Results 1  10
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25
Eta invariants as sliceness obstructions and their relation to CassonGordon invariants
, 2004
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Order in the concordance group and Heegaard Floer homology
 GEOM TOPOL
, 2006
"... We use the HeegaardFloer homology correction terms defined by Ozsváth–Szabó to formulate a new obstruction for a knot to be of finite order in the smooth concordance group. This obstruction bears a formal resemblance to that of Casson and Gordon but is sensitive to the difference between the ..."
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Cited by 14 (3 self)
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We use the HeegaardFloer homology correction terms defined by Ozsváth–Szabó to formulate a new obstruction for a knot to be of finite order in the smooth concordance group. This obstruction bears a formal resemblance to that of Casson and Gordon but is sensitive to the difference between the
The sliceribbon conjecture for 3stranded pretzel knots
, 2007
"... Abstract. We determine the (smooth) concordance order of the 3stranded pretzel knots P(p, q, r) with p, q, r odd. We show that each one of finite order is, in fact, ribbon, thereby proving the sliceribbon conjecture for this family of knots. As corollaries we give new proofs of results first obtain ..."
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Cited by 9 (2 self)
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Abstract. We determine the (smooth) concordance order of the 3stranded pretzel knots P(p, q, r) with p, q, r odd. We show that each one of finite order is, in fact, ribbon, thereby proving the sliceribbon conjecture for this family of knots. As corollaries we give new proofs of results first obtained by FintushelStern and CassonGordon.
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.
The concordance genus of a knot
"... Abstract. In knot concordance three genera arise naturally, g(K), g4(K), and gc(K): these are the classical genus, the 4–ball genus, and the concordance genus, defined to be the minimum genus among all knots concordant to K. Clearly 0 ≤ g4(K) ≤ gc(K) ≤ g(K). Casson and Nakanishi gave examples to s ..."
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Cited by 7 (3 self)
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Abstract. In knot concordance three genera arise naturally, g(K), g4(K), and gc(K): these are the classical genus, the 4–ball genus, and the concordance genus, defined to be the minimum genus among all knots concordant to K. Clearly 0 ≤ g4(K) ≤ gc(K) ≤ g(K). Casson and Nakanishi gave examples to show that g4(K) need not equal gc(K). We begin by reviewing and extending their results. For knots representing elements in A, the concordance group of algebraically slice knots, the relationships between these genera are less clear. Casson and Gordon’s result that A is nontrivial implies that g4(K) can be nonzero for knots in A. Gilmer proved that g4(K) can be arbitrarily large for knots in A. We will prove that there are knots K in A with g4(K) = 1 and gc(K) arbitrarily large. Finally, we tabulate gc for all knots with 10 or fewer crossings. This requires the development of further obstructions and the description of previously unnoticed concordances. 1.
L 2 eta invariants and their approximation by unitary eta invariants
, 2003
"... Abstract. Cochran, Orr and Teichner introduced L 2 –eta–invariants to detect highly non–trivial examples of non slice knots. Using a recent theorem by Lück and Schick we show that their metabelian L 2 –eta–invariants can be viewed as the limit of finite dimensional unitary representations. We recall ..."
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Cited by 7 (1 self)
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Abstract. Cochran, Orr and Teichner introduced L 2 –eta–invariants to detect highly non–trivial examples of non slice knots. Using a recent theorem by Lück and Schick we show that their metabelian L 2 –eta–invariants can be viewed as the limit of finite dimensional unitary representations. We recall a ribbon obstruction theorem proved by the author using finite dimensional unitary eta–invariants. We show that if for a knot K this ribbon obstruction vanishes then the metabelian L 2 –eta–invariant vanishes too. The converse has been shown by the author not to be true. 1.
Knot concordance, Whitney towers . . .
, 2003
"... We construct many examples of nonslice knots in 3space that cannot be distinguished from slice knots by previously known invariants. Using Whitney towers in place of embedded disks, we define a geometric filtration of the 3dimensional topological knot concordance group. The bottom part of the filt ..."
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Cited by 6 (2 self)
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We construct many examples of nonslice knots in 3space that cannot be distinguished from slice knots by previously known invariants. Using Whitney towers in place of embedded disks, we define a geometric filtration of the 3dimensional topological knot concordance group. The bottom part of the filtration exhibits all classical concordance invariants, including the CassonGordon invariants. As a first step, we construct an infinite sequence of new obstructions that vanish on slice knots. These take values in the Ltheory of skew fields associated to certain universal groups. Finally, we use the dimension theory of von Neumann algebras to define an L²signature and use this to detect the first unknown step in our obstruction theory.
EXAMPLES IN CONCORDANCE
, 2001
"... Abstract. In this paper we present a series of examples of new phenomena in the classical knot concordance group. First we show that for (almost) every Seifert form there is an infinite family of knots, distinct in concordance, having that form. Next we demonstrate that a number of results that are ..."
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Cited by 5 (1 self)
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Abstract. In this paper we present a series of examples of new phenomena in the classical knot concordance group. First we show that for (almost) every Seifert form there is an infinite family of knots, distinct in concordance, having that form. Next we demonstrate that a number of results that are known to hold in higher dimensional concordance fail in the classical case. These include: (1) examples of knots with Seifert forms that split as direct sums of Seifert forms but the knots are not concordant to corresponding connected sums, and (2) knots with Alexander polynomials that factor as products of Alexander polynomials (with resultant 1) but the knots are not concordant to corresponding connected sums. We also provide examples showing that: (3) for almost every metabolic Seifert form M and for every Seifert form V, there are knots with Seifert form V ⊕ M which are not concordant to knots with Seifert form V, and (4) there are pairs of irreducible algebraically concordant Seifert forms V and W such that there are knots with Seifert form V that are not concordant to any knot with Seifert form W.
Primary decomposition and the fractal nature of knot concordance
, 2009
"... Abstract. For each sequence P = (p1(t), p2(t),...) of polynomials we define a characteristic series of groups, called the derived series localized at P. Given a knot K in S 3, such a sequence of polynomials arises naturally as the orders of certain submodules of a sequence of higherorder Alexander ..."
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Cited by 2 (1 self)
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Abstract. For each sequence P = (p1(t), p2(t),...) of polynomials we define a characteristic series of groups, called the derived series localized at P. Given a knot K in S 3, such a sequence of polynomials arises naturally as the orders of certain submodules of a sequence of higherorder Alexander modules of K. These group series yield filtrations of the knot concordance group that refine the (n)solvable filtration. We show that the quotients of successive terms of these refined filtrations have infinite rank. These results also suggest higherorder analogues of the p(t)primary decomposition of the algebraic concordance group. We use these techniques to give evidence that the set of smooth concordance classes of knots is a fractal set. We also show that no CochranOrrTeichner knot is concordant to any CochranHarveyLeidy knot. 1.
The algebraic concordance order of a knot
, 2008
"... Since the inception of knot concordance, questions related to torsion in the concordance group, C1, have been of particular interest; see for instance [6, 8, 14]. The only known torsion in C1 is 2–torsion, arising from amphicheiral knots, whereas Levine’s analysis of higher dimensional concordance r ..."
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Cited by 2 (1 self)
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Since the inception of knot concordance, questions related to torsion in the concordance group, C1, have been of particular interest; see for instance [6, 8, 14]. The only known torsion in C1 is 2–torsion, arising from amphicheiral knots, whereas Levine’s analysis of higher dimensional concordance revealed far more 2–torsion and also 4–torsion in C2n−1, n> 1. Casson and Gordon [1, 2] demonstrated that Levine’s algebraic classification of concordance does not apply to C1; since then, the basic questions relating to torsion in C1 have remained open. However, many of the deep theoretical tools of 4–dimensional topology, for instance [1, 4, 23], have been applied to this problem, ruling out potential classes of order two and four.