The non-triviality of the Grope filtrations of the knot and link concordance groups

Horn, Peter D.

doi:10.4171/cmh/210

Cited by 7 publications

(4 citation statements)

References 10 publications

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“…The following corollary generalizes [, Proposition 3.4, Corollary 3.14; , Proposition 4.7; , Theorem 3.4]. Recall that we denote the set of

m

component links that bound a grope of height

n

D^{4}

G_{n}^{m}

, and we say that a string link

J \in {scriptG}_{n}^{m}

\hat{J} \in {scriptG}_{n}^{m}

.…”

Section: The Effect Of Satellite Operations and String Link Infectionsmentioning

confidence: 81%

“…Remark There is a natural trade‐off between the height of gropes versus the genus of the first‐stage surface. One can often increase the height of the grope at the expense of increasing the genus of the first or subsequent stages; constructions can be found in, for example, , and our Proposition . For a high

q

parameter a high grope is valued more, whereas for a low

q

parameter a low genus of the first stage has more value.…”

Section: Definition Of the Metricmentioning

confidence: 99%

“…The techniques used in the following proof are very similar to those used in [, Proposition 3.4, Corollary 3.14; , Theorem 3.4]. The difference is that we keep precise track of the genera and the number of copies used in the construction.…”

Section: The Effect Of Satellite Operations and String Link Infectionsmentioning

confidence: 99%

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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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“…The following corollary generalizes [, Proposition 3.4, Corollary 3.14; , Proposition 4.7; , Theorem 3.4]. Recall that we denote the set of

m

component links that bound a grope of height

n

D^{4}

G_{n}^{m}

, and we say that a string link

J \in {scriptG}_{n}^{m}

\hat{J} \in {scriptG}_{n}^{m}

.…”

Section: The Effect Of Satellite Operations and String Link Infectionsmentioning

confidence: 81%

q

parameter a high grope is valued more, whereas for a low

q

parameter a low genus of the first stage has more value.…”

Section: Definition Of the Metricmentioning

confidence: 99%

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Grope metrics on the knot concordance set

Cochran

Harvey

Powell

2017

Journal of Topology

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“…Influenced by these works, the link slicing problem has been studied extensively using various covers of the 0-surgery manifolds of links. For example, [Har08], [CHL08], and [Hor10] used Cheeger-Gromov ρ-invariants from PTFA (poly-torsion-free-abelian) covers. In [Cha10] and [Cha09], Hirzebruch type invariants from iterated prime power fold covers are defined and used.…”

Section: Introductionmentioning

confidence: 99%

Whitney towers, gropes and Casson–Gordon style invariants of links

Kim

2015

Algebr. Geom. Topol.

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In this paper, we prove a conjecture of Friedl and Powell that their Casson-Gordon type invariant of 2-component link with linking number one is actually an obstruction to being height 3.5 Whitney tower/grope concordant to the Hopf Link. The proof employs the notion of solvable cobordism of 3-manifolds with boundary, which was introduced by Cha. We also prove that the Blanchfield form and the Alexander polynomial of links in S 3 give obstructions to height 3 Whitney tower/grope concordance. This generalizes the results of Hillman and Kawauchi.

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Symmetric Whitney tower cobordism for bordered 3-manifolds and links

Choon

2014

Trans. Amer. Math. Soc.

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We introduce the notion of a symmetric Whitney tower cobordism between bordered 3-manifolds, aiming at the study of homology cobordism and link concordance. It is motivated by the symmetric Whitney tower approach to slicing knots and links initiated by T. Cochran, K. Orr, and P. Teichner. We give amenable Cheeger-Gromov ρ-invariant obstructions to bordered 3manifolds being Whitney tower cobordant. Our obstruction is related to and generalizes several prior known results, and also gives new interesting cases. As an application, our method applied to link exteriors reveals new structures on (Whitney tower and grope) concordance between links with nonzero linking number, including the Hopf link.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 3242 JAE CHOON CHA Cheeger-Gromov ρ-invariants due to Orr and the author [CO12]. As a new application that known Whitney tower frameworks do not cover, we study concordance between links with nonzero linking number. In particular we investigate Whitney tower and grope concordance to the Hopf link.Symmetric Whitney tower cobordism of bordered 3-manifolds. First we briefly introduce how we adapt the Whitney tower approach in [COT03] for homology cobordism of bordered 3-manifolds.Recall that a 3-manifold M is bordered by a surface Σ if it is endowed with a marking homeomorphism of Σ onto ∂M . For 3-manifolds M and M bordered by the same surface, one obtains a closed 3-manifold M ∪ ∂ −M by glueing the boundary along the marking homeomorphism. A 4-manifold W is a relative cobordism. Initiated by S. Cappell and J. Shaneson [CS74], understanding homology cobordism of bordered manifolds is essential in the study of manifold embeddings, in particular knot and link concordance. This also relates homology cobordism to other key problems including topological surgery on 4-manifolds.As a surgery theoretic Whitney tower approximation to a homology cobordism, we will define the notion of a height h Whitney tower cobordism W between bordered 3-manifolds (h ∈ 1 2 Z ≥0 ). Roughly speaking, our height h Whitney tower cobordism is a relative cobordism between bordered 3-manifolds, which admits immersed framed 2-spheres satisfying the following: while the 2-spheres may not be embedded, they support a Whitney tower of height h, and form a "lagrangian" in such a way that if the 2-spheres were homotopic to embeddings, then surgery along these would give a homology cobordism. For a more precise description of a Whitney tower cobordism, see Definition 2.7.It turns out that a height h Whitney tower cobordism can be deformed to another type of a relative cobordism satisfying a twisted homology analogue of the above Whitney tower condition, which we call an h-solvable cobordism. Roughly speaking, it is a cobordism which induces an isomorphism on H 1 and admits a certain "lagrangian" and "dual" for the twisted intersection pairing associated to (quotients by) derived subgroups of the fundamental group. See Definition 2.8 and Theorem 2.9 for details. ...

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The non-triviality of the Grope filtrations of the knot and link concordance groups

Cited by 7 publications

References 10 publications

Grope metrics on the knot concordance set

Grope metrics on the knot concordance set

Whitney towers, gropes and Casson–Gordon style invariants of links

Symmetric Whitney tower cobordism for bordered 3-manifolds and links

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