## Chord Diagram Invariants of Tangles and Graphs (1998)

Venue: | Duke Math. J |

Citations: | 16 - 2 self |

### BibTeX

@ARTICLE{Kassel98chorddiagram,

author = {Christian Kassel and Vladimir Turaev},

title = {Chord Diagram Invariants of Tangles and Graphs},

journal = {Duke Math. J},

year = {1998},

volume = {92},

pages = {497--552}

}

### OpenURL

### Abstract

this paper we clarify the relationship between tangles and chord diagrams. It is formulated in terms of categories whose sets of morphisms are spanned by tangles and chord diagrams, respectively. More precisely, we fix a commutative ring R and consider categories T (R) and A(R) whose morphisms are formal linear combinations of framed oriented tangles and chord diagrams with coefficients in R, cf. Section 2. The set of morphisms in T (R) has a canonical filtration given by the powers of an ideal I which we call the augmentation ideal. Functions on morphisms in T (R) vanishing on I

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Citation Context ... of a braided category and to prove that the category b T (R) can be obtained as a pro-unipotent completion. We first recall a few well-known definitions. For more details and further references, see =-=[ML71]-=-, [Tur94], [Kas95]. 3.1. Monoidal categories. Let C be a category and\Omega a covariant functor from C \Theta C to C. This means that for any pair (U; V ) of objects of C there exists an object U\Omeg... |

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Citation Context ...ramed oriented knots by Kontsevich using multiple integrals. 7. Recovering the quantum invariants Given a complex semisimple Lie algebra g and a finite-dimensional g-module V , Reshetikhin and Turaev =-=[RT90]-=- constructed an isotopy invariant F g;V of framed oriented tangles depending on the data (g; V ). Because of the way used to define this invariant, it is often called a quantum invariant. The invarian... |

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Citation Context ...ssion for F g;V (T ) in terms of the invariant Z(T ) of Section 6. Starting from the semisimple Lie algebra g, we can construct two apriori unrelated ribbon categories. (a) Drinfeld [Dri87] and Jimbo =-=[Jim85]-=- constructed a topological Hopf algebra U h g over the ring C[[h]] with the following properties: (i) there is an isomorphism of Hopf algebras: U h g=hU h g = U(g); (ii) there is an isomorphism ! : U ... |

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Citation Context ... a symmetry and r V;W is an endomorphism of V\Omega W . Relations (3.2.3--3.2.4) imply (4.2.1) and (4.2.3) for t V;W = r V;W + oe W;V r W;V oe V;W : (4:2:5) Relations (4.2.1--4.2.3) first appeared in =-=[Dri90], p. 852 o-=-f the English translation. The term "infinitesimal braiding" was coined by Cartier [Car93]. The following example shows that infinitesimal braidings arise naturally in the theory of semisimp... |

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Citation Context ...via quantum groups and R-matrices. In Corollary 7.3 we give an explicit formula computing the quantum invariants of a knot from its chord diagram invariant. This formula appeared as Theorem XX.8.3 in =-=[Kas95]-=- (with a sketch of a proof) and was independently obtained by Le and Murakami [LM94]. The second aim of this paper is to generalize the theory outlined above to embedded graphs in R 3 . This generaliz... |

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Citation Context ...Turaev Institut de Recherche Math'ematique Avanc'ee, Universit'e Louis Pasteur - C.N.R.S., 7 rue Ren'e Descartes, 67084 Strasbourg, France. The notion of a chord diagram emerged from Vassiliev's work =-=[Vas90]-=-, [Vas92] (see also Gusarov [Gus91], [Gus94] and Bar-Natan [BN91], [BN95]). Slightly later, Kontsevich [Kon93] defined an invariant of classical knots taking values in the algebra generated by formal ... |

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Citation Context ...riant F g;V (T ) of a framed oriented tangle T can be expressed as a formal series F g;V (T ) = P m0 F g;V;m (T ) h m with complex coefficients. It was observed by Bar-Natan [BN95] and Birman and Lin =-=[BL93]-=- that F g;V;m is a Vassiliev invariant of degreesm. In this section we shall give an explicit expression for F g;V (T ) in terms of the invariant Z(T ) of Section 6. Starting from the semisimple Lie a... |

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Citation Context ...es, 67084 Strasbourg, France. The notion of a chord diagram emerged from Vassiliev's work [Vas90], [Vas92] (see also Gusarov [Gus91], [Gus94] and Bar-Natan [BN91], [BN95]). Slightly later, Kontsevich =-=[Kon93]-=- defined an invariant of classical knots taking values in the algebra generated by formal linear combinations of chord diagrams modulo the four-term relation. This knot invariant establishes an isomor... |

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Citation Context ...ting the quantum invariants of a knot from its chord diagram invariant. This formula appeared as Theorem XX.8.3 in [Kas95] (with a sketch of a proof) and was independently obtained by Le and Murakami =-=[LM94]-=-. The second aim of this paper is to generalize the theory outlined above to embedded graphs in R 3 . This generalization follows the same lines but involves an additional relation between chord diagr... |

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Citation Context ...one uses categories of tangles, as introduced by Yetter and Turaev (see [Tur94]). Note also that a counterpart of the Kontsevich knot invariant in the theory of braids was discovered earlier by Kohno =-=[Koh85]-=- who considered an algebraic version of chord diagrams. In this paper we clarify the relationship between tangles and chord diagrams. It is formulated in terms of categories whose sets of morphisms ar... |

36 |
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Citation Context ... knots were introduced via the Gauss integral formula without a combinatorial calculation). A combinatorial reformulation of the Kontsevich integral appeared in the works of Bar-Natan [BN94], Cartier =-=[Car93]-=-, Le and Murakami [LM93], Piunikhin [Piu95] (see also [Kas95, Chapter XX]). On the algebraic side, this reformulation uses the notions of braided and infinitesimal symmetric categories as well as the ... |

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Citation Context ...ion of an associator introduced by Drinfeld [Dri89] in his study of quasitriangular quasi-Hopf algebras. On the geometric side, one uses categories of tangles, as introduced by Yetter and Turaev (see =-=[Tur94]-=-). Note also that a counterpart of the Kontsevich knot invariant in the theory of braids was discovered earlier by Kohno [Koh85] who considered an algebraic version of chord diagrams. In this paper we... |

24 |
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Citation Context ...matique Avanc'ee, Universit'e Louis Pasteur - C.N.R.S., 7 rue Ren'e Descartes, 67084 Strasbourg, France. The notion of a chord diagram emerged from Vassiliev's work [Vas90], [Vas92] (see also Gusarov =-=[Gus91]-=-, [Gus94] and Bar-Natan [BN91], [BN95]). Slightly later, Kontsevich [Kon93] defined an invariant of classical knots taking values in the algebra generated by formal linear combinations of chord diagra... |

20 |
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Citation Context ...al formula without a combinatorial calculation). A combinatorial reformulation of the Kontsevich integral appeared in the works of Bar-Natan [BN94], Cartier [Car93], Le and Murakami [LM93], Piunikhin =-=[Piu95]-=- (see also [Kas95, Chapter XX]). On the algebraic side, this reformulation uses the notions of braided and infinitesimal symmetric categories as well as the notion of an associator introduced by Drinf... |

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Citation Context ...Drinfeld proved the existence of solutions of (4.7.3--4.7.6) by constructing a special solution \Phi KZ with complex coefficients reflecting the monodromy of the Knizhnik-Zamolodchikov equations (see =-=[Dri89b]-=-, [Dri90]). Drinfeld also established that there exists an associator with rational coefficients. However, there is no solution with integral coefficients, which is why we need the assumption R oe Q i... |

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Citation Context ...vanc'ee, Universit'e Louis Pasteur - C.N.R.S., 7 rue Ren'e Descartes, 67084 Strasbourg, France. The notion of a chord diagram emerged from Vassiliev's work [Vas90], [Vas92] (see also Gusarov [Gus91], =-=[Gus94]-=- and Bar-Natan [BN91], [BN95]). Slightly later, Kontsevich [Kon93] defined an invariant of classical knots taking values in the algebra generated by formal linear combinations of chord diagrams modulo... |

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Citation Context ...sible to give an analytic expression for the resulting chord diagram invariants of graphs, we confine ourselves to an algebraic approach. For another approach to finite-type invariants of graphs, see =-=[Sta92]-=-. The plan of the paper is as follows. We recall in Section 1 the Vassiliev equivalence of knots and formulate the theorem of Kontsevich computing the algebra of the corresponding equivalence classes ... |

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Citation Context ...ix R h of U h g is related to the central element C h by the relation (R h ) 21 R h = \Delta h (e hCh=2 )(e \GammahC h=2\Omega e \GammahC h=2 ) (7:0:1) where \Delta h is the comultiplication in U h g =-=[Dri89a]-=-; for any finite-dimensional g-module V , there is a unique object e V of U h (g)-Mod fr such that e V =h e V = V as a g-module. Following [RT90], we define F g;V as the unique functor from the tangle... |

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Citation Context ...stitut de Recherche Math'ematique Avanc'ee, Universit'e Louis Pasteur - C.N.R.S., 7 rue Ren'e Descartes, 67084 Strasbourg, France. The notion of a chord diagram emerged from Vassiliev's work [Vas90], =-=[Vas92]-=- (see also Gusarov [Gus91], [Gus94] and Bar-Natan [BN91], [BN95]). Slightly later, Kontsevich [Kon93] defined an invariant of classical knots taking values in the algebra generated by formal linear co... |

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Citation Context ... given prime `. Using work of Grothendieck [Gro84] and Ihara, Drinfeld [Dri90] constructed a group morphism ae : Gal( �� Q=Q) ! GT(Q ` ) where Gal( �� Q=Q) is the absolute Galois group of Q (s=-=ee also [Iha91]-=-, [Iha94]). Let us apply the previous remarks to the category T of framed oriented tangles described in Sections 2 and 3. Consider its pro-unipotent completion b T (Q ` ). Given an element oe of Gal( ... |

11 | Representation of the category of tangles by Kontsevich’s iterated integral - Le, Murakami - 1995 |

5 |
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Citation Context ...sms lim /\Gamma m R[P n ]=I m 0 �� = b Up n and lim /\Gamma m R[B n ]=I m �� = R[S n ]sb Up n : The first isomorphism has been established by Kohno in [Koh85] using rational homotopy theory (s=-=ee also [Hai86]-=-). Proof.--- It follows the same lines as the proof of Theorem 2.4. The main difference is that we replace the categories T and A(R) by smaller ones. We replace T by the categories B and P whose objec... |

3 |
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Citation Context ...teur - C.N.R.S., 7 rue Ren'e Descartes, 67084 Strasbourg, France. The notion of a chord diagram emerged from Vassiliev's work [Vas90], [Vas92] (see also Gusarov [Gus91], [Gus94] and Bar-Natan [BN91], =-=[BN95]-=-). Slightly later, Kontsevich [Kon93] defined an invariant of classical knots taking values in the algebra generated by formal linear combinations of chord diagrams modulo the four-term relation. This... |

1 |
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Citation Context ...ouis Pasteur - C.N.R.S., 7 rue Ren'e Descartes, 67084 Strasbourg, France. The notion of a chord diagram emerged from Vassiliev's work [Vas90], [Vas92] (see also Gusarov [Gus91], [Gus94] and Bar-Natan =-=[BN91]-=-, [BN95]). Slightly later, Kontsevich [Kon93] defined an invariant of classical knots taking values in the algebra generated by formal linear combinations of chord diagrams modulo the four-term relati... |

1 | Natan, Non-associative tangles, preprint - Bar - 1994 |

1 |
On the embedding of Gal( �� Q=Q) into d GT
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(Show Context)
Citation Context ...ime `. Using work of Grothendieck [Gro84] and Ihara, Drinfeld [Dri90] constructed a group morphism ae : Gal( �� Q=Q) ! GT(Q ` ) where Gal( �� Q=Q) is the absolute Galois group of Q (see also [=-=Iha91], [Iha94]). L-=-et us apply the previous remarks to the category T of framed oriented tangles described in Sections 2 and 3. Consider its pro-unipotent completion b T (Q ` ). Given an element oe of Gal( �� Q=Q), ... |

1 | 705--730 Institut de Recherche Math'ematique Avanc'ee Universit'e Louis Pasteur - C.N.R.S. 7 rue Ren'e Descartes 67084 Strasbourg Cedex, France E-mail: kassel@math.u-strasbg.fr, turaev@math.u-strasbg.fr Fax - Math - 1988 |