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Implicit Contact Diffuser: Sequential Contact Reasoning With Latent Point Cloud Diffusion
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It is known that any contact $$3$$-manifold can be obtained by rationally contact Dehn surgery along a Legendrian link $$L$$ in the standard tight contact $$3$$-sphere. We define and study various versions of contact surgery numbers, the minimal number of components of a surgery link $$L$$ describing a given contact $$3$$-manifold under consideration. In the first part of the paper, we relate contact surgery numbers to other invariants in terms of various inequalities. In particular, we show that the contact surgery number of a contact manifold is bounded from above by the topological surgery number of the underlying topological manifold plus three. In the second part, we compute contact surgery numbers of all contact structures on the $$3$$-sphere. Moreover, we completely classify the contact structures with contact surgery number one on $$S^1\times S^2$$, the Poincar\'e homology sphere and the Brieskorn sphere $$\Sigma(2,3,7)$$. We conclude that there exist infinitely many non-isotopic contact structures on each of the above manifolds which cannot be obtained by a single rational contact surgery from the standard tight contact $$3$$-sphere. We further obtain results for the $$3$$-torus and lens spaces. As one ingredient of the proofs of the above results we generalize computations of the homotopical invariants of contact structures to contact surgeries with more general surgery coefficients which might be of independent interest.more » « less
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Barrier potentials gained popularity as a means for robust contact handling in physical modeling and for modeling self-avoiding shapes. The key to the success of these approaches is adherence to geometric constraints, i.e., avoiding intersections, which are the cause of most robustness problems in complex deformation simulation with contact. However, existing barrier-potential methods may lead to spurious forces and imperfect satisfaction of the geometric constraints. They may have strong resolution dependence, requiring careful adaptation of the potential parameters to the object discretizations. We present a systematic derivation of a continuum potential defined for smooth and piecewise smooth surfaces, starting from identifying a set of natural requirements for contact potentials, including the barrier property, locality, differentiable dependence on shape, and absence of forces in rest configurations. Our potential is formulated independently of surface discretization and addresses the shortcomings of existing potential-based methods while retaining their advantages. We present a discretization of our potential that is a drop-in replacement for the potential used in the incremental potential contact formulation [Li et al. 2020], and compare its behavior to other potential formulations, demonstrating that it has the expected behavior. The presented formulation connects existing barrier approaches, as all recent existing methods can be viewed as a variation of the presented potential, and lays a foundation for developing alternative (e.g., higher-order) versions.more » « less
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null (Ed.)We present a novel, low cost framework for reconstructing surface contact movements during in-hand manipulations. Unlike many existing methods focused on hand pose tracking, ours models the behavior of contact patches, and by doing so is the first to obtain detailed contact tracking estimates for multi-contact manipulations. Our framework is highly accessible, requiring only low cost, readily available paint materials, a single RGBD camera, and a simple, deterministic interpolation algorithm. Despite its simplicity, we demonstrate the framework’s effectiveness over the course of several manipulations on three common household items. Finally, we demonstrate the use of a generated contact time series in manipulation learning for a simulated robot hand.more » « less
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Knot filtered embedded contact homology was first introduced by Hutchings in 2015; it has been computed for the standard transverse unknot in irrational ellipsoids by Hutchings and for the Hopf link in lens spaces via a quotient by Weiler. While toric constructions can be used to understand the ECH chain complexes of many contact forms adapted to open books with binding the unknot and Hopf link, they do not readily adapt to general torus knots and links. In this paper, we generalize the definition and invariance of knot filtered embedded contact homology to allow for degenerate knots with rational rotation numbers. We then develop new methods for understanding the embedded contact homology chain complex of positive torus knotted fibrations of the standard tight contact 3‐sphere in terms of their presentation as open books and as Seifert fiber spaces. We provide Morse–Bott methods, using a doubly filtered complex and the energy filtered perturbed Seiberg–Witten Floer theory developed by Hutchings and Taubes, and use them to compute the knot filtered embedded contact homology, for odd and positive.more » « less
