Robotic manipulation of deformable, one-dimensional objects (DOOs) like ropes or cables has important potential applications in manufacturing, agriculture, and surgery. In such environments, the task may involve threading through or avoiding becoming tangled with objects like racks or frames. Grasping with multiple grippers can create closed loops between the robot and DOO, and If an obstacle lies within this loop, it may be impossible to reach the goal. However, prior work has only considered the topology of the DOO in isolation, ignoring the arms that are manipulating it. Searching over possible grasps to accomplish the task without considering such topological information is very inefficient, as many grasps will not lead to progress on the task due to topological constraints. Therefore, we propose a grasp loop signature which categorizes the topology of these grasp loops and show how it can be used to guide planning. We perform experiments in simulation on two DOO manipulation tasks to show that using the signature is faster and succeeds more often than methods that rely on local geometry or finite-horizon planning. Finally, we demonstrate using the signature in the real world to manipulate a cable in a scene with obstacles using a dual-arm robot.