In the paper we introduce new criteria for a handlebody link to be irreducible. These criteria are numerical and can be computed by a code. They are able to detect the irreducibility of all handlebody knots in the Ishii-Kishimoto-Moriuchi-Suzuki knot table and most handlebody links in the link table produced by G. Paolini and the authors. We also prove the existence of irreducible handlebody links of any given type.
