Punctured Torus Groups and 2 Bridge Knot Groups (I) Jorgensens picture of quasifuchsian punctured torus groups Fricke surfaces and PSL(2 ) representations Labeled representations and associated complexes Chain rule and side parameter Special examples Reformulation of Main Theorem 1.3.5 and outline of the proof Openness Closedness Algebraic roots and geometric roots. This monograph is Part 1 of a book project intended to give a full account of Jorgensens theory of punctured torus Kleinian groups and its generalization with application to knot theory. Although Jorgensens original work was not published in complete form it has been a source of inspiration. In particular it has motivated and guided Thurstons revolutionary study of low dimensional geometric topology. In this monograph we give an elementary and self contained description of Jorgensens theory with a complete proof. Through various informative illustrations readers are naturally led to an intuitive synthetic grasp of the theory which clarifies how a very simple fuchsian group evolves into complicated Kleinian groups.