Results 1  10
of
21
The computational Complexity of Knot and Link Problems
 J. ACM
, 1999
"... We consider the problem of deciding whether a polygonal knot in 3dimensional Euclidean space is unknotted, capable of being continuously deformed without selfintersection so that it lies in a plane. We show that this problem, unknotting problem is in NP. We also consider the problem, unknotting pr ..."
Abstract

Cited by 58 (8 self)
 Add to MetaCart
We consider the problem of deciding whether a polygonal knot in 3dimensional Euclidean space is unknotted, capable of being continuously deformed without selfintersection so that it lies in a plane. We show that this problem, unknotting problem is in NP. We also consider the problem, unknotting problem of determining whether two or more such polygons can be split, or continuously deformed without selfintersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in PSPACE. We also give exponential worstcase running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.
Towards the Poincaré conjecture and the classification of 3manifolds
 Notices Amer. Math. Soc
"... ..."
3Manifold invariants and periodicity of homology spheres
, 2001
"... We show how the periodicity of a homology sphere is reflected in the ReshetikhinTuraevWitten invariants of the manifold. These yield a criterion for the periodicity of a homology sphere. ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
We show how the periodicity of a homology sphere is reflected in the ReshetikhinTuraevWitten invariants of the manifold. These yield a criterion for the periodicity of a homology sphere.
Alternating Quadrisecants of Knots
, 2004
"... A knot is a simple closed curve in R³. A secant line is a straight line which intersects the knot in at least two distinct places. Trisecant, quadrisecant and quintisecant lines are straight lines which intersect a knot in at least three, four and five distinct places, respectively. It is clear that ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
A knot is a simple closed curve in R³. A secant line is a straight line which intersects the knot in at least two distinct places. Trisecant, quadrisecant and quintisecant lines are straight lines which intersect a knot in at least three, four and five distinct places, respectively. It is clear that any closed curve has secants. A little thought will reveal that nontrivial knots must have trisecants, but they do not necessarily have quintisecants. The relationship between knots and quadrisecants is not so immediately clear. In 1933, E. Pannwitz proved that nontrivial generic polygonal knots have at least one quadrisecant. In 1994, G. Kuperberg showed that all (nontrivial tame) knots have at least one quadrisecant. Quadrisecants come in three basic types. These are distinguished by comparing the orders of the four points along the knot and along the quadrisecant line. These three types are labeled simple, flipped and alternating. It turns out that alternating quadrisecants capture the knottedness of a knot. The Main Theorem shows that every nontrivial tame knot in R³ has an alternating quadrisecant. This result refines the previous work about quadrisecants and gives greater geometric insight into knots. The Main Theorem provides new proofs to two previously known theorems about the total curvature and second hull of knotted curves. Moreover, essential alternating quadrisecants may be used to dramatically improve the known lower bounds on the ropelength of thick knots.
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 8 (0 self)
 Add to MetaCart
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.
Triangulated Manifolds with Few Vertices: Geometric 3Manifolds.arXiv:math.GT/0311116
"... (without boundary) is with no doubt one of the most prominent tasks in topology ever since Poincaré’s fundamental work [88] on ≪l’analysis situs≫ appeared in 1904. There are various ways for constructing 3manifolds, some of which that are general enough to yield all 3manifolds (orientable or nonor ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
(without boundary) is with no doubt one of the most prominent tasks in topology ever since Poincaré’s fundamental work [88] on ≪l’analysis situs≫ appeared in 1904. There are various ways for constructing 3manifolds, some of which that are general enough to yield all 3manifolds (orientable or nonorientable) and some that produce only particular types or classes of examples. According to Moise [73], all 3manifolds can be triangulated. This implies that there are only countably many distinct combinatorial (and therefore at most so many different topological) types that result from gluing together tetrahedra. Another way to obtain 3manifolds is by starting with a solid 3dimensional polyhedron for which surface faces are pairwise identified (see, e.g., Seifert [98] and Weber and Seifert [118]). Both approaches are rather general and, on the first sight, do not give much control on the kind of manifold we can expect as an outcome. However, if we want to determine the topological type of some given triangulated 3manifold, then small or minimal triangulations
Homotopy invariance of higher signatures and 3manifold
, 2008
"... Abstract. For closed oriented manifolds, we establish oriented homotopy invariance of higher signatures that come from the fundamental group of a large class of orientable 3manifolds, including the “piecewise geometric ” ones in the sense of Thurston. In particular, this class, that will be careful ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Abstract. For closed oriented manifolds, we establish oriented homotopy invariance of higher signatures that come from the fundamental group of a large class of orientable 3manifolds, including the “piecewise geometric ” ones in the sense of Thurston. In particular, this class, that will be carefully described, is the class of all orientable 3manifolds if the Thurston Geometrization Conjecture is true. In fact, for this type of groups, we show that the BaumConnes Conjecture With Coefficients holds. The nonoriented case is also discussed.
CONJUGACY PROBLEM IN GROUPS OF NONORIENTED geometrizable 3manifolds
, 2005
"... We have proved in [Pr] that fundamental groups of oriented geometrizable 3manifolds have a solvable conjugacy problem. We now focus on groups of nonoriented geometrizable 3manifolds in order to conclude that all groups of geometrizable 3manifolds have a solvable conjugacy problem. ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
We have proved in [Pr] that fundamental groups of oriented geometrizable 3manifolds have a solvable conjugacy problem. We now focus on groups of nonoriented geometrizable 3manifolds in order to conclude that all groups of geometrizable 3manifolds have a solvable conjugacy problem.
METRIC STRUCTURES AND PROBABILISTIC COMPUTATION
, 806
"... Abstract. Continuous firstorder logic is used to apply modeltheoretic analysis to analytic structures (e.g. Hilbert spaces, Banach spaces, probability spaces, etc.). Classical computable model theory is used to examine the algorithmic structure of mathematical objects that can be described in clas ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. Continuous firstorder logic is used to apply modeltheoretic analysis to analytic structures (e.g. Hilbert spaces, Banach spaces, probability spaces, etc.). Classical computable model theory is used to examine the algorithmic structure of mathematical objects that can be described in classical firstorder logic. The present paper shows that probabilistic computation (sometimes called randomized computation) can play an analogous role for structures described in continuous firstorder logic. The main result of this paper is an effective completeness theorem, showing that every decidable continuous firstorder theory has a probabilistically decidable model. Later sections give examples of the application of this framework to various classes of structures, and to some problems of computational complexity theory. 1.