lightning talks iii tech topology...
TRANSCRIPT
![Page 1: Lightning Talks III Tech Topology Conferencepeople.math.gatech.edu/.../2017/SunLightningFinal.pdf · Lightning Talks III Tech Topology Conference December 10, 2017. COVERING SPACES,](https://reader034.vdocuments.site/reader034/viewer/2022043003/5f83b9804067e33dd9466d14/html5/thumbnails/1.jpg)
Lightning Talks IIITech Topology Conference
December 10, 2017
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COVERING SPACES, MAPPING CLASS GROUPS, AND THE
SYMPLECTIC REPRESENTATION
Sarah Davis With Laura Stordy, Becca Winarski, Ziyi Zhou Georgia Institute of Technology Tech Topology Conference
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3-Fold Branched Cover
= R
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Symmetric Mapping Class Group
SMod(S2
) = NMod(S2)
(hRi)
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Symplectic Representation
� : Mod(S2) ! Sp(4,Z)
� : R 7! E =
2
664
0 0 �1 00 �1 0 �11 0 �1 00 1 0 0
3
775
Example:
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McMullen’s Question
Question: Is finite index in ?
Venkataramana: Yes, if # branch points ≥ 2*degree
�(SMod(Sg))
NSp(2g,Z)(�(hRdi))
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Main Theorem:
�(SMod(S2)) = NSp(4,Z)(hEi)�(SMod(S2)) = NSp(4,Z)(hEi)
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Main Theorem:
�(SMod(S2)) = NSp(4,Z)(hEi)
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Find such that:
M 2 GL(4,Z)
MEM�1 = E±1
Lemma: It is enough to find M such that:
MEM�1 = E�1
�(SMod(S2)) = NSp(4,Z)(hEi)
![Page 10: Lightning Talks III Tech Topology Conferencepeople.math.gatech.edu/.../2017/SunLightningFinal.pdf · Lightning Talks III Tech Topology Conference December 10, 2017. COVERING SPACES,](https://reader034.vdocuments.site/reader034/viewer/2022043003/5f83b9804067e33dd9466d14/html5/thumbnails/10.jpg)
MATLAB Output
2
664
�z0 z1 � z2 z0 + z3 z1�z4 � z5 �z6 z4 z7 � z6
z3 z1 z0 z2z4 z7 z5 z6
3
775M =
Refine using the symplectic condition
M⌦MT = ⌦
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2
664
2x 1 �x 0�1 0 0 0x 0 �2x �10 0 1 0
3
775 ,
2
664
2x �1 �x 01 0 0 0x 0 �2x 10 0 �1 0
3
775 ,
2
664
0 1 0 0�1 �2x 0 �x
0 0 0 �10 x 1 2x
3
775 ,
2
664
0 �1 0 01 �2x 0 �x
0 0 0 10 x �1 2x
3
775
2
664
x 1 x 10 0 �1 0
2x 1 �x 0�1 0 1 0
3
775 ,
2
664
x �1 x �10 0 1 0
2x �1 �x 01 0 �1 0
3
775 ,
2
664
0 1 0 10 x �1 2x0 1 0 0
�1 x 1 �x
3
775 ,
2
664
0 �1 0 �10 x 1 2x0 �1 0 01 x �1 �x
3
775
2
664
0 x 1 2x0 0 0 11 2x 0 x
0 1 0 0
3
775 ,
2
664
0 x �1 2x0 0 0 �1
�1 2x 0 x
0 �1 0 0
3
775 ,
2
664
0 0 1 0x 0 �2x 11 0 0 0
�2x 1 x 0
3
775 ,
2
664
0 0 �1 0x 0 �2x �1
�1 0 0 0�2x �1 x 0
3
775
MATLAB Output
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�!
Ghaswala-Winarski give generators for . SMod(S2)
�(SMod(S2)) = NSp(4,Z)(hEi)
�( eT↵) =
2
664
1 0 0 02 1 �1 00 0 1 0
�1 0 2 1
3
775
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How can we obtain this matrix from ?
2
664
2x 1 �x 0�1 0 0 0x 0 �2x �10 0 1 0
3
775 = �(eh�
) · �( eT↵
)x · �(ehc
)
�(SMod(S2)) = NSp(4,Z)(hEi)
![Page 14: Lightning Talks III Tech Topology Conferencepeople.math.gatech.edu/.../2017/SunLightningFinal.pdf · Lightning Talks III Tech Topology Conference December 10, 2017. COVERING SPACES,](https://reader034.vdocuments.site/reader034/viewer/2022043003/5f83b9804067e33dd9466d14/html5/thumbnails/14.jpg)
2
664
2x 1 �x 0�1 0 0 0x 0 �2x �10 0 1 0
3
775 = �(eh�
) · �( eT↵
)x · �(ehc
)�⇣fH�
� fT↵
x
� fHc
⌘
H� Hc = H⌘ �H◆T↵
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Main Theorem:
�(SMod(S2)) = NSp(4,Z)(hEi)
Thank you!
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Lightning Talks IIITech Topology Conference
December 10, 2017
![Page 17: Lightning Talks III Tech Topology Conferencepeople.math.gatech.edu/.../2017/SunLightningFinal.pdf · Lightning Talks III Tech Topology Conference December 10, 2017. COVERING SPACES,](https://reader034.vdocuments.site/reader034/viewer/2022043003/5f83b9804067e33dd9466d14/html5/thumbnails/17.jpg)
Salem Number Stretch Factors
Joshua Pankau
University of California, Santa BarbaraAdvisor: Darren Long
12/10/2017
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 1 / 6
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I. Background
Definition of pseudo-Anosov map
A homeomorphism φ from a closed, orientable surface S to itself is calledpseudo-Anosov if there are two transverse, measured foliations, Fu andFs , along with a real number λ > 1, such that φ stretches S along Fu bya factor of λ and contracts S along Fs by a factor of λ−1. The number λis known as the stretch factor of φ.
Theorem (Thurston 1974)
If λ is the stretch factor of a pseudo-Anosov homeomorphism of a genus gsurface, then λ is an algebraic unit such that [Q(λ) : Q] ≤ 6g − 6.
Main Question
Which algebraic units can appear as stretch factors?
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 2 / 6
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I. Background
Definition of pseudo-Anosov map
A homeomorphism φ from a closed, orientable surface S to itself is calledpseudo-Anosov if there are two transverse, measured foliations, Fu andFs , along with a real number λ > 1, such that φ stretches S along Fu bya factor of λ and contracts S along Fs by a factor of λ−1. The number λis known as the stretch factor of φ.
Theorem (Thurston 1974)
If λ is the stretch factor of a pseudo-Anosov homeomorphism of a genus gsurface, then λ is an algebraic unit such that [Q(λ) : Q] ≤ 6g − 6.
Main Question
Which algebraic units can appear as stretch factors?
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 2 / 6
![Page 20: Lightning Talks III Tech Topology Conferencepeople.math.gatech.edu/.../2017/SunLightningFinal.pdf · Lightning Talks III Tech Topology Conference December 10, 2017. COVERING SPACES,](https://reader034.vdocuments.site/reader034/viewer/2022043003/5f83b9804067e33dd9466d14/html5/thumbnails/20.jpg)
I. Background
Definition of pseudo-Anosov map
A homeomorphism φ from a closed, orientable surface S to itself is calledpseudo-Anosov if there are two transverse, measured foliations, Fu andFs , along with a real number λ > 1, such that φ stretches S along Fu bya factor of λ and contracts S along Fs by a factor of λ−1. The number λis known as the stretch factor of φ.
Theorem (Thurston 1974)
If λ is the stretch factor of a pseudo-Anosov homeomorphism of a genus gsurface, then λ is an algebraic unit such that [Q(λ) : Q] ≤ 6g − 6.
Main Question
Which algebraic units can appear as stretch factors?
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 2 / 6
![Page 21: Lightning Talks III Tech Topology Conferencepeople.math.gatech.edu/.../2017/SunLightningFinal.pdf · Lightning Talks III Tech Topology Conference December 10, 2017. COVERING SPACES,](https://reader034.vdocuments.site/reader034/viewer/2022043003/5f83b9804067e33dd9466d14/html5/thumbnails/21.jpg)
II. Constructions
There are several general constructions of pseudo-Anosov maps. Thefollowing two consist of taking products of Dehn twists.
Penner’s ConstructionRestriction: Shin and Strenner showed that stretch factors ofpseudo-Anosov maps coming from Penner’s construction cannot haveGalois conjugates on the unit circle.
Thurston’s ConstructionRestriction: Veech showed that if λ is the stretch factor of apseudo-Anosov map coming from Thurston’s construction thenλ+ λ−1 is a totally real algebraic integer.
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 3 / 6
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II. Constructions
There are several general constructions of pseudo-Anosov maps. Thefollowing two consist of taking products of Dehn twists.
Penner’s Construction
Restriction: Shin and Strenner showed that stretch factors ofpseudo-Anosov maps coming from Penner’s construction cannot haveGalois conjugates on the unit circle.
Thurston’s ConstructionRestriction: Veech showed that if λ is the stretch factor of apseudo-Anosov map coming from Thurston’s construction thenλ+ λ−1 is a totally real algebraic integer.
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 3 / 6
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II. Constructions
There are several general constructions of pseudo-Anosov maps. Thefollowing two consist of taking products of Dehn twists.
Penner’s Construction
Restriction: Shin and Strenner showed that stretch factors ofpseudo-Anosov maps coming from Penner’s construction cannot haveGalois conjugates on the unit circle.
Thurston’s Construction
Restriction: Veech showed that if λ is the stretch factor of apseudo-Anosov map coming from Thurston’s construction thenλ+ λ−1 is a totally real algebraic integer.
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 3 / 6
![Page 24: Lightning Talks III Tech Topology Conferencepeople.math.gatech.edu/.../2017/SunLightningFinal.pdf · Lightning Talks III Tech Topology Conference December 10, 2017. COVERING SPACES,](https://reader034.vdocuments.site/reader034/viewer/2022043003/5f83b9804067e33dd9466d14/html5/thumbnails/24.jpg)
II. Constructions
There are several general constructions of pseudo-Anosov maps. Thefollowing two consist of taking products of Dehn twists.
Penner’s ConstructionRestriction: Shin and Strenner showed that stretch factors ofpseudo-Anosov maps coming from Penner’s construction cannot haveGalois conjugates on the unit circle.
Thurston’s Construction
Restriction: Veech showed that if λ is the stretch factor of apseudo-Anosov map coming from Thurston’s construction thenλ+ λ−1 is a totally real algebraic integer.
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 3 / 6
![Page 25: Lightning Talks III Tech Topology Conferencepeople.math.gatech.edu/.../2017/SunLightningFinal.pdf · Lightning Talks III Tech Topology Conference December 10, 2017. COVERING SPACES,](https://reader034.vdocuments.site/reader034/viewer/2022043003/5f83b9804067e33dd9466d14/html5/thumbnails/25.jpg)
II. Constructions
There are several general constructions of pseudo-Anosov maps. Thefollowing two consist of taking products of Dehn twists.
Penner’s ConstructionRestriction: Shin and Strenner showed that stretch factors ofpseudo-Anosov maps coming from Penner’s construction cannot haveGalois conjugates on the unit circle.
Thurston’s ConstructionRestriction: Veech showed that if λ is the stretch factor of apseudo-Anosov map coming from Thurston’s construction thenλ+ λ−1 is a totally real algebraic integer.
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 3 / 6
![Page 26: Lightning Talks III Tech Topology Conferencepeople.math.gatech.edu/.../2017/SunLightningFinal.pdf · Lightning Talks III Tech Topology Conference December 10, 2017. COVERING SPACES,](https://reader034.vdocuments.site/reader034/viewer/2022043003/5f83b9804067e33dd9466d14/html5/thumbnails/26.jpg)
III. Salem numbers
Salem number
A real algebraic unit, λ > 1, is called a Salem number if λ−1 is a Galoisconjugate, and all other conjugates lie on the unit circle.
Theorem A (P. 2017)
Given a Salem number λ, there are positive integers k , g such that λk isthe stretch factor of a pseudo-Anosov homeomorphism φ : Sg → Sg ,where φ arises from Thurston’s construction. Moreover, g depends only onthe degree of λ over Q.
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 4 / 6
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III. Salem numbers
Salem number
A real algebraic unit, λ > 1, is called a Salem number if λ−1 is a Galoisconjugate, and all other conjugates lie on the unit circle.
Theorem A (P. 2017)
Given a Salem number λ, there are positive integers k , g such that λk isthe stretch factor of a pseudo-Anosov homeomorphism φ : Sg → Sg ,where φ arises from Thurston’s construction. Moreover, g depends only onthe degree of λ over Q.
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 4 / 6
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IV. Connecting Salem numbers to Thurston’s construction
Thurston’s construction requires a collection of curves that cut the surfaceinto disks. The intersection matrix of these curves also plays a crucial role.
Theorem (P. 2017)
Every Salem number λ has a power k such that λk + λ−k is thedominating eigenvalue of an invertible, positive, symmetric, integer matrix.
Theorem (P. 2017)
Given an invertible, positive, integer matrix Q, there is a closed, orientablesurface S along with a collection of curves that cut S into disks, such thatthe intersection matrix of those curves is Q.
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 5 / 6
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IV. Connecting Salem numbers to Thurston’s construction
Thurston’s construction requires a collection of curves that cut the surfaceinto disks. The intersection matrix of these curves also plays a crucial role.
Theorem (P. 2017)
Every Salem number λ has a power k such that λk + λ−k is thedominating eigenvalue of an invertible, positive, symmetric, integer matrix.
Theorem (P. 2017)
Given an invertible, positive, integer matrix Q, there is a closed, orientablesurface S along with a collection of curves that cut S into disks, such thatthe intersection matrix of those curves is Q.
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 5 / 6
![Page 30: Lightning Talks III Tech Topology Conferencepeople.math.gatech.edu/.../2017/SunLightningFinal.pdf · Lightning Talks III Tech Topology Conference December 10, 2017. COVERING SPACES,](https://reader034.vdocuments.site/reader034/viewer/2022043003/5f83b9804067e33dd9466d14/html5/thumbnails/30.jpg)
IV. Connecting Salem numbers to Thurston’s construction
Thurston’s construction requires a collection of curves that cut the surfaceinto disks. The intersection matrix of these curves also plays a crucial role.
Theorem (P. 2017)
Every Salem number λ has a power k such that λk + λ−k is thedominating eigenvalue of an invertible, positive, symmetric, integer matrix.
Theorem (P. 2017)
Given an invertible, positive, integer matrix Q, there is a closed, orientablesurface S along with a collection of curves that cut S into disks, such thatthe intersection matrix of those curves is Q.
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 5 / 6
![Page 31: Lightning Talks III Tech Topology Conferencepeople.math.gatech.edu/.../2017/SunLightningFinal.pdf · Lightning Talks III Tech Topology Conference December 10, 2017. COVERING SPACES,](https://reader034.vdocuments.site/reader034/viewer/2022043003/5f83b9804067e33dd9466d14/html5/thumbnails/31.jpg)
V. Totally real number fields
Methods and results used to prove Theorem A can be adapted to provethe following:
Theorem B (P. 2017)
Every totally real number field is of the form K = Q(λ+ λ−1) where λ isthe stretch factor of a pseudo-Anosov map coming from Thurston’sconstruction.
Thank you!
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 6 / 6
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V. Totally real number fields
Methods and results used to prove Theorem A can be adapted to provethe following:
Theorem B (P. 2017)
Every totally real number field is of the form K = Q(λ+ λ−1) where λ isthe stretch factor of a pseudo-Anosov map coming from Thurston’sconstruction.
Thank you!
J. Pankau (UCSB) Salem Number Stretch Factors 12/10/2017 6 / 6
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Lightning Talks IIITech Topology Conference
December 10, 2017
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Why are there are so many spectralsequences from Khovanov homology?
Adam Saltz (University of Georgia)December 10, 2017
Georgia TechTech Topology Conference
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Spectral sequences galore
Theorem (Ozsvath, Szabo; Bloom; Scaduto; Daemi; Kronhemier, Mrowka)
Let L be a link in S3. Let Σ(L) be the double cover of S3 branchedalong L. There are spectral sequences
Khovanov homology
HeegaardMonopole
Framed instantonPlane
Singular instanton
LeeBar-Natan
Szabo
Floer homologyof Σ(L)
link homology
I am missing a few words like “mirror of” and “reduced.”
![Page 36: Lightning Talks III Tech Topology Conferencepeople.math.gatech.edu/.../2017/SunLightningFinal.pdf · Lightning Talks III Tech Topology Conference December 10, 2017. COVERING SPACES,](https://reader034.vdocuments.site/reader034/viewer/2022043003/5f83b9804067e33dd9466d14/html5/thumbnails/36.jpg)
Spectral sequences galore
Theorem (Ozsvath, Szabo; Bloom; Scaduto; Daemi; Kronhemier, Mrowka)
Let L be a link in S3. Let Σ(L) be the double cover of S3 branchedalong L. There are spectral sequences
Khovanov homology
HeegaardMonopole
Framed instantonPlane
Singular instanton
LeeBar-Natan
Szabo
Floer homologyof Σ(L)
link homology
I am missing a few words like “mirror of” and “reduced.”
![Page 37: Lightning Talks III Tech Topology Conferencepeople.math.gatech.edu/.../2017/SunLightningFinal.pdf · Lightning Talks III Tech Topology Conference December 10, 2017. COVERING SPACES,](https://reader034.vdocuments.site/reader034/viewer/2022043003/5f83b9804067e33dd9466d14/html5/thumbnails/37.jpg)
Khovanov-Floer theories
Definition (Baldwin, Hedden, and Lobb)A Khovanov-Floer theory is a gadget:
Dlink diagram
Ei(D)
spectral sequence
D D′ Fi : Ei(D)→ Ei(D′)map of
spectral sequencesone-handleattachment
• E2(D) = Kh(D)
• F2 agrees with the standard map Kh(D)→ Kh(D′).• Kunneth formula, etc.
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Khovanov-Floer theories: the good
Theorem (Baldwin, Hedden, Lobb)All of the homology theories from the second slide areKhovanov-Floer theories.
Theorem (Baldwin, Hedden, Lobb)Khovanov-Floer theories are
• link invariants.• functorial: they assign maps to isotopy classes of link
cobordisms in S3 × I.
Everything that works for Khovanov homology works forKhovanov-Floer theories because that’s how maps on spectralsequences work.
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Two maps on homology!
one-handle attachment
filtered chain map F
Fi : Ei(D)→ Ei(D′)F∞
F∗
A priori, F∗ 6= F∞!
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A different approach
DefinitionA strong Khovanov-Floer theory is a gadget:
Dlink diagram
K(D)
filtered complex
D D′ F : K(D)→ K(D′)filtered chain maphandle attachment
so that
• For a crossingless diagrams, H(K(D)) agrees with Kh(D) (oranother Frobenius algebra).
• Handle attachment maps satisfy some relations (e.g. swappingdistant handles, Bar-Natan’s S, T, and 4Tu)
• Kunneth formula, etc.
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Strong Khovanov-Floer theories: the good
DefinitionA strong Khovanov-Floer theory is conic if, for D with crossings,
K = cone(h : D0 → D1)
where h is a one-handle attachment map.
Theorem (S.)Conic strong Khovanov-Floer theories are
• link invariants. (chain homotopy type)• functorial: they assign (chain homotopy types of) maps to
isotopy classes of link cobordisms in S3 × I.
Everything that works for Bar-Natan’s cobordism-theoreticconstruction of link homology works for strong Khovanov-Floertheories.
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Strong Khovanov-Floer theories: the good
DefinitionA strong Khovanov-Floer theory is conic if, for D with crossings,
K = cone(h : D0 → D1)
where h is a one-handle attachment map.
Theorem (S.)Conic strong Khovanov-Floer theories are
• link invariants. (chain homotopy type)• functorial: they assign (chain homotopy types of) maps to
isotopy classes of link cobordisms in S3 × I.
Everything that works for Bar-Natan’s cobordism-theoreticconstruction of link homology works for strong Khovanov-Floertheories.
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Strong Khovanov-Floer theories: the good
Theorem (S.)Heegaard Floer homology, singular instanton homology, Szabohomology, and Lee/Bar-Natan homology all produce conic strongKhovanov-Floer theories. (The rest probably are, too.)
Theorem (S.)A conic strong Khovanov-Floer theory yields a Khovanov-Floertheory.
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Strong Khovanov-Floer theories: what’s next
How does this help us understand invariants of transverse linksand contact structures?
What other link homology theories can we use besides Khovanovhomology? (E.g. Lin has constructed a spectral sequence fromBar-Natan-Lee homology to monopole Floer homology)
Can we understand e.g. Heegaard Floer homology via Morsetheory on surfaces?
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Lightning Talks IIITech Topology Conference
December 10, 2017
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Least Dilatation of Pure Surface Braids
Marissa LovingUniversity of Illinois at Urbana-Champaign
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What?
Why?
How?
Who?
When?
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What?
Why?
How?
Who? Me!
When? Now!
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What?
Why?
How? Who? Me!
When? Now!
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• pure mapping classes
• isotopic to the identity on the closed surface
• denoted PBn(Sg)
What are pure surface braids?
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• a real number > 1• associated to a
mapping class 𝑓• denoted 𝜆 𝑓
What is the dilatation?
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What did I prove?
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Theorem (L., 2017)
𝑐 loglog 𝑔𝑛 + 𝑐 ≤ 𝐿 𝑃𝐵1 𝑆3 ≤ 𝑐4 log
𝑔𝑛 + 𝑐4
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Why should we care?
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Theorem (Penner, 1991) L(Mod(Sg)) goes to zero as g goes to infinity.
Theorem (Farb-Leininger-Margalit, 2008)L(Ig) is universally bounded between 0.197 and 4.127.
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Theorem (Dowdall, Aougab—Taylor)
15 log(2𝑔) ≤ 𝐿 𝑃𝐵9 𝑆3 < 4 log(𝑔) + 2 log(24)
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How did I prove it?
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The Upper Bound
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The Upper Bound
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The Upper Bound
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The Lower Bound
𝑃𝐵1(𝑆3) ∋ 𝑓 ↷Imayoshi—Ito—Yamamoto:
“𝜆 𝐹@ ≤ 𝜆(𝑓)”
↶ 𝐹@
↶ 𝐹C@𝑥 𝐹C@(𝑥)
Kra: “max
H𝑑(𝑥, 𝐹C@ 𝑥 ) ≤ 𝜆(𝐹C@)”
𝜋
ΗM
𝜋(𝑥) 𝜋(𝐹C@ 𝑥 )
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The Lower Bound
𝑃𝐵1(𝑆3) ∋ 𝑓 ↷Imayoshi—Ito—Yamamoto:
“𝜆 𝐹@ ≤ 𝜆(𝑓)”
↶ 𝐹@
↶ 𝐹C@𝑥 𝐹C@(𝑥)
Kra: “max
H𝑑(𝑥, 𝐹C@ 𝑥 ) ≤ 𝜆(𝐹C@)”
𝜋
ΗM
𝜋(𝑥) 𝜋(𝐹C@ 𝑥 )
“maxH
d 𝑥, 𝐹@ 𝑥 ≤ 𝜆(𝑓)”
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Theorem (L.—Parlier, 2017)
A filling graph Γ embedded in a surface 𝑆3 has diameter
at least PQR STU
VWX
.
The Lower Bound
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Thank you!
𝑃𝐵1(𝑆3) ∋ 𝑓 ↷
↶ 𝐹@
↶ 𝐹C@𝑥 𝐹C@(𝑥)
𝜋
ΗM
𝜋(𝑥) 𝜋(𝐹C@ 𝑥 )
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Lightning Talks IIITech Topology Conference
December 10, 2017
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Truncated Heegaard Floer homology andconcordance invariants
Linh Truong
Columbia University
Tech Topology Conference, December 2017
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Motivation
Our motivation is to better understand knot concordance.
DefinitionK1 and K2 are concordant if they cobound a smooth cylinder inS3 × [0, 1].
DefinitionThe concordance group is C = {knots in S3/ ∼,#}, whereK1 ∼ K2 if K1 is concordant to K2.
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Open Questions
There are many open questions about knot concordance.
QuestionIs every slice knot a ribbon knot?
Figure: “Square ribbon knot”; figure by David Eppstein, Wikipedia.
The boundary of a self-intersecting disk with only “ribbonsingularities” is called a ribbon knot.
QuestionIs there any torsion in the concordance group C besides 2-torsion?
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Truncated Heegaard Floer homology
Heegaard Floer homology is an invariant for three-manifoldsdefined by Ozsvath and Szabo.
Truncated Heegaard Floer homology, denoted HF n(Y , s)(Ozsvath-Szabo, Ozsvath-Manolescu), is the homology of thekernel CF n(Y , s) of the multiplication map
Un : CF+(Y , s)→ CF+(Y , s)
where n ∈ Z+.
RemarkNote for n = 1, truncated Heegaard Floer homology equalsHF (Y , s).
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Truncated Concordance Invariants
Motivated by the constructions of the Ozsvath-Szabo ν(K ) andHom-Wu ν+(K ), we construct a sequence of knot invariantsνn(K ), n ∈ Z:
DefinitionFor n > 0, define
νn(K ) = min{s ∈ Z | vns : CF n(S3N(K ), ss)→ CF n(S3)
induces a surjection on homology},
where N is sufficiently large so that the Ozsvath-Szabo largeinteger surgery formula holds, and ss denotes the restriction toS3N(K ) of a Spinc structure t on the corresponding 2-handle
cobordism such that
〈c1(t), [F ]〉+ N = 2s,
where F is a capped-off Seifert surface for K .
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Truncated Concordance Invariants, continued...
DefinitionFor n < 0, define
νn(K ) = max{s ∈ Z | vns : CF−n(S3)→ CF−n(S3−N(K ), ss)
induces an injection on homology},
where N is sufficiently large so that the Ozsvath-Szabo largeinteger surgery formula holds, and ss denotes the restriction toS3−N(K ) of a Spinc structure t on the corresponding 2-handle
cobordism such that
〈c1(t), [F ]〉 − N = 2s,
where F is a capped-off Seifert surface for K .For n = 0, we define ν0(K ) = τ(K ).
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Properties of νn(K )
The knot invariants νn(K ), n ∈ Z, satisfy the following properties:
• νn(K ) is a concordance invariant.
• ν1(K ) = ν(K ).
• νn(K ) ≤ νn+1(K ).
• For sufficiently large n, νn(K ) = ν+(K ).
• νn(−K ) = −ν−n(K ), where −K is the mirror of K .
• νn(K ) ≤ g4(K ).
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Homologically thin knots are knots with HFK supported in a singleδ = A−M grading.
TheoremLet K be a homologically thin knot with τ(K ) = τ .
(i) If τ = 0, νn(K ) = 0 for all n.
(ii) If τ > 0,
νn(K ) =
0, for n ≤ −(τ + 1)/2,
τ + 2n + 1, for − τ/2 ≤ n ≤ −1,
τ, for n ≥ 0.
(iii) If τ < 0,
νn(K ) =
τ, for n ≤ 0,
τ + 2n − 1, for 1 ≤ n ≤ −τ/2,
0, for n ≥ (−τ + 1)/2.
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Large Gaps
In fact, the difference between νn(K ) and νn+1(K ) can bearbitrarily big.
TheoremLet Tp,p+1 denote the (p, p + 1) torus knot. For p > 3,
ν−1(Tp,p+1)− ν−2(Tp,p+1) = p.
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Thank you!
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Lightning Talks IIITech Topology Conference
December 10, 2017
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Embedded Exact Lagrangian FillingsImmersed Exact Lagrangian Fillings
Augmentations and Immersed Exact LagrangianFillings
Yu Pan
MIT
Tech Topology ConferenceDec. 10th, 2017
Yu Pan Augmentations and Immersed Exact Lagrangian Fillings
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Embedded Exact Lagrangian FillingsImmersed Exact Lagrangian Fillings
Lagrangian fillingsAugmentations
Exact Lagrangian fillings
An embedded exact Lagrangian filling of Λ is a 2-dimensionalembedded surface L in (Rt × R3, ω = d(etα)) such that
L is cylindrical over Λ when t is bigenough;
there exists a function f : L→ R such
that etα∣∣∣TL
= df and f is constant on
Λ.
t
Λ
L
Yu Pan Augmentations and Immersed Exact Lagrangian Fillings
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Embedded Exact Lagrangian FillingsImmersed Exact Lagrangian Fillings
Lagrangian fillingsAugmentations
Augmentations
By [Ekholm-Honda-Kalman, ’12],an exact Lagrangian filling L =⇒ an augmentation ε of A(Λ)
Λ
L
(A(Λ), ∂
)y(Z2, 0)
ε
Yu Pan Augmentations and Immersed Exact Lagrangian Fillings
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Embedded Exact Lagrangian FillingsImmersed Exact Lagrangian Fillings
Lagrangian fillingsAugmentations
Correspondence
Derived Fukaya Category Augmentation Category
Objects: Exact Lagrangian Fillings Augmentations
However, not all the augmentations of A(Λ) are induced fromembedded exact Lagrangian fillings of Λ.
Yu Pan Augmentations and Immersed Exact Lagrangian Fillings
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Embedded Exact Lagrangian FillingsImmersed Exact Lagrangian Fillings
Lagrangian fillingsAugmentations
Correspondence
Derived Fukaya Category Augmentation Category
Objects: Exact Lagrangian Fillings Augmentations
However, not all the augmentations of A(Λ) are induced fromembedded exact Lagrangian fillings of Λ.
Yu Pan Augmentations and Immersed Exact Lagrangian Fillings
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Embedded Exact Lagrangian FillingsImmersed Exact Lagrangian Fillings
Lagrangian fillingsAugmentations
Correspondence
Derived Fukaya Category Augmentation Category
Objects: Exact Lagrangian Fillings Augmentations
However, not all the augmentations of A(Λ) are induced fromembedded exact Lagrangian fillings of Λ.
Yu Pan Augmentations and Immersed Exact Lagrangian Fillings
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Embedded Exact Lagrangian FillingsImmersed Exact Lagrangian Fillings
Augmentations induced from immersed exact Lagrangian fillings
Immersed Exact Lagrangian fillings
t
Λ
Σ
Yu Pan Augmentations and Immersed Exact Lagrangian Fillings
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Embedded Exact Lagrangian FillingsImmersed Exact Lagrangian Fillings
Augmentations induced from immersed exact Lagrangian fillings
Augmentations induced from immersed exact Lagrangianfillings
Suppose that Σ can be lifted to an embedded Legendrian surface Lin R× R× R3 and A(L) has an augmentation εL.
t
Λ
Σ
(A(Λ), ∂
)(A(L), ∂
)y f
Yu Pan Augmentations and Immersed Exact Lagrangian Fillings
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Embedded Exact Lagrangian FillingsImmersed Exact Lagrangian Fillings
Augmentations induced from immersed exact Lagrangian fillings
Augmentations induced from immersed exact Lagrangianfillings
Suppose that Σ can be lifted to an embedded Legendrian surface Lin R× R× R3 and A(L) has an augmentation εL.
t
Λ
Σ
(A(Λ), ∂
)(A(L), ∂
)yy
(Z2, 0)
εL
fThus ε = εL ◦ f is anaugmentation of A(Λ).
Yu Pan Augmentations and Immersed Exact Lagrangian Fillings
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Embedded Exact Lagrangian FillingsImmersed Exact Lagrangian Fillings
Augmentations induced from immersed exact Lagrangian fillings
Result
Theorem (P.-D. Rutherford)
All the augmentations of A(Λ) are induced from possibly immersedexact Lagrangian fillings of Λ.
Yu Pan Augmentations and Immersed Exact Lagrangian Fillings
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Lightning Talks IIITech Topology Conference
December 10, 2017
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Lightning Talks IIITech Topology Conference
December 10, 2017