2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/73826Integrase proteins acting on circular double-stranded DNA often change its topology by transforming unknotted circles into torus knots and links. Two systems of tangle equations--corresponding to the two initial DNA sequences--arise when modelling this transformation: direct and inverted. With no a priori assumptions on the constituent tangles, we utilize Dehn surgery arguments to completely classify the tangle solutions for each of the two systems. A key step is to combine work of our previous paper with recent results of Kronheimer, Mrowka, Ozsvath and Szabo, and Ernst to show a certain prime tangle must in fact be a generalized Montesinos tangle. These tangle solutions are divided into three classes, common to both systems, plus a fourth class for the inverted system that contains the sole generalized Montesinos tangle. We discuss the possible biological implications of our classification, and of this novel solution.17 pages, 1 figureGeometric TopologyBiomolecules57Mtext