The article draws from chaos theory, catastrophe theory, and topology theory, focusing on psychoanalytic semiotics and three exemplary topological constructions: the Mobius band, the cross-cap, and Borromean knots. The relevance to doing critical criminology is indicated by focusing attention on some selected criminological theories that have affinities for an integrative approach, particularly those on edgework, invitational edge, and the foreground factors. This emergent state is more in tune with modeling and with explaining dynamic, nonlinear, transformational states (complexity theory). The article is concerned with sensitizing criminologists to postmodern methodologies, and offers various concepts and topological constructions in critical inquiry, and suggests directions for further inquiry. Affirmative postmodern methodologies advocate an expansion in presenting ways of describing possibilities. But this cannot take place by theorizing alone. Active struggle creates the decidability of the sign in the intersections of the Symbolic, the Imaginary, and the Real Order. Notes, figures, references, case cited, appendix