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39
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.
Eta invariants as sliceness obstructions and their relation to CassonGordon invariants
, 2004
"... ..."
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 18 (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.
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 13 (2 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
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 9 (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
A survey of classical knot concordance
 In Handbook of knot theory
, 2005
"... In 1926 Artin [3] described the construction of knotted 2–spheres in R 4. The intersection of each of these knots with the standard R 3 ⊂ R 4 is a nontrivial knot in R 3. Thus a natural problem is to identify which knots can occur as such slices of knotted 2–spheres. Initially it seemed possible tha ..."
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Cited by 7 (0 self)
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In 1926 Artin [3] described the construction of knotted 2–spheres in R 4. The intersection of each of these knots with the standard R 3 ⊂ R 4 is a nontrivial knot in R 3. Thus a natural problem is to identify which knots can occur as such slices of knotted 2–spheres. Initially it seemed possible that every knot is such a slice knot and it wasn’t until the early 1960s that Murasugi [84] and Fox and Milnor [24, 25] succeeded at proving that some knots are not slice. Slice knots can be used to define an equivalence relation on the set of knots in S 3: knots K and J are equivalent if K # − J is slice. With this equivalence the set of knots becomes a group, the concordance group of knots. Much progress has been made in studying slice knots and the concordance group, yet some of the most easily asked questions remain untouched. There are two related theories of concordance, one in the smooth category and the other topological. Our focus will be on the smooth setting, though the distinctions and main results in the topological setting will be included. Related topics must be excluded, in particular the study of link concordance. Our focus lies entirely in the classical setting; higher dimensional concordance theory is only mentioned when needed to understand the classical setting. 1.
Splitting the concordance group of algebraically slice knots
, 2008
"... As a corollary of work of Ozsváth and Szabó, it is shown that the classical concordance group of algebraically slice knots has an infinite cyclic summand and in particular is not a divisible group. ..."
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Cited by 7 (4 self)
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As a corollary of work of Ozsváth and Szabó, it is shown that the classical concordance group of algebraically slice knots has an infinite cyclic summand and in particular is not a divisible group.
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 7 (7 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 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.
The Lehmer Polynomial and Pretzel Knots
 Bulletin of Canadian Math. Soc
, 1998
"... The Lehmer polynomial, which is the monic, symmetric polynomial with smallest known Mahler measure, is also the Alexander polynomial for a (7; 3; 2)pretzel knot. In this paper, we nd a formula for the Alexander polynomial for an innite family of pretzel knots which includes the (7; 3; 2)pretzel kn ..."
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Cited by 6 (5 self)
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The Lehmer polynomial, which is the monic, symmetric polynomial with smallest known Mahler measure, is also the Alexander polynomial for a (7; 3; 2)pretzel knot. In this paper, we nd a formula for the Alexander polynomial for an innite family of pretzel knots which includes the (7; 3; 2)pretzel knot. By generalizing this family, we obtain polynomials p1 ;:::;p k (x) among which the Lehmer polynomial has smallest Mahler measure. The polynomials p1 ;:::;p k (x) turn out to be the same as the denominators of growth functions of Coxeter reection groups. 1 Introduction A symmetric polynomial (also called a palindromic or reciprocal polynomial) is a polynomial p(x) of the form p(x) = a 0 + a 1 x + + a k x k + a k 1 x k+1 + + a 0 x 2k : Given a symmetric polynomial p(x), the product of the norms of the roots of p(x) on or outside the unit circle is called the Mahler measure of p(x). A long standing open question is whether the Mahler measure of a monic, irreduc...