Recently researchers have approached the problem of motion planning with topological constraints. In such problems, the inputs to the planner are source and destination points and the output is expected to be a valid path that respects the topological constraints. A concrete example of such problems - and the topic of this paper - is planning for a robot which is connected with a cable of limited length to a fixed point in the operation space. This paper presents a planning method for such problems by examining how the configuration space manifold can be represented efficiently. We introduce a convenient method for generating either parts or the complete atlas for the manifold based on special “cable events”. Generating parts of the configuration space on-the-fly enables improvements over the state of the art: (a)we decompose the environment into cells as needed rather than an off-line global discretization, obtaining competitive time and space complexity for our planner, (b) we are able to exploit topological structure to represent robot-cable configurations concisely, (c) we generalize the representation in order to examine cable-to-cable contacts, which have been widely ignored in the literature until now. Our results show the efficiency of the method and indicate further promise for procedures that represent manifolds via an amalgamation of implicit discrete topological structure and explicit Euclidean cells.