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52
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 58 (8 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.
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.
Isomorphism free lexicographic enumeration of triangulated surfaces and 3manifolds
, 2006
"... We present a fast enumeration algorithm for combinatorial 2 and 3manifolds. In particular, we enumerate all triangulated surfaces with 11 and 12 vertices and all triangulated 3manifolds with 11 vertices. We further determine all equivelar maps on the nonorientable surface of genus 4 as well as a ..."
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Cited by 8 (6 self)
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We present a fast enumeration algorithm for combinatorial 2 and 3manifolds. In particular, we enumerate all triangulated surfaces with 11 and 12 vertices and all triangulated 3manifolds with 11 vertices. We further determine all equivelar maps on the nonorientable surface of genus 4 as well as all equivelar triangulations of the orientable surface of genus 3 and the nonorientable surfaces of genus 5 and 6. 1
L.: Holonomy and Skyrme’s model
 Comm. Math. Phys
"... In this paper we consider two generalizations of the Skyrme model. One is a variational problem for maps from a compact 3manifold to a compact Lie group. The other is a variational problem for flat connections. We describe the path components of the configuration spaces of smooth fields for each of ..."
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In this paper we consider two generalizations of the Skyrme model. One is a variational problem for maps from a compact 3manifold to a compact Lie group. The other is a variational problem for flat connections. We describe the path components of the configuration spaces of smooth fields for each of the variational problems. We prove that the invariants separating the path components are welldefined for (not necessarily smooth) fields with finite Skyrme energy. We prove that for every possible value of these invariants there exists a minimizer of the Skyrme functional. Throughout the paper we emphasize the importance of holonomy in the Skyrme model. Some of the results may be useful in other contexts. In particular, we define the holonomy of a distributionally flat L 2 loc connection; the local developing maps for such connections need not be continuous. 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 5 (4 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
IDEAL TURAEV–VIRO INVARIANTS
, 2005
"... Abstract. A Turaev–Viro invariant is a state sum, i.e., a polynomial that can be read off from a special spine or a triangulation of a compact 3manifold. If the polynomial is evaluated at the solution of a certain system of polynomial equations (Biedenharn–Elliott equations) then the result is a ho ..."
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Cited by 4 (2 self)
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Abstract. A Turaev–Viro invariant is a state sum, i.e., a polynomial that can be read off from a special spine or a triangulation of a compact 3manifold. If the polynomial is evaluated at the solution of a certain system of polynomial equations (Biedenharn–Elliott equations) then the result is a homeomorphism invariant of the manifold (“numerical TuraevViro invariant”). The equation system defines an ideal, and actually the coset of the polynomial with respect to that ideal is a homeomorphism invariant as well (“ideal Turaev–Viro invariant”). It is clear that ideal Turaev–Viro invariants are at least as strong as numerical Turaev–Viro invariants, and we show that there is reason to expect that they are strictly stronger. They offer a more unified approach, since many numerical Turaev–Viro invariants can be captured in a singly ideal Turaev–Viro invariant. Using computer algebra, we obtain computational results on some examples of ideal Turaev–Viro invariants. 1.
Symmetry of Links and Classification of Lens Spaces
, 2000
"... Abstract. We give a concise proof of a classification of lens spaces up to orientationpreserving homeomorphisms. The chief ingredient in our proof is a study of the Alexander polynomial of ‘symmetric ’ links in S 3. Let T1 and T2 be solid tori, and let mi and li be the meridian and longitude of Ti ..."
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Abstract. We give a concise proof of a classification of lens spaces up to orientationpreserving homeomorphisms. The chief ingredient in our proof is a study of the Alexander polynomial of ‘symmetric ’ links in S 3. Let T1 and T2 be solid tori, and let mi and li be the meridian and longitude of Ti (i = 1, 2). The lens space L(p, q) is a 3manifold that is obtained from T1 and T2 by identifying their boundaries in such a way that m2 = pl1 +qm1 and l2 = ql1 +rm1, where (p, q) = 1 and qq − pr = 1. In 1935, Reidemeister classified lens spaces up to orientationpreserving PL homeomorphisms [9]. This classification was generalized to the topological category with the proof of the Hauptvermutung by Moise in 1952 [7]. Meanwhile, Fox had outlined an aproach to classification up to homeomorphisms which would not require the Hauptvermutung; see [4, Problem 2], [5]. This was implemented later by Brody [3]. We refer the reader to [6] for history of classifications of lens spaces. In this paper, we give a concise proof of a classification of lens spaces up to orientationpreserving homeomorphisms. Our method is motivated by that of FoxBrody. While the chief ingredient in their proof was a study of the Alexander polynomial of knots in lens spaces, we study the Alexander polynomial of ‘symmetric’ links in S 3. For an oriented 3manifold M with finite first homology group, the linking form lkM: H1(M; Z) × H1(M; Z) − → Q/Z is defined as follows [1], [2]. Let x and y be 1cycles in M that represent elements [x] and [y] of H1(M; Z) respectively. Suppose that nx bounds a 2chain c for some n ∈ Z. Then c · y lkM([x], [y]) = ∈ Q/Z, n where c · y is the intersection number of c and y. Let ∆K(t) be the Conwaynormalized Alexander polynomial of K, i.e., ∆K(t) = ∇K(t−1/2 − t1/2), where ∇K(z) is the Conway polynomial. Theorem 1. Let ρ: S 3 − → L(p, q) be the pfold cyclic cover and K a knot in L(p, q) that represents a generator of H1(L(p, q); Z). If ∆ ρ −1 (K)(t) = 1, then lk L(p,q)([K], [K]) = q/p or = q/p in Q/Z.
Manifold aspects of the Novikov conjecture
 In Surveys on surgery theory
, 2000
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POINTWISE PARTIAL HYPERBOLICITY IN 3DIMENSIONAL NILMANIFOLDS
"... Abstract. We show the existence of a family of manifolds on which all (pointwise or absolutely) partially hyperbolic systems are dynamically coherent. This family is the set of 3manifolds with nilpotent, nonabelian fundamental group. We further classify the partially hyperbolic systems on these ma ..."
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Abstract. We show the existence of a family of manifolds on which all (pointwise or absolutely) partially hyperbolic systems are dynamically coherent. This family is the set of 3manifolds with nilpotent, nonabelian fundamental group. We further classify the partially hyperbolic systems on these manifolds up to leaf conjugacy. We also classify those systems on the 3torus which do not have an attracting or repelling periodic 2torus. These classification results allow us to prove some dynamical consequences, including existence and uniqueness results for measures of maximal entropy and quasiattractors.