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)
We show that 3-MANIFOLD KNOT GENUS, the problem of deciding whether a polygonal knot in a closed three-dimensional manifold bounds a surface of genus at most g, is NP-complete. We also show that the problem of deciding whether a curve in a PL manifold bounds a surface of area less than a given constant C is NP-hard.
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)
This paper presents some finiteness results for the number of boundary slopes of immersed proper π1-injective surfaces of given genus g in a compact 3-manifold with torus boundary. In the case of hyperbolic 3-manifolds we obtain uniform quadratic bounds in g, independent of the 3-manifold.
Given a closed polygon P having n edges, embedded in R d , we give upper and lower bounds for the minimal number of triangles t needed to form a triangulated PL surface embedded in R d having P as its geometric boundary. More generally we obtain such bounds for a triangu-lated (locally flat) PL surface having P as its boundary which is immersed in R d and… (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.
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)
The classical isoperimetric inequality in R 3 states that the surface of smallest area enclosing a given volume is a sphere. We show that the least area surface enclosing two equal volumes is a double bubble, a surface made of two pieces of round spheres separated by a flat disk, meeting along a single circle at an angle of 120 • .