Vladimir Turaev

Invariants of 3-manifolds via link polynomials and quantum groups

The aim of this paper is to construct new topological invariants of compact oriented 3-manifolds and of framed links in such manifolds. Our invariant of (a link in) a closed oriented 3-manifold is a sequence of complex numbers parametrized by complex roots of 1. For a framed link in S 3 the terms of the sequence are equale to the values of the (suitably parametrized) Jones polynomial of the link in the corresponding roots of 1. In the case of manifolds with boundary our invariant is a (sequence of) finite dimensional complex linear operators. This produces from each root of unity q a 3-dimensional topological quantum field theory

Ribbon graphs and their invariants derived from quantum groups

The generalization of Jones polynomial of links to the case of graphs inR3 is presented. It is constructed as the functor from the category of graphs to the category of representations of the quantum groups.

STATE SUM INVARIANTS OF 3-MANIFOLDS AND QUANTUM 6j -SYMBOLS

IN THE 1980s the topology of low dimensional manifolds has experienced the most remarkable intervention of ideas developed in rather distant areas of mathematics. In the 4dimensional topology this process was initiated by S. Donaldson. He applied the theory of the Yang-Mills equation and instantons to study 4-manifolds. In dimension 3 a similar breakthrough was made by V. Jones. He discovered his famous polynomial of links in 3-sphere S3 via an astonishing use of von Neumann algebras. It has been soon understood that deep notions of statistical mechanics and quantum field theory stay behind the Jones polynomial (see [8], [16], [18]). The relevant basic algebraic structures turn out to be the Yang-Baxter equation, the R-matrices, and the quantum groups (see [S], [6], [7]). This viewpoint, in particular, enables one to generalize the Jones polynomial to links in arbitrary compact oriented 3-manifolds (see [ 131). In this paper we present a new approach to constructing “quantum” invariants of 3-manifolds. Our approach is intrinsic and purely combinatorial. The invariant of a manifold is defined as a certain state sum computed on an arbitrary triangulation of the manifold. The state sum in question is based on the so-called quantum 6j-symbols associated with the quantized universal enveloping algebra U,&(C)) where CJ is a complex root of 1 of a certain degree z > 2 (see [9]). The state sum on a triangulation X of a compact 3-manifold M is defined, roughly speaking, as follows. Assume for simplicity that M is closed, i.e. 8M = @. We consider “colorings” of X which associate with edges of X elements of the set of colors (0, l/2, 1, . . . , (I 2)/2). H avm a coloring of X we associate . g with each 3-simplex of X the q-6j-symbol

Introduction to Combinatorial Torsions

I Algebraic Theory of Torsions.- 1 Torsion of chain complexes.- 2 Computation of the torsion.- 3 Generalizations and functoriality of the torsion.- 4 Homological computation of the torsion.- II Topological Theory of Torsions.- 5 Basics of algebraic topology.- 6 The Reidemeister-Franz torsion.- 7 The Whitehead torsion.- 8 Simple homotopy equivalences.- 9 Reidemeister torsions and homotopy equivalences.- 10 The torsion of lens spaces.- 11 Milnors torsion and Alexanders function.- 12 Group rings of finitely generated abelian groups.- 13 The maximal abelian torsion.- 14 Torsions of manifolds.- 15 Links.- 16 The Fox Differential Calculus.- 17 Computing ?(M3) from the Alexander polynomial of links.- III Refined Torsions.- 18 The sign-refined torsion.- 19 The Conway link function.- 20 Euler structures.- 21 Torsion versus Seiberg-Witten invariants.- References.

Invariants of knots and 3-manifolds from quantum groupoids

Abstract We use the categories of representations of finite-dimensional quantum groupoids (weak Hopf algebras) to construct ribbon and modular categories that give rise to invariants of knots and 3-manifolds.

Homotopy quantum field theory

Homotopy Quantum Field Theory (HQFT) is a branch of Topological Quantum Field Theory founded by E. Witten and M. Atiyah. It applies ideas from theoretical physics to study principal bundles over manifolds and, more generally, homotopy classes of maps from manifolds to a fixed target space. This book is the first systematic exposition of Homotopy Quantum Field Theory. It starts with a formal definition of an HQFT and provides examples of HQFTs in all dimensions. The main body of the text is focused on 2-dimensional and 3-dimensional HQFTs. A study of these HQFTs leads to new algebraic objects: crossed Frobenius group-algebras, crossed ribbon group-categories, and Hopf group-coalgebras. These notions and their connections with HQFTs are discussed in detail. The text ends with several appendices including an outline of recent developments and a list of open problems. Three appendices by M. Müger and A. Virelizier summarize their work in this area. The book is addressed to mathematicians, theoretical physicists, and graduate students interested in topological aspects of quantum field theory. The exposition is self-contained and well suited for a one-semester graduate course. Prerequisites include only basics of algebra and topology.

Modified quantum dimensions and re-normalized link invariants

In this paper we give a re-normalization of the Reshetikhin-Turaev quantum invariants of links, by modified quantum dimensions. In the case of simple Lie algebras these modified quantum dimensions are proportional to the usual quantum dimensions. More interestingly we will give two examples where the usual quantum dimensions vanish but the modified quantum dimensions are non-zero and lead to non-trivial link invariants. The first of these examples is a class of invariants arising from Lie superalgebras previously defined by the first two authors. These link invariants are multivariable and generalize the multivariable Alexander polynomial. The second example, is a hierarchy of link invariants arising from nilpotent representations of quantized sl(2) at a root of unity. These invariants contain Kashaevs quantum dilogarithm invariants of knots.

Integrals for braided Hopf algebras

Let H be a Hopf algebra in a rigid braided monoidal category with split idempotents. We prove the existence of integrals on (in) H characterized by the universal property, employing results about Hopf modules, and show that their common target (source) object IntH is invertible. The fully braided version of Radfords formula for the fourth power of the antipode is obtained. The relationship of integration with cross-product and transmutation is studied. The results apply to topological Hopf algebras which do not have an additive structure, e.g. a torus with a hole.

CHORD DIAGRAM INVARIANTS OF TANGLES AND GRAPHS

The notion of a chord diagram emerged from Vassilievs work Vas90], Vas92] (see also Gusarov Gus91], Gus94] and Bar-Natan BN91], BN95]). Slightly later, Kontsevich Kon93] deened 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 isomorphism of a projective limit of algebras generated by the Vassiliev equivalence classes of knots onto the algebra of chord diagrams. Kontsevich originally deened his invariant of knots via a multiple integral given by an explicit but complicated analytic expression. This expression, however beautiful, does not reveal the combinatorial nature of the invariant. (A similar situation would occur if the linking number of 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 innnitesimal symmetric categories as well as the notion 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 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 x 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 coeecients in R, cf. Section 2. The set of morphisms in T (R) has a canonical ltration given by the powers of an ideal I which we call the augmentation ideal. Functions on morphisms in T (R) vanishing on I m+1 are exactly the Vassiliev invariants of degree m for framed oriented tangles. Completing T (R) at the ideal I, we obtain the pro-unipotent completion b T (R) = lim ?m T (R)=I m+1. Our main result (Corollary 2.5) states that, if R contains the eld Q of rational numbers, then b T (R) is isomorphic to a suitable completion b A(R) of the category …

MODULAR CATEGORIES AND 3-MANIFOLD INVARIANTS

The aim of this paper is to give a concise introduction to the theory of knot invariants and 3-manifold invariants which generalize the Jones polynomial and which may be considered as a mathematical version of the Witten invariants. Such a theory was introduced by N. Reshetikhin and the author on the ground of the theory of quantum groups. Here we use more general algebraic objects, specifically, ribbon and modular categories. Such categories in particular arise as the categories of representations of quantum groups. The notion of modular category, interesting in itself, is closely related to the notion of modular tensor category in the sense of G. Moore and N. Seiberg. For simplicity we restrict ourselves in this paper to the case of closed 3-manifolds.