Michael Polyak

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.

Calculus of clovers and finite type invariants of 3–manifolds

A clover is a framed trivalent graph with some additional structure, embedded in a 3{manifold. We dene surgery on clovers, generalizing surgery on Y{graphs used earlier by the second author to dene a new theory of nite-type invariants of 3{manifolds. We give a systematic exposition of a topological calculus of clovers and use it to deduce some important results about the corresponding theory of nite type invariants. In particular, we give a description of the weight systems in terms of uni-trivalent graphs modulo the AS and IHX relations, reminiscent of the similar results for links. We then compare several denitions of nite type invariants of homology spheres (based on surgery on Y{graphs, blinks, algebraically split links, and boundary links) and prove in a self-contained way their equivalence.

Minimal generating sets of Reidemeister moves

It is well known that any two diagrams representing the same oriented link are related by a finite sequence of Reidemeis- ter moves 1, 2 and 3. Depending on orientations of fragments involved in the moves, one may distinguish 4 different versions of each of the 1 and 2 moves, and 8 versions of the 3 move. We introduce a minimal generating set of 4 oriented Reidemeister moves, which includes two 1 moves, one 2 move, and one 3 move. We then study which other sets of up to 5 oriented moves generate all moves, and show that only few of them do. Some commonly considered sets are shown not to be generating. An unexpected non-equivalence of different 3 moves is discussed.

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.

Invariants of curves and fronts via Gauss diagrams

Abstract We use a notion of chord diagrams to define their representations in Gauss diagrams of plane curves. This enables us to obtain invariants of generic plane and spherical curves in a systematic way via Gauss diagrams. We define a notion of invariants are of finite degree and prove that any Gauss diagram invariants are of finite degree. In this way we obtain elementary combinatorial formulas for the degree 1 invariants J ± and St of generic plane curves introduced by Arnold [1] and for the similar invariants J ± S and St S of spherical curves. These formulas allow a systematic study and an easy computation of the invariants and enable one to answer several questions stated by Arnold. By a minor modification of this technique we obtain similar expressions for the generalization of the invariants J ± and St to the case of Legendrian fronts. Different generalizations of the invariants and their relations to Vassiliev knot invariants are discussed.

On the Algebra of Arrow Diagrams

AbstractThe purpose of this Letter is to develop further the Gauss diagram approach to finite-type link invariants. In this context we introduce Gauss diagrams counterparts to chord diagrams, 4T relation, weight systems arising from Lie algebras, and an algebra of unitrivalent graphs modulo the STU relation. The counterparts, respectively, are arrow diagrams, 6T relation, weights arising from the solutions of the classical Yang–Baxter equation, and an algebra

Elementary Combinatorics of the HOMFLYPT Polynomial

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