Kontsevich Integral

Chmutov, Sergei; Duzhin, S.

doi:10.1016/b0-12-512666-2/00241-8

Cited by 25 publications

(42 citation statements)

References 10 publications

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“…The next lemma is the only one where we go into more detail than Chmutov and Duzhin [6]. We will need a modification of this proof in the graph case, so we felt it was important to use rigorous notation and touch on the fine points.…”

Section: Invariancementioning

confidence: 99%

“…A complete, detailed proof can be found in Chmutov and Duzhin [6] or Bar-Natan [2] and Kontsevich [10].…”

Section: Universalitymentioning

confidence: 99%

“…The proposition then follows from the lemma by taking a limit. (See Chmutov and Duzhin [6] for more details. )…”

Section: Invariancementioning

confidence: 99%

“…The reason for reproducing the proofs is that we need to modify some of them for the results to carry over to graphs, therefore understanding the knot case is crucial for the readability of the paper. Our main reference is a nice exposition by Chmutov and Duzhin [6]. This is the source of any theorems and proofs in Section 2, unless otherwise stated.…”

Section: Introductionmentioning

confidence: 99%

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On the Kontsevich integral for knotted trivalent graphs

Dancso

2010

Algebr. Geom. Topol.

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We construct an extension of the Kontsevich integral of knots to knotted trivalent graphs, which commutes with orientation switches, edge deletions, edge unzips and connected sums. In 1997 Murakami and Ohtsuki [16] first constructed such an extension, building on Drinfel 0 d's theory of associators. We construct a step-by-step definition, using elementary Kontsevich integral methods, to get a one-parameter family of corrections that all yield invariants well behaved under the graph operations above.05C10, 57M15, 57M25, 57M27 IntroductionThe goal of this paper is to construct an extension of the Kontsevich integral Z of knots to knotted trivalent graphs. The extension is a universal finite type invariant of knotted trivalent graphs, which commutes with natural operations that are defined on the space of graphs, as well as on the target space of Z . These operations are changing the orientation of an edge, deleting an edge, unzipping an edge (an analogue of cabling of knots) and connected sum.One reason this is interesting is that several knot properties (such as genus, unknotting number and ribbon property, for example) are definable with short formulas involving knotted trivalent graphs and the above operations. Therefore, such an operationrespecting invariant yields algebraic necessary conditions for these properties, ie equations in the target space of the invariant. This idea is due to Dror Bar-Natan, and is described by him in more detail in [1]. The extension of Z is the first example for such an invariant. Unfortunately, the target space of Z is too complicated for it to be useful in a computational sense. However, we hope that by finding sufficient quotients of the target space more computable invariants could be born.The construction also provides an algebraic description of the Kontsevich integral (of knots and graphs), due to the fact that knotted trivalent graphs are finitely generated, ie there's a finite (small) set of graphs such that any knotted trivalent graph can be obtained from these using the above operations. This is described in more detail by 1318 Zsuzsanna Dancso D P Thurston [18]. Since the extension commutes with the operations, it is enough to compute it for the graphs in the generating set. As knots are special cases of knotted trivalent graphs, this also yields an algebraic description of the Kontsevich integral of knots.The Kontsevich invariant Z was first extended to knotted trivalent graphs by Murakami and Ohtsuki in [16] and by Cheptea and Le in [5]. Both papers extend the combinatorial definition of Z , using q -tangles (aka parenthesized tangles) and building on a significant body of knowledge about Drinfel 0 d's associators to prove that the extension is a well-defined invariant. When one tries to extend Z naively, replacing knots by knotted trivalent graphs, the result is neither convergent nor an isotopy invariant. Thus, one needs to apply renormalizations to make it converge, and corrections to make it invariant. Using q -tangles, [16] and [5] do not have to deal with the c...

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Section: Invariancementioning

confidence: 99%

“…A complete, detailed proof can be found in Chmutov and Duzhin [6] or Bar-Natan [2] and Kontsevich [10].…”

Section: Universalitymentioning

confidence: 99%

“…The proposition then follows from the lemma by taking a limit. (See Chmutov and Duzhin [6] for more details. )…”

Section: Invariancementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the Kontsevich integral for knotted trivalent graphs

Dancso

2010

Algebr. Geom. Topol.

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“…For an introduction to the Kontsevich integral; see for example Bar-Natan [1], Chmutov and Duzhin, [7], Lescop [21] and Ohtsuki [26].…”

Section: Proof Of Theorem 11 Using the Kontsevich Integralmentioning

confidence: 99%

On the Kontsevich integral of Brunnian links

Habiro

Meilhan

2006

Algebr. Geom. Topol.

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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 .nC1/-component Brunnian links can be expressed as a quadratic form on the Milnor x link-homotopy invariants of length n C 1. Second, we describe the structure of the Brunnian part of the degree-2n graded quotient of the Goussarov-Vassiliev filtration for .nC1/-component links. 57M25, 57M27

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