taut foliations, positive 3-braids, and the l-space conjecture
TRANSCRIPT
![Page 1: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/1.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Taut Foliations, Positive 3-Braids, and theL-Space Conjecture
Siddhi Krishna
Boston College
January 17, 2019
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 2: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/2.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Taut FoliationsL-Spaces
The L-Space Conjecture
The L-Space Conjecture:Suppose Y is a closed, irreducible 3-manifold.The following are equivalent:
Y admits a taut foliation
Y is a non-L-space
π1(Y ) is not left orderable
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 3: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/3.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Taut FoliationsL-Spaces
The L-Space Conjecture
The L-Space Conjecture:Suppose Y is a closed, irreducible 3-manifold.The following are equivalent:
Y admits a taut foliation
Y is a non-L-space
π1(Y ) is not left orderable
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 4: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/4.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Taut FoliationsL-Spaces
The L-Space Conjecture
The L-Space Conjecture:Suppose Y is a closed, irreducible 3-manifold.The following are equivalent:
Y admits a taut foliation “geometry”
Y is a non-L-space “Heegaard-Floer homology”
π1(Y ) is not left orderable “algebra”
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 5: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/5.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Taut FoliationsL-Spaces
The L-Space Conjecture
The L-Space Conjecture:Suppose Y is a closed, irreducible 3-manifold.The following are equivalent:
Y admits a taut foliation “geometry”
Y is a non-L-space “Heegaard-Floer homology”
π1(Y ) is not left orderable “algebra”
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 6: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/6.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Taut FoliationsL-Spaces
The L-Space Conjecture
The L-Space Conjecture:Suppose Y is a closed, irreducible 3-manifold.The following are equivalent:
Y admits a taut foliation “geometry”
• Y is a non-L-space “Heegaard-Floer homology”
π1(Y ) is not left orderable “algebra”
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 7: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/7.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Taut FoliationsL-Spaces
Taut Foliations
Definition
A taut foliation is
• a decomposition of a manifold into codim-1 submanifolds, calledleaves, such that
• there exists a simple closed curve meeting each leaf transversely
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 8: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/8.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Taut FoliationsL-Spaces
An Example
Fibered 3-Manifolds
Start with
a compact, connected, oriented surface F
ϕ : F → F , a diffeomorphism of F
This determines:
Mϕ = F × I/ϕ = F × I
/((x , 0) ∼ (ϕ(x), 1))
F × {0}
F × {1}
ϕ
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 9: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/9.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Taut FoliationsL-Spaces
An Example
Fibered 3-Manifolds
Start with
a compact, connected, oriented surface F
ϕ : F → F , a diffeomorphism of F
This determines:
Mϕ = F × I/ϕ = F × I
/((x , 0) ∼ (ϕ(x), 1))
F × {0}
F × {1}
ϕ
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 10: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/10.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Taut FoliationsL-Spaces
An Example
Fibered 3-Manifolds
Start with
a compact, connected, oriented surface F
ϕ : F → F , a diffeomorphism of F
This determines:
Mϕ = F × I/ϕ = F × I
/((x , 0) ∼ (ϕ(x), 1))
F × {0}
F × {1}
ϕ
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 11: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/11.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Taut FoliationsL-Spaces
An Example
Fibered 3-Manifolds
Start with
a compact, connected, oriented surface F
ϕ : F → F , a diffeomorphism of F
This determines:
Mϕ = F × I/ϕ = F × I
/((x , 0) ∼ (ϕ(x), 1))
F × {0}
F × {1}
ϕ
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 12: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/12.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Taut FoliationsL-Spaces
An Example
Fibered 3-Manifolds
Start with
a compact, connected, orientable surface F
ϕ : F → F , a diffeomorphism of F
This determines:
Mϕ = F × I/ϕ = F × I
/((x , 0) ∼ (ϕ(x), 1))
F × {0}
F × {1}
ϕ
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 13: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/13.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Taut FoliationsL-Spaces
An Example
Fibered 3-Manifolds
Start with
a compact, connected, orientable surface F
ϕ : F → F , a diffeomorphism of F
This determines:
Mϕ = F × I/ϕ = F × I
/((x , 0) ∼ (ϕ(x), 1))
F × {0}
F × {1}
ϕ
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 14: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/14.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Taut FoliationsL-Spaces
An Example
Fibered 3-Manifolds
Start with
a compact, connected, orientable surface F
ϕ : F → F , a diffeomorphism of F
This determines:
Mϕ = F × I/ϕ = F × I
/((x , 0) ∼ (ϕ(x), 1))
F × {0}
F × {1}
ϕ
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 15: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/15.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Taut FoliationsL-Spaces
An Example
Fibered 3-Manifolds
Start with
a compact, connected, orientable surface F
ϕ : F → F , a diffeomorphism of F
This determines:
Mϕ = F × I/ϕ = F × I
/((x , 0) ∼ (ϕ(x), 1))
F × {0}
F × {1}
ϕ
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 16: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/16.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Taut FoliationsL-Spaces
An Example
Fibered 3-Manifolds
Start with
a compact, connected, orientable surface F
ϕ : F → F , a diffeomorphism of F
This determines:
Mϕ = F × I/ϕ = F × I
/((x , 0) ∼ (ϕ(x), 1))
F × {0}
F × {1}
ϕ
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 17: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/17.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Taut FoliationsL-Spaces
Taut Foliations in the Literature
Theorem (Thurston ’86):Compact leaves of taut foliations minimize genus in their homologyclass.
Theorem (Gabai ’87):The “Property R” Conjecture is true, i.e.
If K ⊂ S3, and S30 (K ) ≈ S1 × S2, then K ≈ U.
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 18: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/18.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Taut FoliationsL-Spaces
Taut Foliations in the Literature
Theorem (Thurston ’86):Compact leaves of taut foliations minimize genus in their homologyclass.
Theorem (Gabai ’87):The “Property R” Conjecture is true, i.e.
If K ⊂ S3, and S30 (K ) ≈ S1 × S2, then K ≈ U.
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 19: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/19.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Taut FoliationsL-Spaces
The L-Space Conjecture
The L-Space Conjecture:Suppose Y is a closed, irreducible 3-manifold.The following are equivalent:
Y admits a taut foliation “geometry”
Y is a non-L-space “Heegaard-Floer homology”
π1(Y ) is not left orderable “algebra”
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 20: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/20.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Taut FoliationsL-Spaces
L-Spaces
“L-spaces are simple from the perspective ofHeegaard Floer homology”
Definition
A closed, irreducible 3-manifold Y is an L-space if
rk(HF (Y ;Z/2Z)) = |H1(Y ;Z)| <∞
Remark: rk(HF (Y ;Z/2Z)) ≥ |H1(Y ;Z)| holds for QHS3
Examples: Lens Spaces ⊂ Manifolds with Elliptic Geometry
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 21: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/21.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Taut FoliationsL-Spaces
L-Spaces
“L-spaces are simple from the perspective ofHeegaard Floer homology”
Definition
A closed, irreducible 3-manifold Y is an L-space if
rk(HF (Y ;Z/2Z)) = |H1(Y ;Z)| <∞
Remark: rk(HF (Y ;Z/2Z)) ≥ |H1(Y ;Z)| holds for QHS3
Examples: Lens Spaces ⊂ Manifolds with Elliptic Geometry
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 22: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/22.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Taut FoliationsL-Spaces
L-Spaces
“L-spaces are simple from the perspective ofHeegaard Floer homology”
Definition
A closed, irreducible 3-manifold Y is an L-space if
rk(HF (Y ;Z/2Z)) = |H1(Y ;Z)| <∞
Remark: rk(HF (Y ;Z/2Z)) ≥ |H1(Y ;Z)| holds for QHS3
Examples: Lens Spaces ⊂ Manifolds with Elliptic Geometry
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 23: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/23.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Taut FoliationsL-Spaces
L-Spaces
“L-spaces are simple from the perspective ofHeegaard Floer homology”
Definition
A closed, irreducible 3-manifold Y is an L-space if
rk(HF (Y ;Z/2Z)) = |H1(Y ;Z)| <∞
Remark: rk(HF (Y ;Z/2Z)) ≥ |H1(Y ;Z)| holds for QHS3
Examples: Lens Spaces ⊂ Manifolds with Elliptic Geometry
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 24: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/24.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Taut FoliationsL-Spaces
The L-Space Conjecture Revisited
The L-Space Conjecture:Suppose Y is a closed, irreducible 3-manifold.The following are equivalent:
Y admits a taut foliation “geometry”
Y is a non-L-space “Heegaard-Floer homology”
π1(Y ) is not left orderable “algebra”
These manifolds are “extra”.
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 25: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/25.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Taut FoliationsL-Spaces
The L-Space Conjecture Revisited
The L-Space Conjecture:Suppose Y is a closed, irreducible 3-manifold.The following are equivalent:
Y admits a taut foliation “geometry”
Y is a non-L-space “Heegaard-Floer homology”
π1(Y ) is not left orderable “algebra”
These manifolds are “extra”.
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 26: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/26.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
EvidenceProducing Non-L-SpacesA New Theorem
Evidence for the LSC
Theorem (Ozsvath and Szabo):If Y admits a taut foliation, then Y is a non-L-space.
Theorem: The L-Space Conjecture is true for graph manifolds.Proof: Input by many!
Eisenbud, Hirsch, Neumann, Naimi, Calegari, Walker, Boyer,Gordon, Clay, Watson, Lisca, Stipsicz, J. & S. Rasmussen,Hanselman, and many more!
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 27: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/27.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
EvidenceProducing Non-L-SpacesA New Theorem
Evidence for the LSC
Theorem (Ozsvath and Szabo):If Y admits a taut foliation, then Y is a non-L-space.
Theorem: The L-Space Conjecture is true for graph manifolds.
Proof: Input by many!
Eisenbud, Hirsch, Neumann, Naimi, Calegari, Walker, Boyer,Gordon, Clay, Watson, Lisca, Stipsicz, J. & S. Rasmussen,Hanselman, and many more!
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 28: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/28.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
EvidenceProducing Non-L-SpacesA New Theorem
Evidence for the LSC
Theorem (Ozsvath and Szabo):If Y admits a taut foliation, then Y is a non-L-space.
Theorem: The L-Space Conjecture is true for graph manifolds.Proof: Input by many!
Eisenbud, Hirsch, Neumann, Naimi, Calegari, Walker, Boyer,Gordon, Clay, Watson, Lisca, Stipsicz, J. & S. Rasmussen,Hanselman, and many more!
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 29: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/29.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
EvidenceProducing Non-L-SpacesA New Theorem
Focusing on Foliations
For Y is a closed, irreducible 3-manifold:
Y admits a taut foliation Y is a non-L-space
Ozsvath-Szabo
Questions: How do we(1) identify non-L-spaces, and(2) build taut foliations in them?
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 30: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/30.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
EvidenceProducing Non-L-SpacesA New Theorem
Focusing on Foliations
For Y is a closed, irreducible 3-manifold:
Y admits a taut foliation Y is a non-L-space
Ozsvath-Szabo
Questions: How do we(1) identify non-L-spaces?(2) build taut foliations in them?
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 31: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/31.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
EvidenceProducing Non-L-SpacesA New Theorem
Focusing on Foliations
For Y is a closed, irreducible 3-manifold:
Y admits a taut foliation Y is a non-L-space
Ozsvath-Szabo
Questions: How do we(1) identify non-L-spaces?
(2) build taut foliations in them?
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 32: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/32.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
EvidenceProducing Non-L-SpacesA New Theorem
Focusing on Foliations
For Y is a closed, irreducible 3-manifold:
Y admits a taut foliation Y is a non-L-space
Ozsvath-Szabo
Questions: How do we(1) identify non-L-spaces?(2) build taut foliations in them?
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 33: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/33.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
EvidenceProducing Non-L-SpacesA New Theorem
Focusing on Foliations
For Y is a closed, irreducible 3-manifold:
Y admits a taut foliation Y is a non-L-space
Ozsvath-Szabo
Questions: How do we(1) identify non-L-spaces?(2) build taut foliations in them?
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 34: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/34.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
EvidenceProducing Non-L-SpacesA New Theorem
Producing Non-L-Spaces via Dehn Surgery
Definition
A non-trivial knot K ⊂ S3 is an L-Space knot if K admits anon-trivial surgery to an L-space.
Examples: Torus knots
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 35: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/35.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
EvidenceProducing Non-L-SpacesA New Theorem
Producing Non-L-Spaces via Dehn Surgery
Definition
A non-trivial knot K ⊂ S3 is an L-Space knot if K admits anon-trivial surgery to an L-space.
Examples: Torus knots
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 36: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/36.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
EvidenceProducing Non-L-SpacesA New Theorem
Producing Non-L-Spaces via Dehn Surgery
Suppose K ⊂ S3.
Then either:
(1) K isn’t an L-space knot:For all r ∈ Q, S3
r (K ) is a non-L-space.
(2) K is an L-space knot:
Theorem: (Kronheimer-Mrowka-Ozsvath-Szabo; J+S Rasmussen):
For r ∈ Q, S3r (K ) =
{non-L-space r < 2g(K )− 1
L-space r ≥ 2g(K )− 1
If r < 2g(K )− 1, then S3r (K ) is always a non-L-space.
Therefore, the LSC predicts it admits a taut foliation.
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 37: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/37.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
EvidenceProducing Non-L-SpacesA New Theorem
Producing Non-L-Spaces via Dehn Surgery
Suppose K ⊂ S3. Then either:
(1) K isn’t an L-space knot:For all r ∈ Q, S3
r (K ) is a non-L-space.
(2) K is an L-space knot:
Theorem: (Kronheimer-Mrowka-Ozsvath-Szabo; J+S Rasmussen):
For r ∈ Q, S3r (K ) =
{non-L-space r < 2g(K )− 1
L-space r ≥ 2g(K )− 1
If r < 2g(K )− 1, then S3r (K ) is always a non-L-space.
Therefore, the LSC predicts it admits a taut foliation.
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 38: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/38.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
EvidenceProducing Non-L-SpacesA New Theorem
Producing Non-L-Spaces via Dehn Surgery
Suppose K ⊂ S3. Then either:
(1) K isn’t an L-space knot:For all r ∈ Q, S3
r (K ) is a non-L-space.
(2) K is an L-space knot:
Theorem: (Kronheimer-Mrowka-Ozsvath-Szabo; J+S Rasmussen):
For r ∈ Q, S3r (K ) =
{non-L-space r < 2g(K )− 1
L-space r ≥ 2g(K )− 1
If r < 2g(K )− 1, then S3r (K ) is always a non-L-space.
Therefore, the LSC predicts it admits a taut foliation.
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 39: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/39.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
EvidenceProducing Non-L-SpacesA New Theorem
Producing Non-L-Spaces via Dehn Surgery
Suppose K ⊂ S3. Then either:
(1) K isn’t an L-space knot:For all r ∈ Q, S3
r (K ) is a non-L-space.
(2) K is an L-space knot:
Theorem: (Kronheimer-Mrowka-Ozsvath-Szabo; J+S Rasmussen):
For r ∈ Q, S3r (K ) =
{non-L-space r < 2g(K )− 1
L-space r ≥ 2g(K )− 1
If r < 2g(K )− 1, then S3r (K ) is always a non-L-space.
Therefore, the LSC predicts a taut foliation.
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 40: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/40.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
EvidenceProducing Non-L-SpacesA New Theorem
Producing Non-L-Spaces via Dehn Surgery
Suppose K ⊂ S3. Then either:
(1) K isn’t an L-space knot:For all r ∈ Q, S3
r (K ) is a non-L-space.
(2) K is an L-space knot:
Theorem (Kronheimer-Mrowka-Ozsvath-Szabo; J+S Rasmussen):
For r ∈ Q, S3r (K ) =
{non-L-space r < 2g(K )− 1
L-space r ≥ 2g(K )− 1
If r < 2g(K )− 1, then S3r (K ) is always a non-L-space.
Therefore, the LSC predicts a taut foliation.
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 41: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/41.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
EvidenceProducing Non-L-SpacesA New Theorem
Producing Non-L-Spaces via Dehn Surgery
Suppose K ⊂ S3. Then either:
(1) K isn’t an L-space knotFor all r ∈ Q, S3
r (K ) is a non-L-space.
(2) K is an L-space knot
Theorem (Kronheimer-Mrowka-Ozsvath-Szabo; J+S Rasmussen):
For r ∈ Q, S3r (K ) =
{non-L-space r < 2g(K )− 1
L-space r ≥ 2g(K )− 1
If r < 2g(K )− 1, then S3r (K ) is always a non-L-space.
Therefore, the LSC predicts a taut foliation.
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 42: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/42.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
EvidenceProducing Non-L-SpacesA New Theorem
A New Theorem
Theorem (K.)
Suppose K ⊂ S3 is the closure of a positive 3-braid.Then for all rational r < 2g(K )− 1, S3
r (K ) admits a taut foliation.
Example: The P(−2, 3, 7) (Fintushel-Stern) Pretzel Knot
7 77
7
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 43: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/43.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
EvidenceProducing Non-L-SpacesA New Theorem
A New Theorem
Theorem (K.)
Suppose K ⊂ S3 is the closure of a positive 3-braid.Then for all rational r < 2g(K )− 1, S3
r (K ) admits a taut foliation.
Example: The P(−2, 3, 7) (Fintushel-Stern) Pretzel Knot
7 77
7
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 44: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/44.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
EvidenceProducing Non-L-SpacesA New Theorem
A New Theorem
Theorem (K.)
Suppose K ⊂ S3 is the closure of a positive 3-braid.Then for all rational r < 2g(K )− 1, S3
r (K ) admits a taut foliation.
Example: The P(−2, 3, 7) (Fintushel-Stern) Pretzel Knot
7 77
7
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 45: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/45.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
EvidenceProducing Non-L-SpacesA New Theorem
A New Theorem
Theorem (K.)
Suppose K ⊂ S3 is the closure of a positive 3-braid.Then for all rational r < 2g(K )− 1, S3
r (K ) admits a taut foliation.
q qq
q
Remark: Applying (Lidman-Moore ’16), we get the first example of
Y is a non-L-space ⇐⇒ Y admits a taut foliation
for every non-L-space obtained by Dehn surgery along an infinitefamily of hyperbolic L-space knots.
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 46: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/46.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
EvidenceProducing Non-L-SpacesA New Theorem
A New Theorem
Theorem (K.)
Suppose K ⊂ S3 is the closure of a positive 3-braid.Then for all rational r < 2g(K )− 1, S3
r (K ) admits a taut foliation.
q qq
q
Remark: Applying (Lidman-Moore ’16), we get the first example of
Y is a non-L-space ⇐⇒ Y admits a taut foliation
for every non-L-space obtained by Dehn surgery along an infinitefamily of hyperbolic L-space knots.
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 47: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/47.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Branched SurfacesProof Sketch
What’s Known, Revisited
For Y is a closed, irreducible 3-manifold:
Y admits a taut foliation Y is a non-L-space
Ozsvath-Szabo
Questions: How do we(1) identify non-L-spaces?(2) build taut foliations in them?
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 48: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/48.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Branched SurfacesProof Sketch
What’s Known, Revisited
For Y is a closed, irreducible 3-manifold:
Y admits a taut foliation Y is a non-L-space
Ozsvath-Szabo
Questions: How do we(1) identify non-L-spaces? Dehn Surgery(2) build taut foliations in them?
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 49: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/49.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Branched SurfacesProof Sketch
What’s Known, Revisited
For Y is a closed, irreducible 3-manifold:
Y admits a taut foliation Y is a non-L-space
Ozsvath-Szabo
Questions: How do we(1) identify non-L-spaces? Dehn Surgery(2) build taut foliations in them? Branched Surfaces
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 50: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/50.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Branched SurfacesProof Sketch
Branched Surfaces
Definition
A branched surface is a “co-oriented 2-complex”, locallymodelled by:
Branched surfaces can encode data about(1) surfaces in 3-manifolds (Oertel; Floyd-Oertel)(2) laminations in 3-manifolds (Li)(3) foliations of 3-manifolds (Roberts)
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 51: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/51.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Branched SurfacesProof Sketch
Branched Surfaces
Definition
A branched surface is a “co-oriented 2-complex”, locallymodelled by:
Branched surfaces can encode data about(1) surfaces in 3-manifolds (Oertel; Floyd-Oertel)(2) laminations in 3-manifolds (Li)(3) foliations of 3-manifolds (Roberts)
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 52: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/52.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Branched SurfacesProof Sketch
Branched Surfaces
Definition
A branched surface is a “co-oriented 2-complex”, locallymodelled by:
Branched surfaces can encode data about(1) surfaces in 3-manifolds (Oertel; Floyd-Oertel)(2) laminations in 3-manifolds (Li)(3) foliations of 3-manifolds (Roberts)
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 53: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/53.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Branched SurfacesProof Sketch
Branched Surfaces
Definition
A branched surface is a “co-oriented 2-complex”, locallymodelled by:
Branched surfaces can encode data about(1) surfaces in 3-manifolds (Oertel; Floyd-Oertel)(2) laminations in 3-manifolds (Li)(3) foliations of 3-manifolds (Roberts)
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 54: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/54.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Branched SurfacesProof Sketch
Branched Surfaces and Taut Foliations
Theorem (K.)
Suppose K ⊂ S3 is the closure of a positive 3-braid.Then for all rational r < 2g(K )− 1, S3
r (K ) admits a taut foliation.
Proof Sketch:(1) Identify a fiber surface for K in
XK = S3 − ◦ν(K ) = F × I
/ϕ
(2) Use the fibration to build a branched surface B in XK
(3) Use B to build a taut foliation in XK
(4) Understand how F meets ∂XK
(5) Dehn fill to produce a taut foliation in S3r (K )
Branched surfaces give a recipe for building taut foliations.
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 55: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/55.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Branched SurfacesProof Sketch
Branched Surfaces and Taut Foliations
Theorem (K.)
Suppose K ⊂ S3 is the closure of a positive 3-braid.Then for all rational r < 2g(K )− 1, S3
r (K ) admits a taut foliation.
Proof Sketch:(1) Identify a fiber surface for K in
XK = S3 − ◦ν(K ) = F × I
/ϕ
(2) Use the fibration to build a branched surface B in XK
(3) Use B to build a taut foliation in XK
(4) Understand how F meets ∂XK
(5) Dehn fill to produce a taut foliation in S3r (K )
Branched surfaces give a recipe for building taut foliations.
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 56: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/56.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Branched SurfacesProof Sketch
Branched Surfaces and Taut Foliations
Theorem (K.)
Suppose K ⊂ S3 is the closure of a positive 3-braid.Then for all rational r < 2g(K )− 1, S3
r (K ) admits a taut foliation.
Proof Sketch:(1) Identify a fiber surface for K in
XK = S3 − ◦ν(K ) = F × I
/ϕ
(2) Use the fibration to build a branched surface B in XK
(3) Use B to build a taut foliation in XK
(4) Understand how F meets ∂XK
(5) Dehn fill to produce a taut foliation in S3r (K )
Branched surfaces give a recipe for building taut foliations.
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 57: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/57.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Branched SurfacesProof Sketch
Branched Surfaces and Taut Foliations
Theorem (K.)
Suppose K ⊂ S3 is the closure of a positive 3-braid.Then for all rational r < 2g(K )− 1, S3
r (K ) admits a taut foliation.
Proof Sketch:(1) Identify a fiber surface for K in
XK = S3 − ◦ν(K ) = F × I
/ϕ
(2) Use the fibration to build a branched surface B in XK
(3) Use B to build a taut foliation in XK
(4) Understand how F meets ∂XK
(5) Dehn fill to produce a taut foliation in S3r (K )
Branched surfaces give a recipe for building taut foliations.
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 58: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/58.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Branched SurfacesProof Sketch
Branched Surfaces and Taut Foliations
Theorem (K.)
Suppose K ⊂ S3 is the closure of a positive 3-braid.Then for all rational r < 2g(K )− 1, S3
r (K ) admits a taut foliation.
Proof Sketch:(1) Identify a fiber surface for K in
XK = S3 − ◦ν(K ) = F × I
/ϕ
(2) Use the fibration to build a branched surface B in XK
(3) Use B to build a taut foliation in XK
(4) Understand how F meets ∂XK
(5) Dehn fill to produce a taut foliation in S3r (K )
Branched surfaces give a recipe for building taut foliations.
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 59: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/59.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Branched SurfacesProof Sketch
Branched Surfaces and Taut Foliations
Theorem (K.)
Suppose K ⊂ S3 is the closure of a positive 3-braid.Then for all rational r < 2g(K )− 1, S3
r (K ) admits a taut foliation.
Proof Sketch:(1) Identify a fiber surface for K in
XK = S3 − ◦ν(K ) = F × I
/ϕ
(2) Use the fibration to build a branched surface B in XK
(3) Use B to build a taut foliation in XK
(4) Understand how F meets ∂XK
(5) Dehn fill to produce a taut foliation in S3r (K )
Branched surfaces give a recipe for building taut foliations.
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors
![Page 60: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture](https://reader031.vdocuments.us/reader031/viewer/2022012510/6187897c7f46b559c45632f9/html5/thumbnails/60.jpg)
The L-Space ConjectureInvestigating the LSC
Proof Strategy
Branched SurfacesProof Sketch
Branched Surfaces and Taut Foliations
Theorem (K.)
Suppose K ⊂ S3 is the closure of a positive 3-braid.Then for all rational r < 2g(K )− 1, S3
r (K ) admits a taut foliation.
Proof Sketch:(1) Identify a fiber surface for K in
XK = S3 − ◦ν(K ) = F × I
/ϕ
(2) Use the fibration to build a branched surface B in XK
(3) Use B to build a taut foliation in XK
(4) Understand how F meets ∂XK
(5) Dehn fill to produce a taut foliation in S3r (K )
Branched surfaces give a recipe for building taut foliations.
Siddhi Krishna Constructing Taut Foliations in Positive 3-Braid Exteriors