We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, ie., capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, UNKNOTTING PROBLEM is in NP. We also consider the problem, SPLITTING PROBLEM of determining whether two or more such polygons… (More)
Here we present the results of the NSF-funded Workshop on Computational Topol-ogy, which met on June 11 and 12 in Miami Beach, Florida. This report identifies important problems involving both computation and topology.
By applying displacement maps to slightly perturb two free–form surfaces, one can ensure exact agreement between the images in 3 of parameter– domain approximations to their curve of intersection. Thus, at the expense of slightly altering the surfaces in the vicinity of their intersection, a perfect matching of the surface trimming curves is guaranteed.… (More)
We describe a method that serves to simultaneously determine the topological configuration of the intersection curve of two parametric surfaces and generate compatible decomposi-tions of their parameter domains, that are amenable to the application of existing perturbation schemes ensuring exact topological consistency of the trimmed surface… (More)
We show that the problem of deciding whether a polygonal knot in a closed three-dimensional manifold bounds a surface of genus at most <i>g</i> is <b>NP</b>-complete.
We present a sequence of diagrams of the unknot for which the minimum number of Reidemeister moves required to pass to the trivial diagram is quadratic with respect to the number of crossings. These bounds apply both in S 2 and in R 2 .
For each integer n ≥ 0, there is a closed, unknotted, polygonal curve Kn in R 3 having less than 10n + 9 edges, with the property that any Piecewise-Linear triangulated disk spanning the curve contains at least 2 n−1 triangles.
We present a method for establishing correspondences between human cortical surfaces that exactly matches the positions of given point landmarks, while attaining the global minimum of an objective function that quantifies how far the mapping deviates from conformality. On each surface, a conformal transformation is applied to the Euclidean distance metric,… (More)
We study random knots and links in R 3 using the Petaluma model, which is based on the petal projections developed in . In this model we obtain a formula for the distribution of the linking number of a random two-component link. We also obtain formulas for the expectations and the higher moments of the Casson invariant and the order-3 knot invariant v3.… (More)
Topologically consistent algorithms for the intersection and trimming of free-form parametric surfaces are of fundamental importance in computer-aided design, analysis, and manufacturing. Since the intersection of (for example) two bicubic tensor-product surface patches is not a rational curve, it is usually described by approximations in the parameter… (More)