Vassily Olegovich Manturov

Virtual knots: the state of the art

Basic Definitions and Notions Virtual Knots and Three-Dimensional Topology Quandles (Distributive Groupoids) in Virtual Knot Theory The Jones Polynomial. Atoms Khovanov Homology Virtual Braids Vassilievs Invariants Parity in Knot Theory. Free-Knots. Cobordisms Theory of Graph-Links.

KAUFFMAN-LIKE POLYNOMIAL AND CURVES IN 2-SURFACES

In the present paper, we construct a virtual knot invariant with values in the free infinite-dimensional module over Z[a, a-1]. The restriction of this invariant to the set of classical knots coincides with the Jones–Kauffman polynomial. It distinguishes virtual knots stronger than the generalised Jones-Kauffman polynomial proposed in [Kau].

The Khovanov Complex for Virtual Links

One of the most outstanding achievements of modern knot theory is Khovanov’s categorification of Jones polynomials. In the present paper, we construct the homology theory for virtual knots. An important obstruction to this theory (unlike the case of classical knots) is the nonorientability of “atoms”; an atom is a two-dimensional combinatorial object closely related with virtual link diagrams. The problem is solved directly for the field ℤ2 and also by using some geometrical constructions applied to atoms. We discuss a generalization proposed by Khovanov; he modifies the initial homology theory by using the Frobenius extension. We construct analogs of these theories for virtual knots, both algebraically and geometrically (by using atoms).

On Two Categorifications of the Arrow Polynomial for Virtual Knots

Two categorifications are given for the arrow polynomial, an extension of the Kauffman bracket polynomial for virtual knots. The arrow polynomial extends the bracket polynomial to infinitely many variables, each variable corresponding to an integer arrow number calculated from each loop in an oriented state summation for the bracket. The categorifications are based on new gradings associated with these arrow numbers, and give homology theories associated with oriented virtual knots and links via extra structure on the Khovanov chain complex. Applications are given to the estimation of virtual crossing number and surface genus of virtual knots and links.

VASSILIEV INVARIANTS FOR VIRTUAL LINKS, CURVES ON SURFACES AND THE JONES–KAUFFMAN POLYNOMIAL

We discuss the strong invariant of virtual links proposed in [23]. This invariant is obtained as a generalization of the Jones–Kauffman polynomial (generalized Kauffmans bracket) by adding to the sum some equivalence classes of curves in two-dimensional surfaces. Thus, the invariant is valued in the infinite-dimensional free module over Z[q,q-1]. We prove that this invariant can be decomposed into finite type Vassiliev invariant of virtual links (in Kauffmans sense); thus we present new infinite series of Vassiliev invariants. It is also proved that this invariant is strictly stronger than the Jones–Kauffman polynomial for virtual knots proposed by Kauffman. Some examples when the invariant can recognize virtual knots that can not be recognized by other invariants are given.

ON VIRTUAL CROSSING NUMBER ESTIMATES FOR VIRTUAL LINKS

We address the question of detecting minimal virtual diagrams with respect to the number of virtual crossings. This problem is closely connected to the problem of detecting the minimal number of additional intersection points for a generic immersion of a singular link in

MULTI-VARIABLE POLYNOMIAL INVARIANTS FOR VIRTUAL LINKS

R^{2}

Additional Gradings in Khovanov Homology

. We tackle this problem by the so-called

GRAPH-VALUED INVARIANTS OF VIRTUAL AND CLASSICAL LINKS AND MINIMALITY PROBLEM

\xi

A Graphical Construction of the sl(3) Invariant for Virtual Knots

-polynomial whose leading (lowest) degree naturally estimates the virtual crossing number. Several sufficient conditions for minimality together with infinite series of new examples are given. We also state several open questions about