daniel mathews- chord diagrams, topological quantum field theory, and the sutured floer homology of...

8/3/2019 Daniel Mathews- Chord diagrams, topological quantum field theory, and the sutured Floer homology of solid tori

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Background Contact elements in SFH(T) Contact geometry applications Idea of proof

Chord diagrams,

topological quantum field theory,

and the sutured Floer homology of solid tori

arXiv:0903.1453v1 [math.GT]

Daniel Mathews

Stanford University

[email protected]

Columbia University

17 April 2009
http://find/http://goback/


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Outline

1 BackgroundSutured Floer homology, contact elements, TQFT

Solid tori, Catalan, Narayana

2 Contact elements in SFH(T)

Computation, addition of contact elementsCreation operators, basis of contact elements

Partial order, main theorem

3 Contact geometry applications

StackabilityContact 2-category

4 Idea of proof of main theorems

Comparable pairs and bypass systems


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Outline











4/110


Sutured manifolds & SFHJuhsz 2006, Holomorphic discs and sutured manifolds

Sutured manifold (M,)

M 3-manifold with boundary.

collection of disjoint simple closed curves on boundary,dividing M into positive/negative regions.

(Balanced.)

(M,) SFH(M, )

Take sutured Heegaard decomposition, symmetric product

of Heegaard surface.Chain complex generated by intersection points of , tori.

Differential counts certain holomorphic curves in

symmetric product with certain boundary conditions.

Invariant of (balanced) sutured manifold.


5/110


Sutured manifolds & SFHJuhsz 2006, Holomorphic discs and sutured manifolds

Sutured manifold (M,)

M 3-manifold with boundary.

collection of disjoint simple closed curves on boundary,dividing M into positive/negative regions.

(Balanced.)

(M,) SFH(M, )

Take sutured Heegaard decomposition, symmetric product

of Heegaard surface.Chain complex generated by intersection points of , tori.

Differential counts certain holomorphic curves in

symmetric product with certain boundary conditions.

Invariant of (balanced) sutured manifold.

C S ( ) C f f


6/110


Contact elementsClosed case OzsvthSzab 2005, HondaKazezMatic 2007; sutured case

HondaKazezMatic 2007, The contact invariant in sutured Floer homology

Contact structutre on sutured manifold

contact structure on (M, ):

M convex

dividing set

Positive/negative regions.

Theorem (HondaKazezMatic)

A contact structure on(M, ) gives a well-definedcontact element c() SFH(M, ).

We take Z2 coefficients throughout.

WithZ

coefficients, c() subset of form {x}.

B k d C t t l t i SFH(T ) C t t t li ti Id f f


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M convex

dividing set



A contact structure on(M, ) gives a well-definedcontact elementc() SFH(M, ).


WithZ

coefficients, c() subset of form {x}.

Background Contact elements in SFH(T ) Contact geometry applications Idea of proof


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M convex

dividing set



A contact structure on(M, ) gives a well-definedcontact element c() SFH(M, ).


With Z coefficients, c(

)subset of form

{x

}.



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Properties of SFH

Contact element properties:

(HKM 2007, The contact invariant in sutured Floer homology)

overtwisted c() = 0

(M, , ) embeds in closed (N, ) withc() = 0 c() = 0.

Every generator of chain complex has a spin-c structure s.

SFH splits over spin-c structures:

SFH(M, ) =s

SFH(M,, s).



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Properties of SFH

Contact element properties:

(HKM 2007, The contact invariant in sutured Floer homology)

overtwisted c() = 0

(M, , ) embeds in closed (N, ) withc() = 0 c() = 0.

Every generator of chain complex has a spin-c structure s.

SFH splits over spin-c structures:

SFH(M, ) =s

SFH(M,, s).



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TQFT propertyHondaKazezMatic 2008, Contact structures, sutured Floer homology and TQFT

Theorem

Given

(M,) (M, ) inclusion of sutured manifolds.

contact structure on (M M

,

)there is a natural map

SFH(M,) SFH(M,).

Further c() c( ).

TQFT-inclusion.

(Actually SFH(M, ) Vm where m is number of isolated

components).



12/110


TQFT propertyHondaKazezMatic 2008, Contact structures, sutured Floer homology and TQFT

Theorem

Given

(M,) (M, ) inclusion of sutured manifolds.

contact structure on(M

M,

)there is a natural map

SFH(M,) SFH(M,).

Further c() c( ).

TQFT-inclusion.

(Actually SFH(M, ) Vm where m is number of isolated

components).



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g ( ) g y pp p

The question

Motivating question:

How do contact elements lie in SFH?



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g ( ) g y pp p

Outline












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Solid tori

Only sutured 3-manifolds we consider are solid tori.

Sutured manifold (T, n)

Solid torus D2 S1

Convex boundary D2 S1

Longitudinal dividng set F S1,F finite, |F| = 2n.

(Notational cover-up: (T, n) = ((D2 S1), (F S1)).)

Part of the (1 + 1)-dimensional TQFT discussed in HKM2008.

To classify contact structures:

consider dividing sets on convex meridian disc and

boundary torus


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Convex surfacesGiroux 1991, Convexit en topologie de contact

Generic property for embedded surface in contact 3-manifold.

Convex surface S

There exists a contact vector field X transverse to S.

Invariant vertical direction.Dividng set

= {x S : X(x) }.

Where is perpendicular.Dividing set divides S into positive/negative regions S.

Euler class evaluation:

e()[S] = (S+) (S).



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Convex surface S



= {x S : X(x) }.



e()[S] = (S+) (S).



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Convex surface S



= {x S : X(x) }.



e()[S] = (S+) (S).



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Contact structure near a convex surfaceGiroux 1991, Convexit en topologie de contact;

Honda 2000, On the classification of tight contact structures. I

Theorem (Giroux)

The dividing set essentially determines the contact structure

near a convex surface.

Given S, , is the nearby contact structure tight?

For S = S2:

Contact structure is tight iff has no contractible components.

For S = S2:

Contact structure is tight iff has one component.

If S2 = B3, tight contact structure near boundary extends

uniquely over ball (Eliashberg).



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Contact structure near a convex surfaceGiroux 1991, Convexit en topologie de contact;

Honda 2000, On the classification of tight contact structures. I

Theorem (Giroux)

The dividing set essentially determines the contact structure

near a convex surface.

Given S, , is the nearby contact structure tight?

For S = S2:

Contact structure is tight iff has no contractible components.

For S = S2:

Contact structure is tight iff has one component.

If S2 = B3, tight contact structure near boundary extends

uniquely over ball (Eliashberg).



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Corners & roundingHonda 2000, On the classification of tight contact structures. I

When convex surfaces meet transversely along a

legendrian curve, dividing sets interleave.

Corners can be rounded in a standard way.

Figure: Rounding corners.



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Contact structures on (T, n) and chord diagrams

Dividing set on meridional disc (convex, leg. bdy)

Interleaves with sutures F S1 on boundary; 2nendpoints.

For tight contact structure, has no closed components.

Chord diagram

Collection of disjoint properly embedded arcs on disc.

Up to homotopy rel endpoints.

E.g.

+

++

+



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Contact structures on (T, n) and chord diagrams

Dividing set on meridional disc (convex, leg. bdy)

Interleaves with sutures F S1 on boundary; 2nendpoints.

For tight contact structure, has no closed components.

Chord diagram

Collection of disjoint properly embedded arcs on disc.

Up to homotopy rel endpoints.

E.g.

+

++

+



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Euler class of chord diagram

Chord diagram has relative euler class e.

|e| n 1, e+ n 1 mod 2.

+

-

+

-+

-

+

Figure: Basepoint, convention for signs of regions.



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Contact structures on (T, n) arechord diagramsHonda 2000, On the classification of tight contact structures. II;

Honda 2002, Gluing tight contact structures;

Giroux 2001, Structures de contact sur les varits fibres en cercles...

Chord diagram determines at most one tight contact structure

on D2 S1:

Cut into solid cylinder, round corners of D3

For solid tori in general:Chord diagrams on D may give overtwisted contact

structure on D2 S1

Distinct chord diagrams may give isotopic contact

structures.

However with longitudinal suturesof (T, n), neither occurs.

Theorem (Honda, Giroux)

Tight contact structures

on(T, n)

Chord diagrams

with n chords



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on D2 S1:



structure on D2 S1


structures.




on(T, n)

Chord diagrams

with n chords



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on D2 S1:



structure on D2 S1


structures.




on(T, n)

Chord diagrams

with n chords



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Catalan and Naryana numbers

#

Tight contactstructures on (T, n)

= #

Chord diagrams

n chords

Catalan numbers Cn = 1, 1, 2, 5, 14, 42, 132, 429, . . .

#

Tight contact

structures on (T, n)euler class e

= #

Chord diagrams

n chords

euler class e

Narayana numbers Cen:

1

1 1

1 3 1

1 6 6 1

1 10 20 10 1



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Catalan and Naryana numbers

#

Tight contactstructures on (T, n)

= #

Chord diagrams

n chords

Catalan numbers Cn = 1, 1, 2, 5, 14, 42, 132, 429, . . .

#

Tight contact

structures on (T, n)euler class e

= #

Chord diagrams

n chords

euler class e

Narayana numbers Cen:

1

1 1

1 3 1

1 6 6 1

1 10 20 10 1



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The Catalan diseaseand Narayana symptoms

Catalan:

tight ct. strs on (T, n)

chord diagrams, nchords

pairings of n brackets

Dyck paths length 2n

rooted planar bin. trees

Recursion

Cn+1 =

n1+n2=n

Cn1 Cn2 .

Narayana:

# with euler class e


# with k occurrences of ()

# with k peaks

# with k left leaves

Recursion:

Cen+1 =

n1+n2=n

e1+e2=e

Ce1n1 Ce2n2

.



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The Catalan diseaseand Narayana symptoms

Catalan:

tight ct. strs on (T, n)

chord diagrams, nchords

pairings of n brackets

Dyck paths length 2n

rooted planar bin. trees

Recursion

Cn+1 =

n1+n2=n

Cn1 Cn2 .

Narayana:



# with k occurrences of ()

# with k peaks

# with k left leaves

Recursion:

Cen+1 =

n1+n2=n

e1+e2=e

Ce1n1 Ce2n2

.



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Outline











C S ( )


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Computation of SFH(T, n)Juhsz 2008, Floer homology and surface decompositions

HondaKazezMatic 2008, Contact structures, sutured Floer homology and TQFT

Theorem

SFH(T, n+ 1) = Z2n

2 .

Split over spin-c structures:

SFH(T, n+ 1) =

k

Z(nk)2 .

For with euler class e,

c() Z(nk)2 where k =

e+ n

2

so let

SFH(T, n+ 1, e) =Z

(nk)

2 .


C i f SFH(T )


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Computation of SFH(T, n)Juhsz 2008, Floer homology and surface decompositions

HondaKazezMatic 2008, Contact structures, sutured Floer homology and TQFT

Theorem

SFH(T, n+ 1) = Z2n

2 .

Split over spin-c structures:

SFH(T, n+ 1) =

k

Z(nk)2 .

For with euler class e,

c() Z(nk)2 where k =

e+ n

2

so let

SFH(T, n+ 1, e) =Z

(nk)

2 .



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Categorified Pascal triangle

SFH(T, 1) = SFH(T, 1, 0)SFH(T, 2) = SFH(T, 2, 1) SFH(T, 2, 1)SFH(T, 3) = SFH(T, 3, 2) SFH(T, 3, 0) SFH(T, 3, 2)SFH(T, 4) = SFH(T, 4, 3) SFH(T, 4, 1) SFH(T, 4, 1) SFH(T, 4, 3)

SFH(T, 1) = Z(00)2

SFH(T, 2) = Z(10)2 Z

(11)2

SFH(T, 3) = Z(2

0

)2 Z

(21

)2 Z

(22

)2

SFH(T, 4) = Z(30)2 Z

(31)2 Z

(32)2 Z

(33)2


C l d P l i l


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Catalan and Pascal triangle

Contact elements in each SFH(T, n, e) form a distinguishedsubset of order Cen.

11 1

1 3 1

1 6 6 1

Z1

2Z

12 Z

12

Z12 Z

22 Z

12

Z12 Z

32 Z

32 Z

12

Question:

How do the Cen contact elements lie in Z(nk)2 ?


C l d P l i l


38/110

Catalan and Pascal triangle

Contact elements in each SFH(T, n, e) form a distinguishedsubset of order Cen.

11 1

1 3 1

1 6 6 1

Z1

2Z

12 Z

12

Z12 Z

22 Z

12

Z12 Z

32 Z

32 Z

12

Question:

How do the Cen contact elements lie in Z(nk)2 ?


Additi d b


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Addition and bypasses

Are the contact elements a subgroup?

No.Closure under addition described by bypasses.

c

Figure: Upwards bypass surgery along arc c.

c

Figure: Downwards bypass surgery along arc c.


Additi d b


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c


c



Addition and b passes


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c


c



The bypass relation


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The bypass relation

Bypass-related chord diagrams naturally come in triples.Proposition

Suppose a, b SFH(T, n, e) are contact elements.Then a+ b is a contact element if and only if a, b are related by

a bypass surgery.In this case, a+ b is the third chord diagram in the triple.

+ + = 0

Figure: Bypass relation.


The bypass relation


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The bypass relation

Bypass-related chord diagrams naturally come in triples.Proposition

Suppose a, b SFH(T, n, e) are contact elements.Then a+ b is a contact element if and only if a, b are related by

a bypass surgery.In this case, a+ b is the third chord diagram in the triple.

+ + = 0

Figure: Bypass relation.


Reduction to kindergarten


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In fact one can show:

Proposition (SFH is combinatorial)

SFH(T, n, e) = Z2 Chord diags, n chords, euler class eBypass relation

Also:

There is a basis of contact elements.

Distinct contact structures / chord diagrams all give distinct

contact elements.




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In fact one can show:

Proposition (SFH is combinatorial)

SFH(T, n, e) = Z2 Chord diags, n chords, euler class eBypass relation

Also:

There is a basis of contact elements.

Distinct contact structures / chord diagrams all give distinct

contact elements.


Outline


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Outline







Stackability

Contact 2-category




Creation operators


47/110

Creation operators

A well-defined way to create create chords, enclosingpositive/negative regions at the basepoint.

- +

B B+

Figure: Creation operators.

We obtain maps

B : SFH(T, n, e) SFH(T, n+ 1, e 1).


Creation operators


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Creation operators

A well-defined way to create create chords, enclosingpositive/negative regions at the basepoint.

- +

B B+

Figure: Creation operators.

We obtain maps

B : SFH(T, n, e) SFH(T, n+ 1, e 1).


Origin of creation operators


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Origin of creation operators

B arise from TQFT-inclusion

(T, n) (T, n+ 1)

with intermediate contact structure

+

B+B

Figure: Creation operator inclusion.


Annihilation operators


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Annihilation operators

Similarly

A : SFH(T, n+ 1, e) SFH(T, n, e 1)

A A+

Figure: Annihilation operators.

A

+

A+

Figure: Annihilation operator inclusion.


Morphisms in Pascals triangle


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p gFull categorification

Z(00)2

B

A+

A

B+

Z(1

0)2 Z(1

1)2B

A+

A

B+

B

A+

A

B+

Z(20)2 Z

(21)2 Z

(22)2

B

A+

A

B+

B

A+

A

B+

B

A+

A

B+

Z(30)2 Z

(31)2 Z

(32)2 Z

(33)2


Operator relations, Pascal recursion


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Proposition (Creation/annihilation relations)

A+ B = A B+ = 1A+ B+ = A B = 0

Proposition (Categorification of Pascal recursion)

SFH(T, n+ 1, e) = B+SFH(T, n, e 1) BSFH(T, n, e+ 1)

I.e.:

Z(n+1k )2 = B+Z

( nk1)2 BZ

(nk)2




53/110

p ,

Proposition (Creation/annihilation relations)

A+ B = A B+ = 1A+ B+ = A B = 0

Proposition (Categorification of Pascal recursion)

SFH(T, n+ 1, e) = B+SFH(T, n, e 1) BSFH(T, n, e+ 1)

I.e.:

Z(n+1k )2 = B+Z

( nk1)2 BZ

(nk)2


QFT analogy


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gy

SFH(T, n+ 1, e) = n-particle states of charge e

Figure: The vacuum v SFH(T, 1, 0) = Z2.

Basis: apply creation operators to the vacuum

W(n, n+) = {Words on {,+}, n signs, n+ + signs}

For w W(n, n+), form vw SFH(T, n+ 1, e).

(n = n + n+, e = n+ n.)

Background Contact elements in SFH(

T)

Contact geometry applications Idea of proof

QFT analogy


55/110

gy

SFH(T, n+ 1, e) = n-particle states of charge e

Figure: The vacuum v SFH(T, 1, 0) = Z2.

Basis: apply creation operators to the vacuum

W(n, n+) = {Words on {,+}, n signs, n+ + signs}

For w W(n, n+), form vw SFH(T, n+ 1, e).

(n = n + n+, e = n+ n.)


QFT basis


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E.g.

v+++ = BB+BB+B+v SFH(T, 6, 1).

Proposition

The vw, w W(n, n+), form a basis for SFH(T, n+ 1, e).


QFT basis


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E.g.

v+++ = BB+BB+B+v SFH(T, 6, 1).

Proposition



QFT basis


58/110

E.g.

v+++ = BB+BB+B+v SFH(T, 6, 1).

+

Proposition



QFT basis


59/110

E.g.

v+++ = BB+BB+B+v SFH(T, 6, 1).

+

Proposition



QFT basis


60/110

E.g.

v+++ = BB+BB+B+v SFH(T, 6, 1).

+

+

Proposition



QFT basis


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E.g.

v+++ = BB+BB+B+v SFH(T, 6, 1).

+

+ +

Proposition



QFT basis


62/110

E.g.

v+++ = BB+BB+B+v SFH(T, 6, 1).

+

+ +

Proposition



QFT basis


63/110

E.g.

v+++ = BB+BB+B+v SFH(T, 6, 1).

+

+ +

Proposition



Basis decomposition


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+

+

+

-

--


Basis decomposition


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+

+

+

-

--

++

+

-

--

+

+

+

-

--

+=


Basis decomposition


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+

+

+

-

--

++

+

-

--

+

+

+

-

--

+=

+

+

+ --

+

+

-

-

-

= B + B+


Basis decomposition


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+

+

+

-

--

++

+

-

--

+

+

+

-

--

+=

+

+

+ --

+

+

-

-

-

= B + B+

+

++

-

-

+

++

-

-

++ - -= B + B + B+B


Basis decomposition


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+

+

+

-

--

++

+

-

--

+

+

+

-

--

+=

+

+

+ --

+

+

-

-

-

= B + B+

+

++

-

-

+

++

-

-

++ - -= B + B + B+B

= BBB+B+v + BB+B+Bv + B+B(BB+v + B+Bv)= v++ + v++ + v++ + v++


Examples of basis decomposition


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++




70/110

++

+-

-

+-

++




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++

+-

-

+-

++

+

+

+

-

--

+ + + + + ++ +




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++

+-

-

+-

++

+

+

+

-

--

+ + + + + ++ +

+

++

+

+ + + + + + + + + + + + + + ++ + +

+

+

++ + + + + + + + +


Outline


73/110

1 Background

Sutured Floer homology, contact elements, TQFT


2 Contact elements in SFH(T)Computation, addition of contact elements

Creation operators, basis of contact elements



Stackability

Contact 2-category




Orderings on W(n, n+)


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Lexicographic ordering: Total order.

Partial order : All minus signs move right (or stay wherethey are).

E.g.

++ + +

but

+ +, + + not comparable.

Theorem

Write a contact element v as a sum of basis vectors

v =

w

vw, w W(n, n+).

Let w, w+ be (lex.) first and last words occurring.

Then for all words w in decomposition, w w w+.




75/110



E.g.

++ + +

but


Theorem


v =

w

vw, w W(n, n+).






76/110



E.g.

++ + +

but


Theorem


v =

w

vw, w W(n, n+).




Chord diagram = comparable pair


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Now have

: {Contact elements} {Comparable pairs of words}

v =

w

vw (w, w+)

Proposition

These sets have the same cardinality.

I.e. # comparable pairs of words = Cen.

Theorem

is a bijection.

I.e. for any w w+ ! contact element with vw

first, vw+ last.




78/110

Now have


v =

w

vw (w, w+)

Proposition

These sets have thesame cardinality.


Theorem

is a bijection.


first, vw+ last.




79/110

Now have


v =

w

vw (w, w+)

Proposition

These sets have the same cardinality.


Theorem

is abijection.


first, vw+ last.


Other properties of contact elements


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Notation v = [w, w+].

Proposition

The number of terms in the basis decomposition of a contact

element v is

1 if v is a basis element.even otherwise.

Theorem (Not much comparability)

Suppose vw occurs in the basis decomposition of the contact

element v = [w, w+].Suppose w is comparable with every other element in the

decomposition.

Then w = w or w+.




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Proposition


element v is





decomposition.

Then w = w or w+.




82/110


Proposition


element v is





decomposition.

Then w = w or w+.



N i [ ]


83/110


Proposition


element v is





decomposition.

Then w = w or w+.




84/110

++

+-

-

+-

++

+

+

+

-

--

+ + + + + ++ +

+

++

+

+ + + + + + + + + + + + + + ++ + + + + +

+

+ +

+ + + + + +


Summary of results


85/110

Distinct chord diagrams/contact structures give distinct

contact elements.

Contact elements not a subgroup, but addition means

bypasses.Can give a basis for each SFH(T, n, e) consisting of chorddiagrams / contact elements.

There is a partial order on each basis.

Chord diagrams / contact structures correspond precisely

to comparable pairs of basis elements.


Outline


86/110

1 Background

Sutured Floer homology, contact elements, TQFTSolid tori, Catalan, Narayana


Creation operators, basis of contact elementsPartial order, main theorem


Stackability

Contact 2-category




Stacking construction

Given chord diagrams consider M( ):


87/110

Given 0, 1 chord diagrams, consider M(0,1):

sutured solid cylinder D Ii sutures along D {i}

Vertical interleaving sutures along D I.

1

0

Figure: M(0, 1).

M(0, 1) is tight if it admits a tight contact structure.I.e. after rounding corners, sutures form single component.





88/110




1

0

Figure: M(0, 1).






89/110




1

0

Figure: M(0, 1).



Stackability constructions

Proposition (Stackability map)


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There is a linear map

m : SFH(T, n, e) SFH(T, n, e) Z2

taking(0, 1) to1 ifM(0,1) is tight, and0 if overtwisted.

Proposition (Contact interpretation of )

M(w0 , w1 ) is tight iff w0 w1.

Proposition (General stackability)0, 1 chord diagrams, n chords, euler class e.

M(0, 1) is tight #

(w0, w1) :

w0 w1wi occurs ini

is odd.





91/110







Proposition (General stackability)

0, 1 chord diagrams, n chords, euler class e.

M(0, 1) is tight #

(w0, w1) :

w0 w1wi occurs ini

is odd.





92/110







Proposition (General stackability)

0, 1 chord diagrams, n chords, euler class e.

M(0, 1) is tight #

(w0, w1) :

w0 w1wi occurs ini

is odd.


93/110


Outline

B k d


94/110

1 Background





Stackability

Contact 2-category




Contact categoryHonda (unpublished...)


95/110

surface.

Contact category C()

Objects:

Dividing sets on (= Contact structures near )

Morphisms 0 1:Contact structures on I with {i} = i.

Properties:

Behaves functorially w.r.t. SFH.

Obeys some of the axioms of a triangulated category:

Distinguished triangles = bypass triplesOctahedral axiom 6 contact elements inSFH(T, 4, 1) = Z32.



f


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surface.


Objects:



Properties:






f


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surface.


Objects:



Properties:






f


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surface.


Objects:



Properties:





A 2-category

On D2 have C(D2, n, e)


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C( , , )

(restrict to chord diagrams, n chords, euler class e):Objects in C(D2, n, e) (= chord diagrams) given by partialorder [w, w+].

A partial order is a category.

Contact 2-category C(n+ 1, e)

Objects = words in W(n, n+) = basis chord diagrams

1-morphisms = {partial order } = chord diagrams

2-morphisms = contact structures on M(0, 1).

Proposition

This is a 2-category.


A 2-category



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( , , )







Proposition



A 2-category



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( , , )







Proposition



Outline

1 Background


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1 Background





Stackability

Contact 2-category




An explicit construction

Prove correspondence


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Chord diagrams

Comparable pairs

of words

Essential idea:

Given w1 w2, construct a chord diagram whosedecomposition has w1 first and w2 last.

Along the way, show that every other word w in the

decomposition has w1 w w2.Elementary combinatorics gives

# {pairs (w1 w2)} = Cen.

Done.





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Chord diagrams

Comparable pairs

of words

Essential idea:



decomposition has w1 w w2.Elementary combinatorics gives


Done.





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Chord diagrams

Comparable pairs

of words

Essential idea:



decomposition has w1

w

w2

.

Elementary combinatorics gives


Done.


Bypass systems

Take w1 ,w2 basis chord diagrams, w1 w2.


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Proposition1 Onw1 there exists a bypass system FBS(w1 ,w2 ) such

that performingupwardsbypass moves along it givesw2 .

2 Onw2 there exists a bypass system BBS(w1 , w2 ) such

that performingdownwardsbypass moves givesw1 .

Proposition

Performing either:

1

downwardsbypass moves onw1 along FBS(w1, w2 ), or2 upwardsbypass moves onw2 along BBS(w1, w2 )

gives a chord diagram containing w1, w2 in decomposition and:

for all words w in the decomposition, w1 w w2.


Bypass systems

Take w1 ,w2 basis chord diagrams, w1 w2.


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Proposition1 Onw1 there exists a bypass system FBS(w1 ,w2 ) such

that performingupwardsbypass moves along it givesw2 .

2 Onw2 there exists a bypass system BBS(w1 , w2 ) such

that performingdownwardsbypass moves givesw1 .

Proposition

Performing either:

1

downwardsbypass moves onw1 along FBS(w1, w2 ), or2 upwardsbypass moves onw2 along BBS(w1, w2 )

gives a chord diagram containing w1, w2 in decomposition and:

for all words w in the decomposition, w1 w w2.


Proof by increasingly difficult exampleEasy level


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Elementary move on word = Bypass move on attaching arc.

+

+

+

+

+ +

+

++

+

Up

Figure: Upwards move from ++++ to ++++.


Proof by incresingly difficult exampleMedium level


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Generalized elementary

move on word

=

Bypass moves ongeneralized attaching arc

Up

+

+

+

+

+

-

-

-

-

+

+

+

-

-

-

- +

+

-

Figure: Upwards moves from ++++ to ++++.


Proof by increasingly difficult exampleHard level


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Nicely ordered sequenceof generalized elementary

moves on word

=

Bypass moves onwell placed sequence of

generalized attaching arcs

Up

+

+

+

+

-

-

-

-

+

+

+

+

-

-

-

-

Figure: Upwards moves from +++ to +++.

daniel mathews- chord diagrams, topological quantum field theory, and the sutured floer homology of...

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