power series invariant). Rozansky conjectured and the authors proved a Rationality Conjecture for the Kontsevich integral. In this note, we explain how the Ratio ..."
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Abstract. On the level of Lie algebras, the Kontsevich integral of a knot (a graphvalued invariant) becomes the colored Jones function (a power series invariant). Rozansky conjectured and the authors proved a Rationality Conjecture for the Kontsevich integral. In this note, we explain how the Rationality Conjecture is related to Lie groups. 1.
The Topological Ihx Relation
, 1998
"... We give an exposition of the classification of finite type invariants of homology 3spheres. A new conceptuallybased proof of the topological IHX relation, needed to show the welldefinedness of diagrams to manifolds, is given. ..."
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We give an exposition of the classification of finite type invariants of homology 3spheres. A new conceptuallybased proof of the topological IHX relation, needed to show the welldefinedness of diagrams to manifolds, is given.
A Reappearence of Wheels
, 1997
"... . Recently, a number of authors [KS, Oh2, Ro] have independently shown that the universal finite type invariant of rational homology 3spheres on the level of sl 2 can be recovered from the ReshetikhinTuraev sl 2 invariant. An important role in Ohtsuki's proof [Oh3] plays a map j 1 (which join ..."
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. Recently, a number of authors [KS, Oh2, Ro] have independently shown that the universal finite type invariant of rational homology 3spheres on the level of sl 2 can be recovered from the ReshetikhinTuraev sl 2 invariant. An important role in Ohtsuki's proof [Oh3] plays a map j 1 (which joins the legs of 2legged chinese characters) and its relation to a map ff in terms of a power series (on the level of sl 2 ). The purpose of the present note is to give a universal formula of the map ff in terms of a power series F of wheel chinese characters. The above formula is similar to universal formulas of wheel chinese characters considered in [BGRT1] and leads to a simple conceptual proof of the above mentioned relation between the maps j 1 and ff on the level of sl 2 . Contents 1. Introduction 1 1.1. History 1 1.2. Statement of the results 2 1.3. Some Questions 3 1.4. Acknowledgment 4 2. Proofs 4 2.1. Proof of Proposition 1.3 4 2.2. Proof of Proposition 1.2 5 References 7 1. Introduction...