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Untangling ropes, wires, and cables is a challenging task for robots due to the high-dimensional configuration space, visual homogeneity, self-occlusions, and complex dynamics. We consider dense (tight) knots that lack space between self-intersections and present an iterative approach that uses learned geometric structure in configurations. We instantiate this into an algorithm, HULK: Hierarchical Untangling from Learned Keypoints, which combines learning-based perception with a geometric planner into a policy that guides a bilateral robot to untangle knots. To evaluate the policy, we perform experiments both in a novel simulation environment modelling cables with varied knot types and textures and in a physical system using the da Vinci surgical robot. We find that HULK is able to untangle cables with dense figure-eight and overhand knots and generalize to varied textures and appearances. We compare two variants of HULK to three baselines and observe that HULK achieves 43.3% higher success rates on a physical system compared to the next best baseline. HULK successfully untangles a cable from a dense initial configuration containing up to two overhand and figure-eight knots in 97.9% of 378 simulation experiments with an average of 12.1 actions per trial. In physical experiments, HULK achieves 61.7% untangling success, averaging 8.48 actions per trial. Supplementary material, code, and videos can be found at https://tinyurl.com/y3a88ycu.