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60
Interactive Topological Drawing
, 1998
"... The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in threedimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues ..."
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Cited by 21 (1 self)
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The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in threedimensional 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 threedimensional 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 selfintersections. 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
RealTime Knot Tying Simulation
"... While rope is arguably a simpler system to simulate than cloth, the realtime simulation of rope, and knot tying in particular, raise unique and difficult issues in contact detection and management. Some practical knots can only be achieved by complicated crossings of the rope, yielding multiple sim ..."
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Cited by 12 (0 self)
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While rope is arguably a simpler system to simulate than cloth, the realtime simulation of rope, and knot tying in particular, raise unique and difficult issues in contact detection and management. Some practical knots can only be achieved by complicated crossings of the rope, yielding multiple simultaneous contacts, especially when the rope is pulled tight. This paper describes a simulator allowing a user to grasp and smoothly manipulate a virtual rope and to tie arbitrary knots, including knots around other objects, in realtime. One component of the simulator precisely detects selfcollisions in the rope, as well as collisions with other objects. Another component manages collisions to prevent penetration, while making the rope slide with some friction along itself and other objects, so that knots can be pulled tight in believable manner. An additional module uses recent results from knot theory to identify which topological knots have been tied, also in realtime. This work was motivated by surgical suturing, but simulation in other domains, such as sailing and rock climbing, could benefit from it.
Estimating Jones polynomials is a complete problem for one clean qubit
, 2007
"... It is known that evaluating a certain approximation to the Jones polynomial for the plat closure of a braid is a BQPcomplete problem. That is, this problem exactly captures the power of the quantum circuit model[12, 3, 1]. The one clean qubit model is a model of quantum computation in which all but ..."
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Cited by 11 (4 self)
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It is known that evaluating a certain approximation to the Jones polynomial for the plat closure of a braid is a BQPcomplete problem. That is, this problem exactly captures the power of the quantum circuit model[12, 3, 1]. The one clean qubit model is a model of quantum computation in which all but one qubit starts in the maximally mixed state. One clean qubit computers are believed to be strictly weaker than standard quantum computers, but still capable of solving some classically intractable problems [20]. Here we show that evaluating a certain approximation to the Jones polynomial at a fifth root of unity for the trace closure of a braid is a complete problem for the one clean qubit complexity class. That is, a one clean qubit computer can approximate these Jones polynomials in time polynomial in both the number of strands and number of crossings, and the problem of simulating a one clean qubit computer is reducible to approximating the Jones polynomial of the trace closure of a braid.
The size of spanning disks for polygonal curves
 Discrete Comput. Geom
"... Abstract. 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 PiecewiseLinear triangulated disk spanning the curve contains at least 2 n−1 triangles. 1. Introduction. Let K be a closed polygonal curve in R3 consi ..."
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Cited by 9 (1 self)
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Abstract. 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 PiecewiseLinear triangulated disk spanning the curve contains at least 2 n−1 triangles. 1. Introduction. Let K be a closed polygonal curve in R3 consisting of n line segments. Assume that K is unknotted, so that it is the boundary of an embedded disk in R3. This paper considers the question: How many triangles are needed to triangulate a PiecewiseLinear (PL) spanning disk of K? The main result, Theorem 1 below,
The disjoint curve property
, 2004
"... A Heegaard splitting of a closed, orientable threemanifold satisfies the disjoint curve property if the splitting surface contains an essential simple closed curve and each handlebody contains an essential disk disjoint from this curve [Thompson, 1999]. A splitting is full if it does not have the d ..."
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Cited by 9 (0 self)
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A Heegaard splitting of a closed, orientable threemanifold satisfies the disjoint curve property if the splitting surface contains an essential simple closed curve and each handlebody contains an essential disk disjoint from this curve [Thompson, 1999]. A splitting is full if it does not have the disjoint curve property. This paper shows that in a closed, orientable threemanifold all splittings of sufficiently large genus have the disjoint curve property. From this and a solution to the generalized Waldhausen conjecture it would follow that any closed, orientable three manifold contains only finitely many full splittings.
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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Cited by 8 (0 self)
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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.
Protein similarity from knot theory and geometric convolution
 J Comput Biol
, 2004
"... interpreted as representing the official policies, either expressed or implied, of the Pennsylvania Department ..."
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Cited by 8 (0 self)
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interpreted as representing the official policies, either expressed or implied, of the Pennsylvania Department
Almost Normal Heegaard Splittings
, 2001
"... The study of threemanifolds via their Heegaard splittings was initiated by Poul Heegaard in 1898 in his thesis. Our approach to the subject is through almost normal surfaces, as introduced by Hyam Rubinstein [28] and distance, as introduced by John Hempel [12]. Among the results presented... ..."
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Cited by 8 (4 self)
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The study of threemanifolds via their Heegaard splittings was initiated by Poul Heegaard in 1898 in his thesis. Our approach to the subject is through almost normal surfaces, as introduced by Hyam Rubinstein [28] and distance, as introduced by John Hempel [12]. Among the results presented...
Computing Linking Numbers of a Filtration
 In Algorithms in Bioinformatics (LNCS 2149
, 2001
"... We develop fast algorithms for computing the linking number of a simplicial complex within a filtration. We give experimental results in applying our work toward the detection of nontrivial tangling in biomolecules, modeled as alpha complexes. ..."
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Cited by 7 (5 self)
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We develop fast algorithms for computing the linking number of a simplicial complex within a filtration. We give experimental results in applying our work toward the detection of nontrivial tangling in biomolecules, modeled as alpha complexes.