Covering link calculus and the bipolar filtration of topologically slice links

Choon, Jae; Powell, Mark

doi:10.2140/gt.2014.18.1539

Cited by 9 publications

(6 citation statements)

References 25 publications

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“…Items (5) and (6) are equivalent by the definition of κ − and items (6), (7) and (8) are equivalent by Theorem B. Item (5) implies item (9) as discussed previously, by 'blowing up' at the kinks of a kinky disk.…”

mentioning

confidence: 74%

“…This is currently known for knots at n ≤ 1 [9,10]. For links of two or more components, this is known for all n by work of Cha-Powell [6].…”

Section: Introductionmentioning

confidence: 99%

“…by changing only positive crossings. Therefore, item (2) implies item (6). However, it is known that knots with positive projections necessarily have (strictly) negative signatures [8,34,45] while knots (such as the figure eight knot) with zero signature may bound kinky disks with only positive kinks.…”

mentioning

confidence: 99%

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Casson towers and filtrations of the smooth knot concordance group

Ray

2015

Algebr. Geom. Topol.

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The n-solvable filtration $\{\mathcal{F}_n\}_{n=0}^\infty$ of the smooth knot concordance group (denoted by $\mathcal{C}$), due to Cochran-Orr-Teichner, has been instrumental in the study of knot concordance in recent years. Part of its significance is due to the fact that certain geometric characterizations of a knot imply membership in various levels of the filtration. We show the counterpart of this fact for two new filtrations of $\mathcal{C}$ due to Cochran-Harvey-Horn, the positive and negative filtrations, denoted by $\{\mathcal{P}_n\}_{n=0}^\infty$ and $\{\mathcal{N}_n\}_{n=0}^\infty$ respectively. In particular, we show that if a knot K bounds a Casson tower of height n+2 in the 4-ball with only positive (resp. negative) kinks in the base-level kinky disk, then K is in $\mathcal{P}_n$ (resp. $\mathcal{N}_n$). En route to this result we show that if a knot K bounds a Casson tower of height n+2 in the 4-ball, it bounds an embedded (symmetric) grope of height n+2, and is therefore, n-solvable (this also implies that topologically slice knots bound arbitrarily tall gropes in the 4-ball). We also define a variant of Casson towers and show that if K bounds a tower of type (2,n) in the 4-ball, it is n-solvable. If K bounds such a tower with only positive (resp. negative) kinks in the base-level kinky disk then K is in $\mathcal{P}_n$ (resp. $\mathcal{N}_n$). Our results show that either every knot which bounds a Casson tower of height three is topologically slice or there exists a knot which is not topologically slice but lies in each $\mathcal{F}_n$. We also give a 3-dimensional characterization, up to concordance, of knots which bound kinky disks in the 4-ball with only positive (resp. negative) kinks; such knots form a subset of $\mathcal{P}_0$ (resp. $\mathcal{N}_0$).Comment: 30 pages, 21 figures; version 2 has 34 pages and 22 figures, more detailed discussion and better exposition at several places due to comments from an anonymous referee, added the word `smooth' in the title, to appear in Algebraic & Geometric Topolog

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mentioning

confidence: 74%

“…This is currently known for knots at n ≤ 1 [9,10]. For links of two or more components, this is known for all n by work of Cha-Powell [6].…”

Section: Introductionmentioning

confidence: 99%

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Casson towers and filtrations of the smooth knot concordance group

Ray

2015

Algebr. Geom. Topol.

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“…The notion of rationally 0‐bipolar knots is defined in [, Definition 2.3] as a rational version of 0‐bipolar knots introduced in . For the reader's convenience, we recall the definition here.…”

Section: ν+‐Invariants Of Satellites Do Not Detect Slice Knotsmentioning

confidence: 99%

“…) (B2)A pattern

Q \subset S^{1} \times D^{2}

is called a slice pattern if

Q (U)

is slice. If

K

is a rationally 0‐bipolar knot, then

Q (K)

is rationally 0‐bipolar for any slice pattern

Q

(compare [, Theorem 2.6(6)]). (B3)

V_{0} false(K false) = V_{0} false(- K false) = 0

K

is rationally 0‐bipolar [, Theorem 2.7].…”

Section: ν+‐Invariants Of Satellites Do Not Detect Slice Knotsmentioning

confidence: 99%

On rational sliceness of Miyazaki's fibered, −amphicheiral knots

Kim

2018

Bull. London Math. Soc.

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We prove that fibered, −amphicheiral knots with irreducible Alexander polynomials are rationally slice. This contrasts with the result of Miyazaki that (2n, 1)-cables of these knots are not ribbon. We also show that the concordance invariants ν + and Υ from Heegaard Floer homology vanish for a class of knots that includes rationally slice knots. In particular, the ν +and Υ-invariants vanish for these cable knots.

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The bipolar filtration of topologically slice knots

Choon

Kim

2021

Advances in Mathematics

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Covering link calculus and the bipolar filtration of topologically slice links

Cited by 9 publications

References 25 publications

Casson towers and filtrations of the smooth knot concordance group

Casson towers and filtrations of the smooth knot concordance group

On rational sliceness of Miyazaki's fibered, −amphicheiral knots

The bipolar filtration of topologically slice knots

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