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 and show that the restriction of any such invariant of degree n to classical knots is an invariant of degree ≤ n in the classical sense. A universal invariant of degree ≤ n is defined via a Gauss diagram formula. This machinery is used to obtain explicit formulas for invariants of low degrees. The same technique is also used to prove that any finite type invariant of classical knots is given by a Gauss diagram formula. We introduce the notion of n-equivalence of Gauss diagrams and announce virtual counter-parts of results concerning classical n-equivalence.1991 Mathematics Subject Classification. 57M25.

A clover is a framed trivalent graph with some additional structure, embedded in a 3-manifold. We define surgery on clovers, generalizing surgery on Y-graphs used earlier by the second author to define a new theory of finite-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 finite 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 definitions of finite 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.

Abstract.It is well known that any two diagrams representing the same oriented link are related by a finite sequence of Reidemeister 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.Mathematics Subject Classification (2010). 57M25, 57M27.

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In our previous paper [19] we introduced a new type of combinatorial formulas for Vassiliev knot invariants and presented lots of formulas of this type. To the best of our knowledge, these formulas are by far the simplest and the most practical for computational purposes. Since then Goussarov has proved the main conjecture formulated in [19]: any Vassiliev knot invariant can be described by such a formula, see [10].In [19] the examples of formulas were presented in a formal way, without proofs or even explanations of the ideas. We promised to interpret the invariants as degrees of some maps in a forthcoming paper and mentioned that it was this viewpoint that motivated the whole our investigations and appeared to be a rich source of various special formulas.In a sense, this viewpoint was not new. Quite the contrary, this is the most classical way to think on knot invariants. Indeed, a classical definition of a knot invariant runs as follows: some geometric construction gives an auxiliary space and then the machinery of algebraic topology is applied to this space to produce a number (or a quadratic form, a group, etc.). This scheme was almost forgotten in the eighties, when quantum invariants appeared. An auxiliary space was replaced by a combinatorial object (like knot diagram or closed braid presentation of a knot), while algebraic topology was replaced by representation theory and statistical mechanics. Vassiliev invariants and calculation of the quantum invariants in terms of Vassiliev invariants recovered the role of algebraic topology, but it is applied to the space of all knots, rather than to a space manufactured from a single knot. Presentations of Vassiliev invariants as degrees of maps would completely rehabilitate the classical approach.However, this is not our main intention. Various presentations for Vassiliev invariants reveal a rich geometric contents. The usual benefits of presenting some quantity as a degree of a map are that a degree is easy to calculate by various methods and, furthermore, degrees are manifestly invariant under various kinds of deformations.We were primarily motivated by the well-known case of the linking number. It is the simplest Vassiliev invariant of links. The linking number can be computed in many different ways, see e.g [22]. However, all formulas can be obtained from a single one: the linking number of a pair of circles is the degree of the map of a configuration space of pairs (a point on one circle, a point on the other circle) to S 2 defined by (x, y) → x−y |x−y| . Both the Gauss integral formula, and the combinatorial formulas in terms of a diagram are deduced from this interpretation via various methods for calculation of a degree.We discovered that this situation is reproduced in the case of Vassiliev invariants of higher degree. Both integral formulas found by Kontsevich [12] and Bar-Natan 1991 Mathematics Subject Classification. 57M25.