A circular knitting machine (1) for knitwear or hosiery, comprising a bearing structure, needle cylinder (C), a plurality of needles, and a dial group. The dial group comprises a support ring (4), a hook plate (5), and a thread feeding and cutting organ. The knitting machine comprises rotation transmission means (10) comprising: a drive pulley (11), a hook plate shaft (12), a flange (13) and an engaging mechanism (20) which operates between an engaged configuration, in which it constrains the pulley and the flange to one another, and a disengaged configuration, in which the flange is free from constraints. The engaging mechanism comprises an engaging organ (21) and actuating means (30) which enable the passage from the engaged configuration to the disengaged configuration in any angular position assumed by the flange or the pulley, and the passage from the disengaged configuration to the engaged configuration with the engaging organ positioned at a predetermined number of angular engaging and disengaging positions (40).