New constructions of slice links

Cochran, Tim D.; Friedl, Stefan; Teichner, Peter

doi:10.4171/cmh/175

Cited by 21 publications

(30 citation statements)

References 29 publications

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“…But in @ C C we have seen that this capped-off † is homologous (by surgering along the disks) to the embedded 2-sphere to which the 3-handle is attached. This gives property (6). We have thus shown that…”

Section: A Cobordism Between M K and M @K=@mmentioning

confidence: 60%

“…For example, rather than merely being obstructions to a knot's being a slice knot, the second-order signatures obstruct a knot's being .2:5/-solvable. Here we refer to the .n/-solvable filtration, fF .n/ g, of the knot concordance group due to Cochran-Orr-Teichner [13,Section 7,8].…”

Section: Theorem 72mentioning

confidence: 99%

“…In particular, in this paper, if we draw a band passing through a box labelled by a knot (as for example the box labeled by K 1 in Figure 1) then this means that entire band is to be tied into that knot as shown in Figure 2 for a trefoil knot. In this paper we also need to consider infection by a string link as discussed in [12;5,Section 10;6]. We need only the following special case.…”

Section: Theorem 72mentioning

confidence: 99%

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Derivatives of knots and second-order signatures

Cochran

Harvey

Leidy

2010

Algebr. Geom. Topol.

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We define a set of "second-order" L .2/ -signature invariants for any algebraically slice knot. These obstruct a knot's being a slice knot and generalize Casson-Gordon invariants, which we consider to be "first-order signatures". As one application we prove: If K is a genus one slice knot then, on any genus one Seifert surface †, there exists a homologically essential simple closed curve J of self-linking zero, which has vanishing zero-th order signature and a vanishing first-order signature. This extends theorems of Cooper and Gilmer. We introduce a geometric notion, that of a derivative of a knot with respect to a metabolizer. We also introduce a new relation, generalizing homology cobordism, called null-bordism.57M25; 57M10

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Section: A Cobordism Between M K and M @K=@mmentioning

confidence: 60%

Section: Theorem 72mentioning

confidence: 99%

Section: Theorem 72mentioning

confidence: 99%

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Derivatives of knots and second-order signatures

Cochran

Harvey

Leidy

2010

Algebr. Geom. Topol.

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“…The Z-caps with i = 1 have signatures equal to zero or ρ 0 (K 1, j ) by Corollary 6.4. Therefore |c 1 | is less than or equal to n 11 which was less than m 11 . This concludes the proof that R α is injective on the claimed subgroup, which is the first claim of the theorem.…”

Section: Fig 15 W Nmentioning

confidence: 99%

Primary decomposition and the fractal nature of knot concordance

2010

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For each sequence of polynomials, P=(p_1(t),p_2(t),...), 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 the sequence of higher-order Alexander modules of K. These group series yield new filtrations of the knot concordance group that refine the (n)-solvable filtration of Cochran-Orr-Teichner. We show that the quotients of successive terms of these refined filtrations have infinite rank. These results also suggest higher-order 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 Cochran-Orr-Teichner knot is concordant to any Cochran-Harvey-Leidy knot.Comment: 60 pages, added 4 pages to introduction, minor corrections otherwise; Math. Annalen 201

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“…Given an

m

‐component string link

L

, remove the image of

double-struckH

from

S^{3}

, and replace it by

E_{L}

, identifying the

i^{t h}

longitude of

L

with the

i^{t h}

longitude of

T

and the

i^{t h}

meridian of

L

with the

i^{t h}

meridian of

T

. The resulting 3‐manifold is again homeomorphic to

S^{3}

(see [, Definition 2.2] for more details)

f : cl false(S^{3} ∖ ψ (H) false) \cup E_{L} \overset{≅}{\to} S^{3} .

Denote the image

f (R)

R (L, ψ)

, the output of the string link operator

R (-, ψ)

acting on

L

. When it is clear, we may drop the

ψ

and write

R (L)

.…”

Section: Definitions On String Linksmentioning

confidence: 99%

Grope metrics on the knot concordance set

Cochran

Harvey

Powell

2017

Journal of Topology

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Abstract. To a special type of grope embedded in 4-space, that we call a branchsymmetric grope, we associate a length function for each real number q ≥ 1. This gives rise to a family of pseudo-metrics d q , refining the slice genus metric, on the set of concordance classes of knots, as the infimum of the length function taken over all possible grope concordances between two knots. We investigate the properties of these metrics. The main theorem is that the topology induced by this metric on the knot concordance set is not discrete for all q > 1. The analogous statement for links also holds for q = 1. In addition we translate much previous work on knot concordance into distance statements. In particular, we show that winding number zero satellite operators are contractions in many cases, and we give lower bounds on our metrics arising from knot signatures and higher order signatures. This gives further evidence in favor of the conjecture that the knot concordance group has a fractal structure.

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New constructions of slice links

Cited by 21 publications

References 29 publications

Derivatives of knots and second-order signatures

Derivatives of knots and second-order signatures

Primary decomposition and the fractal nature of knot concordance

Grope metrics on the knot concordance set

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