ABSTRACT
A.A. Markov has described the theorem about necessary and sufficient conditions for braids to represent isotopic links. He left this problem to N.M. Weinberg, who died soon after his first publication on the subject. This chapter discusses the Yang-Baxter equation which is closely connected with braid groups and knot invariants. In his work, G.S. Makanin proposed a nice refinement of the Alexander and Markov theorems: he proved that all knots can be obtained as closures of so-called unary braids. Besides, he proved that for any two unary braids representing the same knot, there is a change of Markov moves from one to the other that lies in the class of unary braids. The Yang-Baxter equation was first developed by physicists. However, these equations turn out to be very convenient in many areas of mathematics.
