Pseudo-characters of groups have recently found applications in the theory of classical knots and links in ℝ³. More precisely, there is a connection between pseudo-characters of Artin's braid groups and properties of links represented by braids. In the present work, this connection is investigated and the notion of kernel pseudo-characters of braid groups is introduced. It is proved that a kernel pseudo-character Φ and a braid β satisfy Φ(β) > C_Φ, where C_Φ is the defect of Φ, then β represents a prime link (i.e., a link that is noncomposite, nonsplit, and nontrivial). Furthermore, the space of braid group pseudo-characters is studied and a way to obtain nontrivial kernel pseudo-characters from an arbitrary braid group pseudo-character that is not a homomorphisrn is described. This allows one to use an arbitrary nontrivial braid group pseudo-character for recognition of prime knots and links. Bibliography: 17 titles.