ABSTRACT
This chapter describes some properties of Jones polynomials and ways that this polynomial can be applied for solving some problems in knot theory. The Murasugi theorem was a great step in the classification of alternating links. The classification problem for alternating links is reduced to the case of links with the same number of vertices. In an interesting construction was suggested. This construction describes a topological Khovanov complex, a formal chain complex, in which linear combinations of labeled sets of circles in the plane play the role of chains, where linear combinations of cobordisms are differentials. In the classification and tabulation of knots the important step is to describe diagrams having a minimal number of crossings. One of the main achievements in the development of knot theory is the Kauffman-Murasugi– Thistlethwaite theorem and the classification of alternating links by W. Menasco and M. Thistlethwaite following from this theorem.
