Primary decomposition and the fractal nature of knot concordance

Cochran, Tim D.; Harvey, Shelly; Leidy, Constance

doi:10.1007/s00208-010-0604-5

Cited by 47 publications

(134 citation statements)

References 41 publications

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“…We show that winding number zero operators are often contraction operators. This gives further evidence that

scriptC

has the structure of a fractal space as conjectured in .…”

Section: Introductionsupporting

confidence: 73%

“…Obstructions to a knot or link being

(n + 2)

‐solvable are obstructions to the knot or link bounding a grope of height

n

, so the preceding proposition translates many results from the literature on the solvable filtration into statements about the distance between knots in our grope metric. For the convenience of the reader we recall the definition of

n

‐solvability for knots, originating from , and reformulated as given below in [, Definition 2.3]. Definition We say that a knot

K

(n)

‐solvable if the zero surgery manifold

M_{K}

bounds a compact oriented 4‐manifold

W

with the inclusion induced map

H_{i} false(M_{K}; double-struckZ false) \to H_{i} false(W; double-struckZ false)

an isomorphism for

i = 0, 1

, and such that

H_{2} false(W; double-struckZ false)

has a basis consisting of

2 k

embedded, connected, compact, oriented surfaces

L_{1}, \dots, L_{k}, D_{1}, \dots, D_{k}

with trivial normal bundles satisfying: (i)

π_{1} false(L_{i} false) \subset π_{1} {(W)}^{(n)}

and

π_{1} false(D_{j} false) \subset π_{1} {(W)}^{(n)}

for all

i, j = 1, \dots, k

; (ii)the geometric intersection numbers are

L_{i} \cdot L_{j} = 0 = D_{i}

…”

Section: Definition Of the Metricmentioning

confidence: 99%

“…It only remains to show that, if

m ⩾ 3

K_{n}^{m} \notin {scriptG}_{n + 3}

. First, we claim that each

(R^{m}, η_{m})

is a robust doubling operator in the sense of [, Definition 7.2]. This requires that

A (R^{m})

is cyclic with order

p false(t false) p false(t^{- 1} false)

where

p (t)

is prime, generated by

η

.…”

Section: Examples Exhibiting the Non‐discrete Behaviormentioning

confidence: 99%

“…We have these properties. Moreover, we must verify that, for each isotropic submodule

P \subset A (R^{m})

, either

P

is a Lagrangian arising from the kernel of the inclusion to a ribbon disk exterior, or else the corresponding first‐order

L^{false(2 false)}

‐ signature of

R^{m}

ρ (M_{R}, ϕ_{P})

is non‐zero [, Definition 7.1]. In our case

R^{m}

has two Lagrangian subspaces corresponding to two ribbon disks and only one other isotropic submodule, namely

P = 0

.…”

Section: Examples Exhibiting the Non‐discrete Behaviormentioning

confidence: 99%

“…In this case, the corresponding first‐order signature is denoted

ρ^{1} false(R^{m} false)

. The details to justify these statements are in [, Example 7.3]. The fact that if

m ⩾ 3

then

ρ^{1} false(R^{m} false) < 0

is shown in [, Theorem 7.2.1].…”

Section: Examples Exhibiting the Non‐discrete Behaviormentioning

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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“…We show that winding number zero operators are often contraction operators. This gives further evidence that

scriptC

has the structure of a fractal space as conjectured in .…”

Section: Introductionsupporting

confidence: 73%

“…Obstructions to a knot or link being

(n + 2)

‐solvable are obstructions to the knot or link bounding a grope of height

n

n

‐solvability for knots, originating from , and reformulated as given below in [, Definition 2.3]. Definition We say that a knot

K

(n)

‐solvable if the zero surgery manifold

M_{K}

bounds a compact oriented 4‐manifold

W

with the inclusion induced map

H_{i} false(M_{K}; double-struckZ false) \to H_{i} false(W; double-struckZ false)

an isomorphism for

i = 0, 1

, and such that

H_{2} false(W; double-struckZ false)

has a basis consisting of

2 k

embedded, connected, compact, oriented surfaces

L_{1}, \dots, L_{k}, D_{1}, \dots, D_{k}

with trivial normal bundles satisfying: (i)

π_{1} false(L_{i} false) \subset π_{1} {(W)}^{(n)}

and

π_{1} false(D_{j} false) \subset π_{1} {(W)}^{(n)}

for all

i, j = 1, \dots, k

; (ii)the geometric intersection numbers are

L_{i} \cdot L_{j} = 0 = D_{i}

…”

Section: Definition Of the Metricmentioning

confidence: 99%

“…It only remains to show that, if

m ⩾ 3

K_{n}^{m} \notin {scriptG}_{n + 3}

. First, we claim that each

(R^{m}, η_{m})

is a robust doubling operator in the sense of [, Definition 7.2]. This requires that

A (R^{m})

is cyclic with order

p false(t false) p false(t^{- 1} false)

where

p (t)

is prime, generated by

η

.…”

Section: Examples Exhibiting the Non‐discrete Behaviormentioning

confidence: 99%

“…We have these properties. Moreover, we must verify that, for each isotropic submodule

P \subset A (R^{m})

, either

P

is a Lagrangian arising from the kernel of the inclusion to a ribbon disk exterior, or else the corresponding first‐order

L^{false(2 false)}

‐ signature of

R^{m}

ρ (M_{R}, ϕ_{P})

is non‐zero [, Definition 7.1]. In our case

R^{m}

has two Lagrangian subspaces corresponding to two ribbon disks and only one other isotropic submodule, namely

P = 0

.…”

Section: Examples Exhibiting the Non‐discrete Behaviormentioning

confidence: 99%

“…In this case, the corresponding first‐order signature is denoted

ρ^{1} false(R^{m} false)

. The details to justify these statements are in [, Example 7.3]. The fact that if

m ⩾ 3

then

ρ^{1} false(R^{m} false) < 0

is shown in [, Theorem 7.2.1].…”

Section: Examples Exhibiting the Non‐discrete Behaviormentioning

confidence: 99%

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Grope metrics on the knot concordance set

Cochran

Harvey

Powell

2017

Journal of Topology

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On the Upsilon invariant and satellite knots

2018

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We study the effect of satellite operations on the Upsilon invariant of Ozsváth-Stipsicz-Szabó. We obtain results concerning when a knot and its satellites are independent; for example, we show that the setis a basis for an infinite rank summand of the group of smooth concordance classes of topologically slice knots, for D the positive clasped untwisted Whitehead double of any knot with positive τ-invariant, e.g. the right-handed trefoil. We also prove that the image of the Mazur satellite operator on the smooth knot concordance group contains an infinite rank subgroup of topologically slice knots.

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Satellites of infinite rank in the smooth concordance group

Hedden

Pinzón-Caicedo

2021

Invent. math.

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Primary decomposition and the fractal nature of knot concordance

Cited by 47 publications

References 41 publications

Grope metrics on the knot concordance set

Grope metrics on the knot concordance set

On the Upsilon invariant and satellite knots

Satellites of infinite rank in the smooth concordance group

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