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Research on robotic manipulation has mainly focused on manipulating rigid objects so far. However, many important application domains require manipulating deformable objects, especially deformable linear objects (DLOs), such as ropes, cables, and sutures. Such objects are far more challenging to handle, as they can exhibit a much greater diversity of behaviors, and their manipulation almost inevitably requires two robotic arms, or more, performing well-coordinated motions. This paper describes a new motion planner for manipulating DLOs and tying knots (both self-knots and knots around simple static objects) using two cooperating robotic arms. This planner blends new ideas with preexisting concepts and techniques from knot theory, robot motion planning, and computational modeling. Unlike in traditional motion planning problems, the goal to be achieved by the planner is a topological state of the world, rather than a geometric one. To search for a manipulation path, the planner constructs a topologically biased probabilistic roadmap in the configuration space of the DLO. During roadmap construction, it uses inverse kinematics to determine the successive robot configurations implied by the DLO configurations and tests their feasibility. Also, inspired by the real life, the planner uses static ldquoneedlesrdquo (by analogy to the needles used in knitting) for maintaining the stability of the DLO during manipulation and to make the resulting manipulation plan robust to imperfections in the physical model of the DLO. The implemented planner has been tested both in graphic simulation and on a dual-PUMA-560 hardware platform to achieve various knots, like bowline, neck-tie, bow (shoe-lace), and stun-sail.