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18
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 ..."
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Cited by 55 (6 self)
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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
, 2003
"... The Poincaré Conjecture was posed ninetynine years ago and may possibly have been proved in the last few months. This note will be an account of some of the major results over the past hundred years which have paved the way towards a proof and towards the even more ambitious project of classifying a ..."
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Cited by 27 (0 self)
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The Poincaré Conjecture was posed ninetynine years ago and may possibly have been proved in the last few months. This note will be an account of some of the major results over the past hundred years which have paved the way towards a proof and towards the even more ambitious project of classifying all compact 3dimensional manifolds. The final paragraph provides a brief description of the latest developments, due to Grigory Perelman. A more serious discussion of Perelman’s work will be provided in a subsequent note by Michael Anderson.
3Manifold invariants and periodicity of homology spheres. Algebraic and Geometric Topology
 Comment. Math. Helv
, 1983
"... Abstract. 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. 1. ..."
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Cited by 10 (1 self)
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Abstract. 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. 1.
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 ..."
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Cited by 7 (3 self)
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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 ..."
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Cited by 6 (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.
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 ..."
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Cited by 4 (3 self)
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(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 ..."
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Cited by 3 (0 self)
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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.
ALPS’07  Groups and Complexity
, 2007
"... The connection between groups and recursive (un)decidability has a long history, going back to the early 1900s. Also, various polynomialtime algorithms have been known in group theory for a long time. However the impact of more general computational complexity (e.g., NPcompleteness or PSpacecompl ..."
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The connection between groups and recursive (un)decidability has a long history, going back to the early 1900s. Also, various polynomialtime algorithms have been known in group theory for a long time. However the impact of more general computational complexity (e.g., NPcompleteness or PSpacecompleteness) has been relatively small and recent. These lectures review a sampling of older facts about algorithmic problems in group theory, and then present more recent results about the connection with complexity: isoperimetric functions and NP; Thompson groups, boolean circuits, and coNP; Thompson monoids and circuit complexity; Thompson groups, reversible computing, and #P; distortion of Thompson groups within Thompson monoids, and oneway permutations. We are especially interested in deep connections between computational complexity and group theory. By “connection ” we do not just mean analyzing the computational complexity of algorithms about groups. We are more interested in algebraic characterizations of complexity classes in terms of group theory, i.e., in finding a “mirror image” of all of complexity theory within group theory. Conversely, we are interested in the computational nature of concepts that appear at first purely algebraic.