Oleg Viro

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

Finite-type invariants of classical and virtual knots

We observe that any knot invariant extends to virtual knots. The isotopy classification problem for virtual knots is reduced to an algebraic problem formulated in terms of an algebra of arrow diagrams. We introduce a new notion of finite type invariant an

Gauss Diagram Formulas for Vassiliev Invariants

Although the original definitions of Vassiliev knot invariants are rather far from explicit formulas, recently several types of formulas for them have been found; see [1], [2], [7], [4], [5]. However, most of them were designed more for the sake of general theory than for actual computations. Our initial goal was to provide more practical formulas. We were motivated by the well-known case of the linking number. It is the simplest of Vassiliev invariants for links. It can be computed in many different ways; see, e.g., [8]. Integral formulas of Bar-Natan [2] and Kontsevich [1] for Vassiliev invariants generalize the Gauss integral formula for the linking number. As is known, the Gauss integral formula has simple combinatorial counterparts. In this paper we present a similar transition to combinatorial formulas for higher-order Vassiliev invariants. As in the case of the linking number, both integral and combinatorial formulas may be obtained from an interpretation of the invariants as degrees of some maps. It was this viewpoint that motivated the whole of our investigations and appeared to be a rich source of various special formulas. We plan to discuss this phenomenon in detail in a forthcoming paper.

Dequantization of Real Algebraic Geometry on Logarithmic Paper

On logarithmic paper some real algebraic curves look like smoothed broken lines. Moreover, the broken lines can be obtained as limits of those curves. The corresponding deformation can be viewed as a quantization, in which the broken line is a classical object and the curves are quantum. This generalizes to a new connection between algebraic geometry and the geometry of polyhedra, which is more straight-forward than the other known connections and gives a new insight into constructions used in the topology of real algebraic varieties.

ON THE CASSON KNOT INVARIANT

We study the Vassiliev knot invariant v_2 of degree 2. We present it via the degrees of maps of various configuration spaces related to a knot to products of spheres. This gives rise to numerous geometrical and combinatorial formulas for this invariant.

Quantum relatives of the Alexander polynomial

The multivariable Conway function is generalized to oriented framed trivalent graphs equipped with additional structure (coloring). This is done via refinements of Reshetikhin-Turaev functors based on irreducible representations of quantized gl(1|1) and sl(2). The corresponding face state sum models for the generalized Conway function are presented.

Elementary Topology: Problem Textbook

General topology: Structures and spaces Continuity Topological properties Topological constructions Topological algebra Elements of algebraic topology: Fundamental group Covering spaces and calculation of fundamental groups Fundamental group and maps Cellular techniques Hints, comments, advices, solutions, and answers Bibliography Index.

TWISTED ACYCLICITY OF A CIRCLE AND SIGNATURES OF A LINK

Homology of the circle with non-trivial local coefficients is trivial. From this well-known fact we deduce geometric corollaries involving codimension-two links. In particular, the Murasugi–Tristram signatures are extended to invariants of links formed of arbitrary oriented closed codimension two submanifolds of an odd-dimensional sphere. The novelty is that the submanifolds are not assumed to be disjoint, but are transversal to each other, and the signatures are parametrized by points of the whole torus. Murasugi–Tristram inequalities and their generalizations are also extended to this setup.