lightning talks iii tech topology...

Lightning Talks IIITech Topology Conference

December 10, 2017

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

3-Fold Branched Cover

Symmetric Mapping Class Group

SMod(S2

) = NMod(S2)

Symplectic Representation

� : Mod(S2) ! Sp(4,Z)

� : R 7! E =

0 0 �1 00 �1 0 �11 0 �1 00 1 0 0

Example:

McMullen’s Question

Question: Is finite index in ?

Venkataramana: Yes, if # branch points ≥ 2*degree

�(SMod(Sg))

NSp(2g,Z)(�(hRdi))

Main Theorem:

�(SMod(S2)) = NSp(4,Z)(hEi)�(SMod(S2)) = NSp(4,Z)(hEi)

Main Theorem:

�(SMod(S2)) = NSp(4,Z)(hEi)

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

MATLAB Output

�z0 z1 � z2 z0 + z3 z1�z4 � z5 �z6 z4 z7 � z6

z3 z1 z0 z2z4 z7 z5 z6

775M =

Refine using the symplectic condition

M⌦MT = ⌦

2x 1 �x 0�1 0 0 0x 0 �2x �10 0 1 0

2x �1 �x 01 0 0 0x 0 �2x 10 0 �1 0

0 1 0 0�1 �2x 0 �x

0 0 0 �10 x 1 2x

0 �1 0 01 �2x 0 �x

0 0 0 10 x �1 2x

x 1 x 10 0 �1 0

2x 1 �x 0�1 0 1 0

x �1 x �10 0 1 0

2x �1 �x 01 0 �1 0

0 1 0 10 x �1 2x0 1 0 0

�1 x 1 �x

0 �1 0 �10 x 1 2x0 �1 0 01 x �1 �x

0 x 1 2x0 0 0 11 2x 0 x

0 1 0 0

0 x �1 2x0 0 0 �1

�1 2x 0 x

0 �1 0 0

0 0 1 0x 0 �2x 11 0 0 0

�2x 1 x 0

0 0 �1 0x 0 �2x �1

�1 0 0 0�2x �1 x 0

MATLAB Output

Ghaswala-Winarski give generators for . SMod(S2)

�( eT↵) =

1 0 0 02 1 �1 00 0 1 0

�1 0 2 1

How can we obtain this matrix from ?

2x 1 �x 0�1 0 0 0x 0 �2x �10 0 1 0

775 = �(eh�

) · �( eT↵

)x · �(ehc

2x 1 �x 0�1 0 0 0x 0 �2x �10 0 1 0

775 = �(eh�

) · �( eT↵

)x · �(ehc

)�⇣fH�

� fT↵

� fHc

H� Hc = H⌘ �H◆T↵

Main Theorem:

Thank you!

December 10, 2017

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

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?

I. Background

Main Question

I. Background

Main Question

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.

II. Constructions

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.

II. Constructions

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.

II. Constructions

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.

II. Constructions

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.

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.

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.

Theorem (P. 2017)

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!

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!

December 10, 2017

Why are there are so many spectralsequences from Khovanov homology?

Adam Saltz (University of Georgia)December 10, 2017

Georgia TechTech Topology Conference

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

Floer homologyof Σ(L)

link homology

I am missing a few words like “mirror of” and “reduced.”

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

Floer homologyof Σ(L)

link homology

I am missing a few words like “mirror of” and “reduced.”

Khovanov-Floer theories

Definition (Baldwin, Hedden, and Lobb)A Khovanov-Floer theory is a gadget:

Dlink diagram

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.

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.

Two maps on homology!

one-handle attachment

filtered chain map F

Fi : Ei(D)→ Ei(D′)F∞

A priori, F∗ 6= F∞!

A different approach

DefinitionA strong Khovanov-Floer theory is a gadget:

Dlink diagram

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.

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.

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.

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.

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?

December 10, 2017

Least Dilatation of Pure Surface Braids

Marissa LovingUniversity of Illinois at Urbana-Champaign

Who? Me!

When? Now!

How? Who? Me!

When? Now!

• pure mapping classes

• isotopic to the identity on the closed surface

• denoted PBn(Sg)

What are pure surface braids?

• a real number > 1• associated to a

mapping class 𝑓• denoted 𝜆 𝑓

What is the dilatation?

What did I prove?

Theorem (L., 2017)

𝑐 loglog 𝑔𝑛 + 𝑐 ≤ 𝐿 𝑃𝐵1 𝑆3 ≤ 𝑐4 log

𝑔𝑛 + 𝑐4

Why should we care?

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.

Theorem (Dowdall, Aougab—Taylor)

15 log(2𝑔) ≤ 𝐿 𝑃𝐵9 𝑆3 < 4 log(𝑔) + 2 log(24)

How did I prove it?

The Upper Bound

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(𝑥) 𝜋(𝐹C@ 𝑥 )

The Lower Bound

𝑃𝐵1(𝑆3) ∋ 𝑓 ↷Imayoshi—Ito—Yamamoto:

“𝜆 𝐹@ ≤ 𝜆(𝑓)”

↶ 𝐹@

Kra: “max

H𝑑(𝑥, 𝐹C@ 𝑥 ) ≤ 𝜆(𝐹C@)”

“maxH

d 𝑥, 𝐹@ 𝑥 ≤ 𝜆(𝑓)”

Theorem (L.—Parlier, 2017)

A filling graph Γ embedded in a surface 𝑆3 has diameter

at least PQR STU

The Lower Bound

Thank you!

𝑃𝐵1(𝑆3) ∋ 𝑓 ↷

↶ 𝐹@

December 10, 2017

Truncated Heegaard Floer homology andconcordance invariants

Linh Truong

Columbia University

Tech Topology Conference, December 2017

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.

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?

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).

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 .

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 ).

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 ).

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.

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.

Thank you!

December 10, 2017

Embedded Exact Lagrangian FillingsImmersed Exact Lagrangian Fillings

Augmentations and Immersed Exact LagrangianFillings

Yu Pan

Tech Topology ConferenceDec. 10th, 2017

Yu Pan Augmentations and Immersed 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

Augmentations

By [Ekholm-Honda-Kalman, ’12],an exact Lagrangian filling L =⇒ an augmentation ε of A(Λ)

(A(Λ), ∂

)y(Z2, 0)

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.

Correspondence

Augmentations induced from immersed exact Lagrangian fillings

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.

(A(Λ), ∂

)(A(L), ∂

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.

(A(Λ), ∂

)(A(L), ∂

(Z2, 0)

fThus ε = εL ◦ f is anaugmentation of A(Λ).

Result

Theorem (P.-D. Rutherford)

All the augmentations of A(Λ) are induced from possibly immersedexact Lagrangian fillings of Λ.

December 10, 2017

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