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- V. O. Manturov
- 2005

In the last few years, knot theory has enjoyed a rapidly developing generalisation, the Virtual knot theory, proposed by Louis Kauffman in 1996, see [Kau2]. A virtual link is a combinatorial generalisation of the notion of classical links: we consider planar diagrams with a new crossing type allowed; this new crossing (called virtual and marked by a circle)… (More)

- V. O. Manturov
- Eur. J. Comb.
- 2002

To date, many ways for encoding links have been discovered, see, e.g., [1, 5], and [3]. One of them is the encoding by closures of braids (Alexander’s theorem, see, e.g., [6]). Thus, it is important to find a ‘good’ class of braids that encodes all link isotopy types. In the present paper, we prove that all link isotopy classes can be encoded by a very… (More)

- V. O. Manturov
- 2004

The VA-polynomial proposed in the author’s earlier paper (Acta Appl. Math. 72 (2002), 295–309) for virtual knots and links is considered in this paper. One goal here is to refine the definition of this polynomial to the case of the ring Z in place of the field Q. Moreover, the approach in the paper mentioned makes it possible to recognize “long virtual… (More)

- V. O. Manturov
- 2004

- V. O. Manturov
- 2006

We construct explicitly the Khovanov homology theory for virtual links with arbitrary coefficients by using the twisted coefficients method. This method also works for constructing Khovanov homology for “non-oriented virtual knots” in the sense of [Viro], in particular, for knots in RP . Virtual knots were introduced in mid-nineties by Lou Kauffman, see… (More)

- V. O. Manturov
- 2009

The aim of the present paper is to study free knots, a dramatic simplification of the notion of virtual knots also connected to finite type invariants, curves on surfaces and other objects of low-dimensional topology. It turns out that the objects of such sort (a factorization of arbitrary graphs modulo three formal Reidemeister moves) can be studied by… (More)

This paper is an introduction to the subject of virtual knot theory, combined with a discussion of some specific new theorems about virtual knots. The new results are as follows: We prove, using a 3-dimensional topology approach that if a connected sum of two virtual knots K1 and K2 is trivial, then so are both K1 and K2. We establish an algorithm, using… (More)

- V. O. Manturov
- 2009

Both classical and virtual knots arise as formal Gauss diagrams modulo some abstract moves corresponding to Reidemeister moves. If we forget about both over/under crossings structure and writhe numbers of knots modulo the same Reidemeister moves, we get a dramatic simplification of virtual knots, which kills all classical knots. However, many virtual knots… (More)

- V. O. Manturov
- 2005

We show that if a classical knot diagram satisfies a certain combinatorial condition then it is minimal with respect to the number of classical crossings. This statement is proved by using the Kauffman bracket and the construction of atoms and knots.

- V. O. Manturov
- 2008

We prove that a virtual link diagrams satisfying two conditions on the Khovanov homology is minimal, that is, there is no virtual diagram representing the same link with smaller number of crossings. This approach works for both classical and virtual links For definitions of the Jones polynomial, Kauffman bracket, and the Khovanov homology, we send the… (More)