We consider the interplay of a quark state and a hadronic threshold in the framework of the P-matrix formalism, which is reviewed and extended for use together with conventional methods of computing quark-gluon dynamics. We provide a quantitative dynamical interpretation of the reduced R or K matrices and their poles that suggests a natural classification of threshold phenomena. At a threshold with a quark state close to it up to three S-matrix poles can be found. The scattering amplitudes for the corresponding cases are discussed. Our analysis is applied to make an outlook for experimental observation of the doubly strange H-dibaryon if it is not stable to strong decays.