Results 1  10
of
74
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 ..."
Abstract

Cited by 31 (3 self)
 Add to MetaCart
(Show Context)
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
Using motion planning for knot untangling
 THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH 2004; 23; 797
, 2004
"... In this paper we investigate the application of motion planning techniques to the untangling of mathematical knots. Knot untangling can be viewed as a highdimensional planning problem in reparametrizable configuration spaces. In the past, simulated annealing and other energy minimization methods ha ..."
Abstract

Cited by 31 (6 self)
 Add to MetaCart
In this paper we investigate the application of motion planning techniques to the untangling of mathematical knots. Knot untangling can be viewed as a highdimensional planning problem in reparametrizable configuration spaces. In the past, simulated annealing and other energy minimization methods have been used to find knot untangling paths. We have developed a probabilistic planner that is capable of untangling knots described by over 400 variables. We have tested on known difficult benchmarks in this area and untangled them more quickly than has been achieved with minimization in the literature. In this work, the use of motion planning techniques is critical for the untangling. Our planner defines local goals and makes combined use of energy minimization and randomized treebased planning. We also show how to produce candidates with a minimal number of segments for a given knot. The planner developed in this work is novel in
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 ..."
Abstract

Cited by 22 (4 self)
 Add to MetaCart
(Show Context)
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 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 ..."
Abstract

Cited by 17 (0 self)
 Add to MetaCart
(Show Context)
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.
Hardness of embedding simplicial complexes in R^d
, 2009
"... Let EMBEDk→d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into d? Known results easily imply polynomiality of EMBEDk→2 (k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBEDk→2k for a ..."
Abstract

Cited by 15 (5 self)
 Add to MetaCart
(Show Context)
Let EMBEDk→d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into d? Known results easily imply polynomiality of EMBEDk→2 (k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBEDk→2k for all k ≥ 3. We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5sphere implies that EMBEDd→d and EMBED (d−1)→d are undecidable for each d ≥ 5. Our main result is NPhardness of EMBED2→4 and, more generally, of EMBEDk→d for all k, d with d ≥ 4 and d ≥ k ≥ (2d −2)/3. These dimensions fall outside the metastable range of a theorem of Haefliger and Weber, which characterizes embeddability using the deleted product obstruction. Our reductions are based on examples, due to Segal, Spie˙z, Freedman, Krushkal, Teichner, and Skopenkov, showing that outside the metastable range the deleted product obstruction is not sufficient to characterize embeddability.
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 ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
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.
The disjoint curve property
 Geom. Topol
"... 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 ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
(Show Context)
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 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 ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
(Show Context)
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,