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Volume: 31 | Article ID: art00009
Visualizing Mathematical Knot Equivalence
  DOI :  10.2352/ISSN.2470-1173.2019.1.VDA-683  Published OnlineJanuary 2019

We present a computer interface to visualize and interact with mathematical knots, i.e., the embeddings of closed circles in 3-dimensional Euclidean space. Mathematical knots are slightly different than everyday knots in that they are infinitely stretchy and flexible when being deformed into their topological equivalence. In this work, we design a visualization interface to depict mathematical knots as closed node-link diagrams with energies charged at each node, so that highly-tangled knots can evolve by themselves from high-energy states to minimal (or lower) energy states. With a family of interactive methods and supplementary user interface elements, out tool allows one to sketch, edit, and experiment with mathematical knots, and observe their topological evolution towards optimal embeddings. In addition, out interface can extract from the entire knot evolution those key moments where successive terms in the sequence differ by critical change; this provides a clear and intuitive way to understand and trace mathematical evolution with a minimal number of visual frames. Finally out interface is adapted and extended to support the depiction of mathematical links and braids, whose mathematical concepts and interactions are just similar to our intuition about knots. All these combine to show a mathematically rich interface to help us explore and understand a family of fundamental geometric and topological problems.

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Juan Lin, Hui Zhang, "Visualizing Mathematical Knot Equivalencein Proc. IS&T Int’l. Symp. on Electronic Imaging: Visualization and Data Analysis,  2019,  pp 683-1 - 683-11,

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