Introduction
1.Groups. Small Cancellations. Greendlinger Theorem
1.1.2The notion of a diagram of a group
1.2.1Small cancellation conditions
1.3Algorithmic problems and the Dehn algorithm
2.1Definitions of the braid group
2.2The stable braid group and the pure braid group
2.3The curve algorithm for braids recognition
2.3.1Construction of the invariant
2.3.2Algebraic description of the invariant
2.4.1Definitions of virtual braids
2.4.2Invariants of virtual braids
3.Curves on Surfaces. Knots and Virtual Knots
3.1Basic notions of knot theory
3.2Curve reduction on surfaces
3.2.2Minimal curves in an annulus
3.2.3Proof of Theorems 3.3 and 3.4
3.2.4Operations on curves on a surface
3.3.1Classical case
3.3.3An analogue of Markov’s theorem in the virtual case
4.Two-dimensional Knots and Links
4.3Other types of 2-dimensional knotted surfaces
4.4Smoothing on 2-dimensional knots
4.4.2The smoothing process in terms of the framing change
Parity Theory
5.Parity in Knot Theories. The Parity Bracket
5.1The Gaußian parity and the parity bracket
5.1.2Smoothings of knot diagrams
5.1.3The parity bracket invariant
5.1.4The bracket invariant with integer coefficients
5.3Parity in terms of category theory
5.5Parities on 2-knots and links
5.6Parity Projection. Weak Parity
5.6.1Gaußian parity and parity projection
5.6.2The notion of weak parity
5.6.3Functorial mapping for Gaußian parity