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The computational complexity of knot genus and spanning area
 electronic), arXiv: math.GT/0205057. MR MR2219001
"... Abstract. We show that the problem of deciding whether a polygonal knot in a closed threedimensional manifold bounds a surface of genus at most g is NPcomplete. 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 NPha ..."
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Abstract. We show that the problem of deciding whether a polygonal knot in a closed threedimensional manifold bounds a surface of genus at most g is NPcomplete. 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 NPhard. 1.
Area inequalities for embedded disks spanning unknotted curves
 2003, arXiv:math.DG/0306313. EFFICIENTLY BOUND 4MANIFOLDS 43
"... We show that a smooth unknotted curve in R 3 satisfies an isoperimetric inequality that bounds the area of an embedded disk spanning the curve in terms of two parameters: the length L of the curve and the thickness r (maximal radius of an embedded tubular neighborhood) of the curve. For fixed length ..."
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We show that a smooth unknotted curve in R 3 satisfies an isoperimetric inequality that bounds the area of an embedded disk spanning the curve in terms of two parameters: the length L of the curve and the thickness r (maximal radius of an embedded tubular neighborhood) of the curve. For fixed length, the expression giving the upper bound on the area grows exponentially in 1/r 2. In the direction of lower bounds, we give a sequence of length one curves with r→0for which the area of any spanning disk is bounded from below by a function that grows exponentially with 1/r. In particular, given any constant A, there is a smooth, unknotted length one curve for which the area of a smallest embedded spanning disk is greater than A. 1
Preserving computational topology by subdivision of quadratic & cubic Bézier curves
 Computing
"... Nonselfintersection is both a topological and a geometric property. It is known that nonselfintersecting regular Bézier curves have nonselfintersecting control polygons, after sufficiently many uniform subdivisions. Here a sufficient condition is given within R 3 for a nonselfintersecting, r ..."
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Nonselfintersection is both a topological and a geometric property. It is known that nonselfintersecting regular Bézier curves have nonselfintersecting control polygons, after sufficiently many uniform subdivisions. Here a sufficient condition is given within R 3 for a nonselfintersecting, regular C 2 cubic Bézier curve to be ambient isotopic to its control polygon formed after sufficiently many subdivisions. The benefit of using the control polygon as an approximant for scientific visualization is presented in this paper.
Emerging Challenges in Computational Topology
 Results of the NFS Workshop on Computational Topology
, 1999
"... Here we present the results of the NSFfunded Workshop on Computational Topology, which met on June 11 and 12 in Miami Beach, Florida. This report identifies important problems involving both computation and topology. Author affiliations: Marshall Bern, Xerox Palo Alto Research Ctr., bern@parc. ..."
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Here we present the results of the NSFfunded Workshop on Computational Topology, which met on June 11 and 12 in Miami Beach, Florida. This report identifies important problems involving both computation and topology. Author affiliations: Marshall Bern, Xerox Palo Alto Research Ctr., bern@parc.xerox.com. David Eppstein, Univ. of California, Irvine, Dept. of Information & Computer Science, eppstein@ics.uci.edu. Pankaj K. Agarwal, Duke Univ., Dept. of Computer Science, pankaj@cs.duke.edu. Nina Amenta, Univ. of Texas, Austin, Dept. of Computer Sciences, amenta@cs.utexas.edu. Paul Chew, Cornell Univ., Dept. of Computer Science, chew@cs.cornell.edu. Tamal Dey, Ohio State Univ., Dept. of Computer and Information Science, tamaldey@cis.ohiostate.edu. David P. Dobkin, Princeton Univ., Dept. of Computer Science, dpd@cs.princeton.edu. Herbert Edelsbrunner, Duke Univ., Dept. of Computer Science, edels@cs.duke.edu. Cindy Grimm, Brown Univ., Dept. of Computer Science, cmg@cs.brown.edu. Leonid...
Computational topology for geometric design and molecular design
 in Mathematics for Industry: Challenges and Frontiers
, 2005
"... The nascent field of computational topology holds great promise for resolving several longstanding industrial design modeling challenges. Geometric modeling has become commonplace in industry as manifested by the critical use of Computer Aided Geometric Design (CAGD) systems within the automotive, ..."
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The nascent field of computational topology holds great promise for resolving several longstanding industrial design modeling challenges. Geometric modeling has become commonplace in industry as manifested by the critical use of Computer Aided Geometric Design (CAGD) systems within the automotive, aerospace, shipbuilding and consumer product industries. Commercial CAGD packages depend upon complementary geometric and topological algorithms. The emergence of geometric modeling for molecular simulation and pharmaceutical design presents new challenges for supportive topological software within Computer Aided Molecular Design (CAMD) systems. For both CAGD and CAMD systems, splines provide relatively mature geometric technology. However, there remain pernicious issues regarding the ‘topology ’ of these models, particularly for support of robust simulations which rely upon the topological characteristics of adjacency, connectivity and nonselfintersection. This paper presents current challenges and frontiers for reliable simulation and approximation of topology for geometric models. The simultaneous consideration of CAGD and CAMD is important to provide unifying abstractions to benefit both domains. In engineering applications it is a common requirement that topological equivalence be preserved during geometric modifications, but in molecular simulations attention is focused upon where topological changes have occurred as indications of important chemical changes. The methods presented here are supportive of both these disciplinary approaches.
The minimal number of triangles needed to span a polygon embedded in R d
 J. GoodmanR. Pollack Festscrift Volume
, 2003
"... 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 triangulated (locally flat) PL surfac ..."
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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 triangulated (locally flat) PL surface having P as its boundary which is immersed in R d and whose interior is disjoint from P. The most interesting case is dimension 3, where the polygon may be knotted. We use the Seifert surface construction to show that for any polygon embedded in R 3 there exists an embedded orientable triangulated PL surface having at most 7n 2 triangles, whose boundary is a subdivision of P. We complement this with a construction of families of polygons with n vertices for which any such embedded surface requires at least 1 2 n2 − O(n) triangles. We also exhibit families of polygons in R 3 for which Ω(n 2) triangles are required in any immersed PL surface of the above kind. In contrast, in dimension 2 and in dimensions d ≥ 5 there always exists an embedded locally flat PL disk having P as boundary that contains at most n triangles. In dimension 4 there always exists an immersed locally flat PL disk of the above kind that contains at most 3n triangles. An unresolved case is that of embedded PL surfaces in dimension 4, where we establish only an O(n 2) upper bound. These results can be viewed as providing qualitative discrete analogues of the isoperimetric inequality for piecewise linear (PL) manifolds. In dimension 3 they imply that the (asymptotic) discrete isoperimetric constant lies between 1/2 and 7. Keywords: isoperimetric inequality, Plateau’s problem, computational complexity
D.: Computing Dehn twists and geometric intersection numbers in polynomial time
, 2005
"... Simple curves on surfaces are often represented as sequences of intersections with a triangulation. However, topologists have much more succinct ways of representing simple curves such as normal coordinates which are exponentially more succinct than intersection sequences. Nevertheless, we show that ..."
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Simple curves on surfaces are often represented as sequences of intersections with a triangulation. However, topologists have much more succinct ways of representing simple curves such as normal coordinates which are exponentially more succinct than intersection sequences. Nevertheless, we show that the following two basic tasks of computational topology, namely performing a Dehntwist of a curve along another curve, and computing the geometric intersection number of two curves, can be solved in polynomial time even in the succinct normal coordinate representation. These are the first algorithms that solve these problems in time polynomial in the succinct representations. As an application we show that a generalized notion of crossing number can be decided in NP, even though the drawings can have exponential complexity. 1
Tracing Compressed Curves in Triangulated Surfaces ∗
, 2012
"... A simple path or cycle in a triangulated surface is normal if it intersects any triangle in a finite set of arcs, each crossing from one edge of the triangle to another. We describe an algorithm to “trace ” a normal curve in O(min{X, n 2 log X}) time, where n is the complexity of the surface triangu ..."
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A simple path or cycle in a triangulated surface is normal if it intersects any triangle in a finite set of arcs, each crossing from one edge of the triangle to another. We describe an algorithm to “trace ” a normal curve in O(min{X, n 2 log X}) time, where n is the complexity of the surface triangulation and X is the number of times the curve crosses edges of the triangulation. In particular, our algorithm runs in polynomial time even when the number of crossings is exponential in n. Our tracing algorithm computes a new cellular decomposition of the surface with complexity O(n); the traced curve appears as a simple path or cycle in the 1skeleton of the new decomposition. curves represented by normal coordinates, which record the number of intersections with each edge of the surface triangulation, and simple geodesics, represented by a starting point and direction in the local coordinate system of some triangle. Our normalcoordinate algorithms are competitive with and conceptually simpler than earlier algorithms by Schaefer, Sedgwick, and Štefankovič [COCOON 2002, CCCG 2008] and by Agol, Hass, and Thurston [Trans. AMS 2005]. Un poète doit laisser des traces de son passage, non des preuves. Seules les traces font rêver. — René Char, La Parole en Archipel (1962) A typical simple closed curve on a surface is complicated, from the point of view of someone tracing out the curve. — William P. Thurston, “On the geometry and dynamics
Algorithms for normal curves and surfaces
 of Lecture Notes in Computer Science
, 2002
"... Abstract. We derive several algorithms for curves and surfaces represented using normal coordinates. The normal coordinate representation is a very succinct representation of curves and surfaces. For embedded curves, for example, its size is logarithmically smaller than a representation by edge inte ..."
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Abstract. We derive several algorithms for curves and surfaces represented using normal coordinates. The normal coordinate representation is a very succinct representation of curves and surfaces. For embedded curves, for example, its size is logarithmically smaller than a representation by edge intersections in a triangulation. Consequently, fast algorithms for normal representations can be exponentially faster than algorithms working on the edge intersection representation. Normal representations have been essential in establishing bounds on the complexity of recognizing the unknot [Hak61, HLP99, AHT02], and string graphs [SS ˇ S02]. In this paper we present efficient algorithms for counting the number of connected components of curves and surfaces, deciding whether two curves are isotopic, and computing the algebraic intersection numbers of two curves. Our main tool are equations over monoids, also known as word equations. 1