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61
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 75 (9 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.
Eta invariants as sliceness obstructions and their relation to CassonGordon invariants
, 2004
"... ..."
A survey of twisted Alexander polynomials
, 2009
"... We give a short introduction to the theory of twisted Alexander polynomials of a 3–manifold associated to a representation of its fundamental group. We summarize their formal properties and we explain their relationship to twisted Reidemeister torsion. We then give a survey of the many applications ..."
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Cited by 26 (15 self)
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We give a short introduction to the theory of twisted Alexander polynomials of a 3–manifold associated to a representation of its fundamental group. We summarize their formal properties and we explain their relationship to twisted Reidemeister torsion. We then give a survey of the many applications of twisted invariants to the study of topological problems. We conclude with a short summary of the theory of higher order Alexander polynomials.
Filtration of the classical knot concordance group and CassonGordon invariants
, 2002
"... Abstract. It is known that if any prime power branched cyclic cover of a knot in S 3 is a homology sphere, then the knot has vanishing CassonGordon invariants. We construct infinitely many examples of (topologically) nonslice knots in S 3 whose prime power branched cyclic covers are homology spher ..."
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Cited by 19 (2 self)
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Abstract. It is known that if any prime power branched cyclic cover of a knot in S 3 is a homology sphere, then the knot has vanishing CassonGordon invariants. We construct infinitely many examples of (topologically) nonslice knots in S 3 whose prime power branched cyclic covers are homology spheres. We show that these knots generate an infinite rank subgroup of F(1.0)/F(1.5) for which CassonGordon invariants vanish in CochranOrrTeichner’s filtration of the classical knot concordance group. As a corollary, it follows that CassonGordon invariants are not a complete set of obstructions to a second layer of Whitney disks. 1.
Knot group epimorphisms
 J. Knot Theory Ramifications
"... Abstract: Let G be a finitely generated group, and let λ ∈ G. If there exists a knot k such that πk = π1(S 3 \k) can be mapped onto G sending the longitude to λ, then there exists infinitely many distinct prime knots with the property. Consequently, if πk is the group of any knot (possibly composite ..."
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Cited by 19 (5 self)
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Abstract: Let G be a finitely generated group, and let λ ∈ G. If there exists a knot k such that πk = π1(S 3 \k) can be mapped onto G sending the longitude to λ, then there exists infinitely many distinct prime knots with the property. Consequently, if πk is the group of any knot (possibly composite), then there exists an infinite number of prime knots k1, k2, · · · and epimorphisms · · · → πk2 → πk1 → πk each perserving peripheral structures. Properties of a related partial order on knots are discussed. 1. Introduction. Suppose that φ: G1
The arithmetic and geometry of Salem numbers
 Bull. Amer. Math. Soc
, 1991
"... Abstract. A Salem number is a real algebraic integer, greater than 1, with the property that all of its conjugates lie on or within the unit circle, and at least one conjugate lies on the unit circle. In this paper we survey some of the recent appearances of Salem numbers in parts of geometry and ar ..."
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Cited by 17 (3 self)
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Abstract. A Salem number is a real algebraic integer, greater than 1, with the property that all of its conjugates lie on or within the unit circle, and at least one conjugate lies on the unit circle. In this paper we survey some of the recent appearances of Salem numbers in parts of geometry and arithmetic, and discuss the possible implications for the ‘minimization problem’. This is an old question in number theory which asks whether the set of Salem numbers is bounded away from 1. Contents
The spectral sequence of an equivariant chain complex and homology with local coefficients
 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
, 2010
"... We study the spectral sequence associated to the filtration by powers of the augmentation ideal on the (twisted) equivariant chain complex of the universal cover of a connected CWcomplex X. In the process, we identify the d_1 differential in terms of the coalgebra structure of H^*(X, k), and the kπ ..."
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Cited by 17 (10 self)
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We study the spectral sequence associated to the filtration by powers of the augmentation ideal on the (twisted) equivariant chain complex of the universal cover of a connected CWcomplex X. In the process, we identify the d_1 differential in terms of the coalgebra structure of H^*(X, k), and the kπ_1(X)module structure on the twisting coefficients. In particular, this recovers in dual form a result of Reznikov, on the mod p cohomology of cyclic pcovers of aspherical complexes. This approach provides information on the homology of all Galois covers of X. It also yields computable upper bounds on the ranks of the cohomology groups of X, with coefficients in a primepower order, rank one local system. When X admits a minimal cell decomposition, we relate the linearization of the equivariant cochain complex of the universal abelian cover to the Aomoto complex, arising from the cupproduct structure of H^*(X, k), thereby generalizing a result of Cohen and Orlik.
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 16 (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.
and Akira Yasuhara. Crosscap number of a knot
 Pacific J. Math
, 1995
"... B. E. Clark defined the crosscap number of a knot to be the minimum number of the first Betti numbers of nonorientable surfaces bounding it. In this paper, we investigate the crosscap numbers of knots. We show that the crosscap number of 74 is equal to 3. This gives an affirmative answer to a ques ..."
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Cited by 15 (0 self)
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B. E. Clark defined the crosscap number of a knot to be the minimum number of the first Betti numbers of nonorientable surfaces bounding it. In this paper, we investigate the crosscap numbers of knots. We show that the crosscap number of 74 is equal to 3. This gives an affirmative answer to a question given by Clark. In general, the crosscap number is not additive under the connected sum. We give a necessary and sufficient condition for the crosscap number to be additive under the connected sum. 0. Introduction. We study knots in the 3sphere S3. The genus g{K) of a knot K is the minimum number of the genera of Seifert surfaces for it [11]. Here a Seifert surface means a connected, orientable surface with boundary K. In 1978, B. E. Clark [3] defined the crosscap number C(K) oϊK to be the minimum num
Euclidean Mahler measure and Twisted Links
, 2005
"... If the twist numbers of a collection of oriented alternating link diagrams are bounded, then the Alexander polynomials of the corresponding links have bounded euclidean Mahler measure (see Definition 1.2). The converse assertion does not hold. Similarly, if a collection of oriented link diagrams, ..."
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Cited by 10 (2 self)
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If the twist numbers of a collection of oriented alternating link diagrams are bounded, then the Alexander polynomials of the corresponding links have bounded euclidean Mahler measure (see Definition 1.2). The converse assertion does not hold. Similarly, if a collection of oriented link diagrams, not necessarily alternating, have bounded twist numbers, then both the Jones polynomials and a parametrization of the 2variable Homflypt polynomials of the corresponding links have bounded Mahler measure.