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Interactive topological drawing Scharein, Robert Glenn


The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in three-dimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues. Because the domain of application is mathematics, topological drawing is also concerned with the correct representation and display of these objects on a computer. By correctness we mean that the essential topological features of objects are maintained during interaction. We have chosen to limit the scope of topological drawing to knot theory, a domain that consists essentially of one class of object (embedded circles in three-dimensional space) yet is rich enough to contain a wide variety of difficult problems of research interest. In knot theory, two embedded circles (knots) are considered equivalent if one may be smoothly deformed into the other without any cuts or self-intersections. This notion of equivalence may be thought of as the heart of knot theory. We present methods for the computer construction and interactive manipulation of a wide variety of knots. Many of these constructions would be difficult using standard computeraided drawing methods. Interactive techniques allow for knot simplification under topological constraints from complicated conformations to simpler embeddings. These methods have proven useful in the investigation of the knot equivalence problem. As a further test of its utility, topological drawing has been used for several knot theoretical applications. The first of these involves finding the stick-number of a knot (the fewest number of straight sticks needed to form the knot). A second application is to the relaxation of knots under a physically-based knot energy (the symmetric energy) that we find effectively simplifies knots to configurations approaching their "canonical form". Finally, our methods have proven useful in the visualization of a class of knots that arise in a study of three-manifold topology. These knots often have complex descriptions (for example, as a huge braid word), but may be simplified greatly through the use of interactive topological drawing. Here, an expert user relies on the visualization in order to steer the computation in a direction that will often significantly improve performance.

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