Operators in the spaces of pseudocharacters of braid groups

Pseudocharacters of groups have recently found application in the theory of classical knots and links in $\mathbb R^3$. More precisely, there is a relationship between pseudocharacters of Artin's braid groups and properties of links represented by braids. In the paper, this relationship is investigated and the notion of kernel pseudocharacters of braid groups is introduced. It is proved that if a kernel pseudocharacter $\phi$ and a braid $\beta$ satisfy $|\phi(\beta)|>C_{\phi}$, where $C_{\phi}$ is the defect of $\phi$, then $\beta$ represents a prime link (i.e., a link that is noncomposite, nonsplit, and nontrivial). Furthermore, the space of braid group pseudocharacters is studied and a way is described to obtain nontrivial kernel pseudocharacters from an arbitrary braid group pseudocharacter that is not a homomorphism. This makes it possible to employ an arbitrary nontrivial braid group pseudocharacter for recognition of prime knots and links.