![Page 1: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/1.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
On annulus twists
Tetsuya Abe1
1OCAMI
May 20, 2016
Tetsuya Abe On annulus twists
![Page 2: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/2.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Outline
Dehn surgery Knot concordance
Heegaard Floer homology
1 Dehn surgery and annulus twists
2 Knot concordance and annulus twists
3 A construction of ribbon disks
Tetsuya Abe On annulus twists
![Page 3: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/3.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Adam Levin’s question.
Dehn surgery Knot concordance
The following question asks a relationbetween Knot concordance and Dehn sugery.
Question (Levine, 2016)
If K is concordant to K ′, then for all n,S3
n(K ) is homology cobordant to S3n(K ′).
Is the converse true?
Tetsuya Abe On annulus twists
![Page 4: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/4.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Dehn surgery 1.
Dehn surgery on knots is a long-standing techniquefor the construction of 3-manifolds.
The following existence theorem is important.
Theorem (Lickorish, Wallace)Any closed, oriented and connected 3-manifold is realized byintegral surgery on a link in S3.
Tetsuya Abe On annulus twists
![Page 5: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/5.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Dehn surgery 1.
Dehn surgery on knots is a long-standing techniquefor the construction of 3-manifolds.
The following existence theorem is important.
Theorem (Lickorish, Wallace)Any closed, oriented and connected 3-manifold is realized byintegral surgery on a link in S3.
Tetsuya Abe On annulus twists
![Page 6: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/6.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Dehn surgery 2.
How about "uniqueness" ???
QuestionIf two framed links L and L′ gives the same 3-manifold, then
L = L′ ?
QuestionIf two framed knots K and K′ gives the same 3-manifold, then
K = K′ ?Tetsuya Abe On annulus twists
![Page 7: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/7.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Dehn surgery 2.
How about "uniqueness" ???
QuestionIf two framed links L and L′ gives the same 3-manifold, then
L = L′ ?
QuestionIf two framed knots K and K′ gives the same 3-manifold, then
K = K′ ?Tetsuya Abe On annulus twists
![Page 8: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/8.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Positive direction : When the question holds ?
QuestionIf two framed knots K and K′ gives the same 3-manifold, then
K = K′ ?Restriction 1 : Fix the knot type.Restriction 2 : Consider only an orientation preserving hom.
Yi Ni and Zhongtao Wu.Cosmetic surgeries on knots in S3,J. Reine Angew. Math., 706, (2015), 1–17.
An application of the d-invariant coming from the HeegaardFloer homology.
Tetsuya Abe On annulus twists
![Page 9: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/9.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Positive direction : When the question holds ?
QuestionIf two framed knots K and K′ gives the same 3-manifold, then
K = K′ ?Restriction 1 : Fix the knot type.Restriction 2 : Consider only an orientation preserving hom.
Yi Ni and Zhongtao Wu.Cosmetic surgeries on knots in S3,J. Reine Angew. Math., 706, (2015), 1–17.
An application of the d-invariant coming from the HeegaardFloer homology.
Tetsuya Abe On annulus twists
![Page 10: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/10.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Positive direction : When the question holds ?
QuestionIf two framed knots K and K′ gives the same 3-manifold, then
K = K′ ?Restriction 1 : Fix the knot type.Restriction 2 : Consider only an orientation preserving hom.
Yi Ni and Zhongtao Wu.Cosmetic surgeries on knots in S3,J. Reine Angew. Math., 706, (2015), 1–17.
An application of the d-invariant coming from the HeegaardFloer homology.
Tetsuya Abe On annulus twists
![Page 11: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/11.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Positive direction : When the question holds ?
QuestionIf two framed knots K and K′ gives the same 3-manifold, then
K = K′ ?Restriction 1 : Fix the knot type.Restriction 2 : Consider only an orientation preserving hom.
Yi Ni and Zhongtao Wu.Cosmetic surgeries on knots in S3,J. Reine Angew. Math., 706, (2015), 1–17.
An application of the d-invariant coming from the HeegaardFloer homology.
Tetsuya Abe On annulus twists
![Page 12: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/12.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Negative direction : Find the worst situation
QuestionIf two framed knots K and K′ gives the same 3-manifold, then
K = K′ ?Problem (Clark, Problem 3.6 in Kirby’s problem list)Let n be an integer.Find a 3-manifold obtained by n-surgery on∞-many knots.
• In 2006, Osoinach solved this problem for n = 0.
• After the work of Teragaito, Takeuchi, Omae, Kohno,it was solved by Abe-Jong-Luecke-Osoinach in 2015.
• Tool : Twisting along an annulus (=annulus twist)
Tetsuya Abe On annulus twists
![Page 13: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/13.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Negative direction : Find the worst situation
QuestionIf two framed knots K and K′ gives the same 3-manifold, then
K = K′ ?Problem (Clark, Problem 3.6 in Kirby’s problem list)Let n be an integer.Find a 3-manifold obtained by n-surgery on∞-many knots.
• In 2006, Osoinach solved this problem for n = 0.
• After the work of Teragaito, Takeuchi, Omae, Kohno,it was solved by Abe-Jong-Luecke-Osoinach in 2015.
• Tool : Twisting along an annulus (=annulus twist)
Tetsuya Abe On annulus twists
![Page 14: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/14.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Negative direction : Find the worst situation
QuestionIf two framed knots K and K′ gives the same 3-manifold, then
K = K′ ?Problem (Clark, Problem 3.6 in Kirby’s problem list)Let n be an integer.Find a 3-manifold obtained by n-surgery on∞-many knots.
• In 2006, Osoinach solved this problem for n = 0.
• After the work of Teragaito, Takeuchi, Omae, Kohno,it was solved by Abe-Jong-Luecke-Osoinach in 2015.
• Tool : Twisting along an annulus (=annulus twist)
Tetsuya Abe On annulus twists
![Page 15: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/15.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Negative direction : Find the worst situation
QuestionIf two framed knots K and K′ gives the same 3-manifold, then
K = K′ ?Problem (Clark, Problem 3.6 in Kirby’s problem list)Let n be an integer.Find a 3-manifold obtained by n-surgery on∞-many knots.
• In 2006, Osoinach solved this problem for n = 0.
• After the work of Teragaito, Takeuchi, Omae, Kohno,it was solved by Abe-Jong-Luecke-Osoinach in 2015.
• Tool : Twisting along an annulus (=annulus twist)
Tetsuya Abe On annulus twists
![Page 16: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/16.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Negative direction : Find the worst situation
QuestionIf two framed knots K and K′ gives the same 3-manifold, then
K = K′ ?Problem (Clark, Problem 3.6 in Kirby’s problem list)Let n be an integer.Find a 3-manifold obtained by n-surgery on∞-many knots.
• In 2006, Osoinach solved this problem for n = 0.
• After the work of Teragaito, Takeuchi, Omae, Kohno,it was solved by Abe-Jong-Luecke-Osoinach in 2015.
• Tool : Twisting along an annulus (=annulus twist)
Tetsuya Abe On annulus twists
![Page 17: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/17.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Osoinach’s result.
Theorem (Osoinach, 2006)
(1) The sequence {Kn} contains∞-many hyperbolic knots.(2) We have the following:
S30(K0) ≈ S3
0(K1) ≈ S30(K2) ≈ S3
0(K3) ≈ · · · .
• K0 is the connected sum of two figure eight knots.• Takioka proved that Kn are mutually distinct for n ≥ 0.
Tetsuya Abe On annulus twists
![Page 18: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/18.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Osoinach’s result.
Theorem (Osoinach, 2006)
(1) The sequence {Kn} contains∞-many hyperbolic knots.(2) We have the following:
S30(K0) ≈ S3
0(K1) ≈ S30(K2) ≈ S3
0(K3) ≈ · · · .
• K0 is the connected sum of two figure eight knots.• Takioka proved that Kn are mutually distinct for n ≥ 0.
Tetsuya Abe On annulus twists
![Page 19: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/19.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Osoinach’s result.
Theorem (Osoinach, 2006)
(1) The sequence {Kn} contains∞-many hyperbolic knots.(2) We have the following:
S30(K0) ≈ S3
0(K1) ≈ S30(K2) ≈ S3
0(K3) ≈ · · · .
• K0 is the connected sum of two figure eight knots.• Takioka proved that Kn are mutually distinct for n ≥ 0.
Tetsuya Abe On annulus twists
![Page 20: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/20.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Basic lemmas
(1)
(2)
(3)
Tetsuya Abe On annulus twists
![Page 21: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/21.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Proof of S30(Kn) ≈ S3
0(K0).
Tetsuya Abe On annulus twists
![Page 22: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/22.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Proof of S30(Kn) ≈ S3
0(K0).
Tetsuya Abe On annulus twists
![Page 23: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/23.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Proof of S30(Kn) ≈ S3
0(K0).
Tetsuya Abe On annulus twists
![Page 24: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/24.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
How to generalize?
Lemma (Osoinach)
S30(K ′
n) ≈ S30(K ′
0).
Tetsuya Abe On annulus twists
![Page 25: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/25.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
How to generalize?
Lemma (Osoinach)
S30(K ′
n) ≈ S30(K ′
0).
Tetsuya Abe On annulus twists
![Page 26: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/26.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
A general case.
m
Tetsuya Abe On annulus twists
![Page 27: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/27.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
A general case.
m
Tetsuya Abe On annulus twists
![Page 28: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/28.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Knot concordance group C.
The knot concordance group was introduced byFox and Milnor in 1966.
Two oriented knots K and K ′ are concordantdef⇐⇒ they cobound a properly embedded annulus
in S3 × I.
The knot concordance group is
C = {Knots}/ ∼
Tetsuya Abe On annulus twists
![Page 29: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/29.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Knot concordance group C.
The knot concordance group was introduced byFox and Milnor in 1966.
Two oriented knots K and K ′ are concordantdef⇐⇒ they cobound a properly embedded annulus
in S3 × I.
The knot concordance group is
C = {Knots}/ ∼
Tetsuya Abe On annulus twists
![Page 30: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/30.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Knot concordance group C.
The knot concordance group was introduced byFox and Milnor in 1966.
Two oriented knots K and K ′ are concordantdef⇐⇒ they cobound a properly embedded annulus
in S3 × I.
The knot concordance group is
C = {Knots}/ ∼
Tetsuya Abe On annulus twists
![Page 31: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/31.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Akbulut-Kirby’s conjecture.
Question (Levine, 2016)
If K is concordant to K ′, then for all n,S3
n(K ) is homology cobordant to S3n(K ′).
Is the converse true?
Conjecture (Akbulut-Kirby)
If two knots K and K ′ have the same 0-surgery,then K and K ′ are concordant.
After Abe-Tagami’s work, Yasui solved this conjecture.
Notation: Xn(K ) is the 2-handleboby obtained from K ⊂ S3
with framing n.Tetsuya Abe On annulus twists
![Page 32: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/32.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Akbulut-Kirby’s conjecture.
Question (Levine, 2016)
If K is concordant to K ′, then for all n,S3
n(K ) is homology cobordant to S3n(K ′).
Is the converse true?
Conjecture (Akbulut-Kirby)
If two knots K and K ′ have the same 0-surgery,then K and K ′ are concordant.
After Abe-Tagami’s work, Yasui solved this conjecture.
Notation: Xn(K ) is the 2-handleboby obtained from K ⊂ S3
with framing n.Tetsuya Abe On annulus twists
![Page 33: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/33.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Akbulut-Kirby’s conjecture.
Question (Levine, 2016)
If K is concordant to K ′, then for all n,S3
n(K ) is homology cobordant to S3n(K ′).
Is the converse true?
Conjecture (Akbulut-Kirby)
If two knots K and K ′ have the same 0-surgery,then K and K ′ are concordant.
After Abe-Tagami’s work, Yasui solved this conjecture.
Notation: Xn(K ) is the 2-handleboby obtained from K ⊂ S3
with framing n.Tetsuya Abe On annulus twists
![Page 34: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/34.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Yasui’s result.
(A picture from Yasui’s paper)
Theorem (Yasui, 2015)
(1) X0(K ) and X0(K ′) are exotic (home. but not diffeo.).(2) τ(K ) 6= τ(K ′).
Tetsuya Abe On annulus twists
![Page 35: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/35.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
A remaining conjecture.
Conjecture
If X0(K ) ≈ X0(K ′), then K and K ′ are concordant.
Theorem (Abe-Jong-Omae-Takeuchi, 2013, Abe-Tagami, 2015)
(1) The 4-manifold X0(K0) and X0(K1) are diffeomorphic.(2) If the slice-ribbon conjecture is true, K0 and K1 are notconcordant.
Tetsuya Abe On annulus twists
![Page 36: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/36.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
A remaining conjecture.
Conjecture
If X0(K ) ≈ X0(K ′), then K and K ′ are concordant.
Theorem (Abe-Jong-Omae-Takeuchi, 2013, Abe-Tagami, 2015)
(1) The 4-manifold X0(K0) and X0(K1) are diffeomorphic.(2) If the slice-ribbon conjecture is true, K0 and K1 are notconcordant.
Tetsuya Abe On annulus twists
![Page 37: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/37.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Another consequence of the slice-ribbon conjecture.
Observation (Abe-Tagami, 2015)
If the slice-ribbon conjecture is true, the set of prime tightfibered knots is linearly independent in Con(S3).
If the slice-ribbon conjecture is true, then following is true.
The set of the algebraic knots is linearly independent.The set of L-space knots is linearly independent.
Conclusion
The slice-ribbon conjecture has information on Con(S3).
Tetsuya Abe On annulus twists
![Page 38: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/38.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Another consequence of the slice-ribbon conjecture.
Observation (Abe-Tagami, 2015)
If the slice-ribbon conjecture is true, the set of prime tightfibered knots is linearly independent in Con(S3).
If the slice-ribbon conjecture is true, then following is true.
The set of the algebraic knots is linearly independent.The set of L-space knots is linearly independent.
Conclusion
The slice-ribbon conjecture has information on Con(S3).
Tetsuya Abe On annulus twists
![Page 39: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/39.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Another consequence of the slice-ribbon conjecture.
Observation (Abe-Tagami, 2015)
If the slice-ribbon conjecture is true, the set of prime tightfibered knots is linearly independent in Con(S3).
If the slice-ribbon conjecture is true, then following is true.
The set of the algebraic knots is linearly independent.The set of L-space knots is linearly independent.
Conclusion
The slice-ribbon conjecture has information on Con(S3).
Tetsuya Abe On annulus twists
![Page 40: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/40.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
Another consequence of the slice-ribbon conjecture.
Observation (Abe-Tagami, 2015)
If the slice-ribbon conjecture is true, the set of prime tightfibered knots is linearly independent in Con(S3).
If the slice-ribbon conjecture is true, then following is true.
The set of the algebraic knots is linearly independent.The set of L-space knots is linearly independent.
Conclusion
The slice-ribbon conjecture has information on Con(S3).
Tetsuya Abe On annulus twists
![Page 41: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/41.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
The definition of a slice knot.
B4 : the 4-ball s.t ∂B4 = S3
A knot K ⊂ S3 is (smoothly) slicedef⇐⇒ it bounds a smoothly embedded disk in B4.
• A knot K is slice iff [K] is the unit element of C.
Tetsuya Abe On annulus twists
![Page 42: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/42.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
The definition of a slice knot.
B4 : the 4-ball s.t ∂B4 = S3
A knot K ⊂ S3 is (smoothly) slicedef⇐⇒ it bounds a smoothly embedded disk in B4.
• A knot K is slice iff [K] is the unit element of C.
Tetsuya Abe On annulus twists
![Page 43: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/43.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
The definition of a slice knot.
B4 : the 4-ball s.t ∂B4 = S3
A knot K ⊂ S3 is (smoothly) slicedef⇐⇒ it bounds a smoothly embedded disk in B4.
• A knot K is slice iff [K] is the unit element of C.
Tetsuya Abe On annulus twists
![Page 44: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/44.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
The definition of a slice knot.
B4 : the 4-ball s.t ∂B4 = S3
A knot K ⊂ S3 is (smoothly) slicedef⇐⇒ it bounds a smoothly embedded disk in B4.
• A knot K is slice iff [K] is the unit element of C.
Tetsuya Abe On annulus twists
![Page 45: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/45.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
The definition of a ribbon disk.
A knot in S3 is ribbondef⇐⇒ it bounds an immersed disk in S3
with only ribbon singularities.
LemmaA ribbon knot R ⊂ S3 bounds a smoothlyembedded disk in B4. In particular, R is slice.
This disk is called a ribbon disk for R.Tetsuya Abe On annulus twists
![Page 46: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/46.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
The definition of a ribbon disk.
A knot in S3 is ribbondef⇐⇒ it bounds an immersed disk in S3
with only ribbon singularities.
LemmaA ribbon knot R ⊂ S3 bounds a smoothlyembedded disk in B4. In particular, R is slice.
This disk is called a ribbon disk for R.Tetsuya Abe On annulus twists
![Page 47: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/47.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
The definition of a ribbon disk.
A knot in S3 is ribbondef⇐⇒ it bounds an immersed disk in S3
with only ribbon singularities.
LemmaA ribbon knot R ⊂ S3 bounds a smoothlyembedded disk in B4. In particular, R is slice.
This disk is called a ribbon disk for R.Tetsuya Abe On annulus twists
![Page 48: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/48.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
The slice-ribbon conjecture.
• Any ribbon knot is slice.
• The slice-ribbon conjecture states that anyslice knot is ribbon.
Using annulus twists technique, we can constructpotential counterexamples of slice-ribbon conjecture.
T. Abe and M. Tange, A construction of slice knots via annulustwists, accepted by the Michigan Mathematical Journal.
Tetsuya Abe On annulus twists
![Page 49: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/49.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
The slice-ribbon conjecture.
• Any ribbon knot is slice.
• The slice-ribbon conjecture states that anyslice knot is ribbon.
Using annulus twists technique, we can constructpotential counterexamples of slice-ribbon conjecture.
T. Abe and M. Tange, A construction of slice knots via annulustwists, accepted by the Michigan Mathematical Journal.
Tetsuya Abe On annulus twists
![Page 50: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/50.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
A construction of ribbon disks.
Let U ⊂ S3 be the unknot, and ∆ ⊂ S3 = ∂B4 a spanning diskfor U. Regard B4 as a 0-handle h0. Consider an m-componentunlink L = L1 t L2 t · · · t Lm such that L ∩∆ = ∅. Regard theunlink L as dotted 1-handles h1
i (i = 1,2, · · · ,m). Attach2-handles h2
i (i = 1,2, · · · ,m) along circles in S3 \ (L ∪ U) sothat, the resulting 2-handlebody
h0⋃
h11 ∪ · · · ∪ h1
m
⋃h2
1 ∪ · · · ∪ h2m
is represented by the handle diagram in Figure 1 after isotopy,where γi is the framing of a 2-handle h2
i (i = 1,2, · · · ,m).γ1 γ2 γm
Figure: A handle diagram.Tetsuya Abe On annulus twists
![Page 51: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/51.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
A construction of ribbon disks.
By the handle canceling theorem, the 2-handlebody
h0⋃
h11 ∪ · · · ∪ h1
m
⋃h2
1 ∪ · · · ∪ h2m
is diffeomorphic to B4. Note that
∆ ⊂ h0 ⊂ h0⋃
h11 ∪ · · · ∪ h1
m
⋃h2
1 ∪ · · · ∪ h2m ≈ B4.
Therefore we obtain a new slice disk D in B4 (which is ∆ inh0 ⋃h1
1 ∪ · · · ∪ h1m⋃
h21 ∪ · · · ∪ h2
m).
Theorem (Abe-Tange, cf. Hitt, Asano-Marumto-Yanagawa)Any slice disk obtained by the above construction is ribbon.Conversely, any ribbon disk is obtained by this construction.
Tetsuya Abe On annulus twists
![Page 52: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/52.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
A schematic picture of this construction for m = 1.
h0
∆h2 D
Figure: The spanning disk ∆ in S3 = ∂B4, the 2-handlebodyh0 ∪h1 ∪h2 which is diffeomorphic to B4, and a new slice disk D in B4.
Tetsuya Abe On annulus twists
![Page 53: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/53.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
A concrete example of this construction for m = 2.
Let n be an integer.
h11
h11
h12h21
h22
∆
Figure: The slice disk Dn.
Then the 2-handlebody
h0⋃
h11 ∪ h1
2
⋃h2
1 ∪ h22
satisfies the condition in this section and is diffeomorphic to B4.A new slice disk, denoted by Dn, is the shaded disk in Figure 3.
Tetsuya Abe On annulus twists
![Page 54: On annulus twistsildt/2016/slides16/Abe.pdfIf K is concordant to K0, then for all n, S3 n(K) is homology cobordant to S n 3(K0). Is the converse true? Tetsuya Abe On annulus twists](https://reader035.vdocument.in/reader035/viewer/2022071508/612905efac773437e409cce5/html5/thumbnails/54.jpg)
Dehn surgery and annulus twistsKnot concordance and annulus twists
A construction of ribbon disks
A concrete example of this construction for m = 2.
Let n be an integer.
Theorem (Abe-Tange)
(1) The exteriors of Dn are the same.(2) Dn is a ribbon disk.(2) ∂Dn = Kn.
Tetsuya Abe On annulus twists