An Introduction to Invariants of Legendrian...

An Introduction toInvariants of Legendrian Knots

Joshua M. Sabloff

Haverford College

AIM Workshop on Legendrian and Transverse Knots

Survey some classical and non-classical invariants of Legendrianknots with an eye towards:

• Combinatorial definitions and geometric motivations for theinvariants

• Connections between the invariants• Extensions of the invariants

• Connections between the invariants

• Extensions of the invariants

Outline

1 Classical Invariants

2 Combinatorial Definitions of InvariantsNormal RulingsThe Chekanov-Eliashberg DGAComputable Invariants from the DGA

3 Geometry of the InvariantsGenerating FamiliesLegendrian Contact Homology

4 Extensions

Outline

4 Extensions

Outline

4 Extensions

Outline

4 Extensions

Where Are We?

4 Extensions

Classical Invariants

Diagrams

• Work in the standard contact 3-space (R3, ξ0 = dz − y dx).

• Use both the Lagrangian (xy ) and front (xz) projections:

xz projection

xy projection

Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 5 / 45

Diagrams

• Work in the standard contact 3-space (R3, ξ0 = dz − y dx).• Use both the Lagrangian (xy ) and front (xz) projections:

xz projection

xy projection

Resolution of Front Diagrams

We can “resolve” a front diagram into a Lagrangian diagram of anisotopic knot:

tb and r

• The Thurston-Bennequin number tbmeasures difference betweenSeifert framing and framing from ξ

• tb may be computed from thewrithe of the xy diagram

• . . . or writhe−#right cusps in thexz diagram

• The rotation number r measurestwisting of K ′(t) inside a trivializationof ξ

• r may be computed from therotation number of the xy diagram

• . . . or #down cusps−#up cuspsin the xz diagram

• tb = −2• r = ±1

tb and r

• tb = −2

• r = ±1

tb and r

• tb = −2

• r = ±1

tb and r

• tb = −2

• r = ±1

tb and r

• tb = −2• r = ±1

tb and r

• tb = −2• r = ±1

Bounds on tb and r

The classical invariants are restricted by the underlying smooth knottype:

• tb(K ) + |r(K )| ≤ 2g(K )− 1 [Bennequin]

• tb(K ) + |r(K )| ≤ 2gs(K )− 1 [Rudolph]

• tb(K ) + |r(K )| ≤ min degaHK (a, z)− 1 [Morton, Franks-Williams,

Fuchs-Tabachnikov, . . . ]

• tb(K ) ≤ min degaFK (a, z)− 1 [Tabachnikov, Fuchs-Tabachnikov,

Chmutov-Goryunov, . . . ]

• And others using τ(K ), s(K ), Khovanov homology, etc.

Bounds on tb and r

Geography

To what extent do the classical invariants classify Legendrian knots ina given knot type?

• tb and r completely classify Legendrian unknots [Eliashberg-Fraser],torus knots and the figure eight knot [Etnyre-Honda], . . . . Call theseknot types simple

• But the classical invariants do not suffice in general

• Still, there are at most finitely many Legendrian knots with a fixedset of classical invariants [Colin-Giroux-Honda]

Geography

Geography via “Mountain Ranges”

Unknot [Eliashberg-Fraser]

0 1 2 3–1–2–3

52 Knot [Chekanov, Epstein-Fuchs-Meyer]

0 1 2 3–1–2–3

(2,3) Cable of (2,3) Torus Knot [Etnyre-Honda]

0 1 2 3–1–2–3

–4 4

Where Are We?

4 Extensions

Combinatorial Definitions of Invariants Normal Rulings

Rulings of Front Diagrams

A ruling of a front is a 1-to-1 correspondence between left and rightcusps together with two paths joining each pair of cusps such that:

1 The interiors of the paths are disjoint and meet only at the cusps2 Any two paths meet only at crossings and cusps

Note that fronts of stabilized knots do not have any rulings

1 The interiors of the paths are disjoint and meet only at the cusps

2 Any two paths meet only at crossings and cuspsNote that fronts of stabilized knots do not have any rulings

The Normality Condition

We don’t want to allow all types of “switches”:

Banning these switches gives a normal ruling of a front

Two Examples

Graded Rulings

We can refine the notion of a normal ruling further by assigning agrading (modulo 2r(K )) to each crossing.

Define a Maslov potential µ on each strand and let the grading of eachcrossing be

µ(\)− µ(/)

A graded ruling is one whose switches all have grading 0.

Graded Rulings

µ(\)− µ(/)

Graded Rulings

µ(\)− µ(/)

Ruling Polynomials

We can organize the set of rulings of a front D into a polynomial!

θ(R) = cusps(D)− switches(R)

ΘD(k) = {rulings R : θ(R) = k}

The ruling polynomial of a front D is:

RD(z) =∑

#ΘD(1− k)zk

Ruling Polynomials

We can organize the set of rulings of a front D into a polynomial!

θ(R) = cusps(D)− switches(R)

ΘD(k) = {rulings R : θ(R) = k}

The ruling polynomial of a front D is:

RD(z) =∑

#ΘD(1− k)zk

The Ruling Invariant

Theorem (Chekanov-Pushkar)

The (graded or ungraded) ruling polynomial is a Legendrian invariant.

Example

The following 52 knots are not Legendrian isotopic.

R(z) = 1+z2 R(z) = 1

The Ruling Invariant

Theorem (Chekanov-Pushkar)

The (graded or ungraded) ruling polynomial is a Legendrian invariant.

Example

The following 52 knots are not Legendrian isotopic.

R(z) = 1+z2 R(z) = 1

A Topological Connection

Theorem (Rutherford)

The ungraded RK (z) is the coefficient of a−tb(K )−1 in FK (a, z), so theKauffman bound is sharp if and only if there is a ruling.

Combinatorial Definitions of Invariants The Chekanov-Eliashberg DGA

The Chekanov-Eliashberg DGA

This invariant takes the form of a differential graded algebra.

First, the algebra: Label the crossings in an xy diagram with {1, . . . , n}.

Let A be the vector space over Z2 generated by labels {q1, . . . , qn},and let A =

⊕∞k=0 A⊗k be the unital tensor algebra over A, graded as

for rulings.

The Differential

To define the differential, first decorate the crossings:

+ +––

+ +––+ +

––

To find contributions to ∂qi , look for smoothly immersed disks withconvex corners whose boundary lies in the knot diagram. There is a +corner at i and − corners elsewhere.

The Differential

+ +––

+ +––+ +

––

The Differential

+ +––

+ +––+ +

––

Record the − corners in counterclockwise order:

∂q1 = q5q4q3

The Differential

+ +––

+ +––+ +

––

∂q1 = q5q4q3 + q5

The Differential

+ +––

+ +––+ +

––

∂q1 = q5q4q3 + q5 + 1 + · · ·

Yes, Immersed!

The following disk would be counted (as a 1) in the differential:

The Main Result

Theorem (Chekanov)

∂ has degree −1, ∂2 = 0, and the Legendrian contact homologyH∗(A, ∂) (denoted LCH∗(K )) is invariant under Legendrian isotopy.

In fact, something more subtle is true: the “stable tame isomorphism”class of (A, ∂) is an invariant. But how do we extract useful informationout of this?

Note that this invariant vanishes on all stabilized knots.

The Main Result

Theorem (Chekanov)

The Main Result

Theorem (Chekanov)

Time for a break!

Combinatorial Definitions of Invariants Computable Invariants from the DGA

Augmentations

An augmentation of (A, ∂) is an algebra map ε : A → Z2 such that:1 ε(1) = 1

2 ε ◦ ∂ = 03 ε is graded if it has support on generators of grading 0

There are only finitely many of these!

Augmentations

An augmentation of (A, ∂) is an algebra map ε : A → Z2 such that:1 ε(1) = 12 ε ◦ ∂ = 0

3 ε is graded if it has support on generators of grading 0There are only finitely many of these!

Augmentations

An augmentation of (A, ∂) is an algebra map ε : A → Z2 such that:1 ε(1) = 12 ε ◦ ∂ = 03 ε is graded if it has support on generators of grading 0

Augmentations

An augmentation of (A, ∂) is an algebra map ε : A → Z2 such that:1 ε(1) = 12 ε ◦ ∂ = 03 ε is graded if it has support on generators of grading 0

Augmentation Numbers

We can form an invariant by counting augmentations as follows:

• Let ak be the number of generators of A of grading k , and define:

χ(A) =∑k≥0

(−1)kak +∑k<0

(−1)k+1ak

• Now let:Aug(A) = 2(−1−χ(A))/2#augmentations

TheoremAug(A) is a Legendrian invariant (now denoted Aug(K ))

χ(A) =∑k≥0

(−1)kak +∑k<0

(−1)k+1ak

χ(A) =∑k≥0

(−1)kak +∑k<0

(−1)k+1ak

χ(A) =∑k≥0

(−1)kak +∑k<0

(−1)k+1ak

Augmentations and Rulings

Theorem (Fuchs, Fuchs-Ishkhanov, JMS)

The Chekanov-Eliashberg DGA of a Legendrian knot K has an(graded) augmentation iff a front diagram of K has a (graded) ruling.

Theorem (Ng-JMS)

Aug(K ) = RK (2−12 )

Augmentations and Rulings

Theorem (Fuchs, Fuchs-Ishkhanov, JMS)

The Chekanov-Eliashberg DGA of a Legendrian knot K has an(graded) augmentation iff a front diagram of K has a (graded) ruling.

Theorem (Ng-JMS)

Aug(K ) = RK (2−12 )

Linearization

Break up ∂:

∂ = ∂0 + ∂1 + ∂2 + · · · , where ∂k : A → A⊗k

Given an augmentation ε, define an automorphism ϕε of A, by:

ϕε(q) = q + ε(q)

We get a new differential ∂ε = ϕε∂(ϕε)−1 with ∂ε0 = 0. Then:

∂2 = 0 =⇒ (∂ε1)

2 = 0.

Linearization

Break up ∂:

∂ = ∂0 + ∂1 + ∂2 + · · · , where ∂k : A → A⊗k

ϕε(q) = q + ε(q)

∂2 = 0 =⇒ (∂ε1)

2 = 0.

Linearization

Break up ∂:

∂ = ∂0 + ∂1 + ∂2 + · · · , where ∂k : A → A⊗k

ϕε(q) = q + ε(q)

∂2 = 0 =⇒ (∂ε1)

2 = 0.

Linearization, ctd.

It follows that (A, ∂ε1) is a finite-dimensional chain complex whose

homology is denoted LCHε∗(K ). Record the homology in a

Poincaré-Chekanov polynomial:

Pε(t) =∞∑

k=−∞tk dim Hk (A, ∂ε

Theorem (Chekanov)

The set{

LCHε∗(K )

}augmentations is an invariant of Legendrian isotopy.

Example

This can also distinguish the 52’s, as they have linearized invariants{2 + t} and {t−2 + t + t2}

Linearization, ctd.

Pε(t) =∞∑

Theorem (Chekanov)

The set{

LCHε∗(K )

Example

Linearization, ctd.

Pε(t) =∞∑

Theorem (Chekanov)

The set{

LCHε∗(K )

Example

Nonlinear Methods

Theorem (JMS)

For any linearization, dim LCHεk = dim LCHε

−k for |k | 6= 1, whiledim LCHε

1 = dim Hε−1 + 1.

More refined computable invariants can be derived from (A, ∂):

• Cup and Massey product structures on the linearization using ∂k ,k ≥ 2 [Etnyre-JMS et al.]

• The characteristic algebra C(A, ∂) = A/im∂ up to isomorphismand adding free generators. [Ng]

Nonlinear Methods

Theorem (JMS)

1 = dim Hε−1 + 1.

Nonlinear Methods

Theorem (JMS)

1 = dim Hε−1 + 1.

Nonlinear Methods

Theorem (JMS)

1 = dim Hε−1 + 1.

Where Are We?

4 Extensions

Geometry of the Invariants Generating Families

Generating Families for Legendrian Knots

One way to produce Legendrian curves in the standard contact R3

(thought of as the 1-jet space J1(R)) is to use graphs of derivatives:

x 7−→ (x , f ′(x), f (x))

We can obtain “non-graphical” Legendrians by extending our functionsto

f : R× RN → R

that are nondegenerate (in some sense) and “linear-quadratic atinfinity”, i.e. equal to the sum of nondegenerate linear and anondegenerate quadratic outside a compact set.

Generating Families for Legendrian Knots

One way to produce Legendrian curves in the standard contact R3

(thought of as the 1-jet space J1(R)) is to use graphs of derivatives:

x 7−→ (x , f ′(x), f (x))

We can obtain “non-graphical” Legendrians by extending our functionsto

f : R× RN → R

that are nondegenerate (in some sense) and “linear-quadratic atinfinity”, i.e. equal to the sum of nondegenerate linear and anondegenerate quadratic outside a compact set.

Generating Families, ctd.

Instead of embedding the whole domain, we look at the fiber critical set

Σf = {(x , e) ∈ R× RN : ∂ef (x , e) = 0}

Then embed Σf → R3 as before:

(x , e) ∈ Σf 7−→ (x , ∂x f , f (x , e))

Theorem (Pushkar?, Fuchs-Rutherford)

Any Legendrian knot with a normal ruling can be obtained this way.

Σf = {(x , e) ∈ R× RN : ∂ef (x , e) = 0}

(x , e) ∈ Σf 7−→ (x , ∂x f , f (x , e))

Σf = {(x , e) ∈ R× RN : ∂ef (x , e) = 0}

(x , e) ∈ Σf 7−→ (x , ∂x f , f (x , e))

An Example

Generating Families and Normal Rulings

A generating family f : R× RN → R induces a normal ruling as follows:

• For all but finitely many fixed x , fx : RN → R is a Morse functionwhose critical points bi have distinct critical values.

• The Morse complex for fx is triangular and acyclic, so (after a bit ofalgebra), there is a pairing bi ↔ bτ(i) of the critical points such that(after a basis change) ∂bi = bτ(i).

• This pairing locally determines the ruling paths, and restrictions onhow these pairings can evolve give rise to the normality condition(via [Barranikov]).

Generating Family Homology

Consider the difference function [Traynor, Jordan-Traynor, Fuchs-Rutherford]:

∆(x , e, e) = f (x , e)− f (x , e)

Note that critical points with positive critical value are in 1-to-1correspondence with crossings of the xy diagram

DefinitionGiven η > 0 such that ∆ has no critical values in (0, η),

GH∗(f ) = H∗+N+1(∆ ≥ η, ∆ = η; Z2)

Generating Family Homology

Consider the difference function [Traynor, Jordan-Traynor, Fuchs-Rutherford]:

∆(x , e, e) = f (x , e)− f (x , e)

Note that critical points with positive critical value are in 1-to-1correspondence with crossings of the xy diagram

DefinitionGiven η > 0 such that ∆ has no critical values in (0, η),

GH∗(f ) = H∗+N+1(∆ ≥ η, ∆ = η; Z2)

Generating Family Homology, ctd.

Theorem (Jordan-Traynor, . . . )

The set of generating family homolgies (taken over all possiblegenerating families) is a Legendrian invariant.

Theorem (Fuchs-Rutherford)

If f generates K and is generic and linear at infinity, then there exists agraded augmentation ε of (AK , ∂) such that

GH∗(f ) = LCHε∗(K )

Generating Family Homology, ctd.

Theorem (Jordan-Traynor, . . . )

The set of generating family homolgies (taken over all possiblegenerating families) is a Legendrian invariant.

Theorem (Fuchs-Rutherford)

If f generates K and is generic and linear at infinity, then there exists agraded augmentation ε of (AK , ∂) such that

GH∗(f ) = LCHε∗(K )

Geometry of the Invariants Legendrian Contact Homology

The Action Functional

An infinite-dimensional analogue of the difference function forgenerating families is the action functional on the space of paths in R3

beginning and ending on K :

A(γ) =

∫γα

Instead of computing relative homology as for GH∗(f ), take motivationfrom the Morse-Witten-Floer theory of A . . .

The Action Functional

An infinite-dimensional analogue of the difference function forgenerating families is the action functional on the space of paths in R3

beginning and ending on K :

A(γ) =

∫γα

Instead of computing relative homology as for GH∗(f ), take motivationfrom the Morse-Witten-Floer theory of A . . .

Geometric preliminaries

The Reeb field R satisfies:

1 dα(R, ·) = 02 α(R) = 1

This is ∂z for the standard contact structure

The symplectization of (R3, α) is the symplectic manifold

(R3 × R, d(etα))

L× R is a Lagrangian submanifold.

There is an almost complex structure J for which ξ is complex and thatpairs the Reeb and symplectization directions.

The Reeb field R satisfies:1 dα(R, ·) = 0

2 α(R) = 1This is ∂z for the standard contact structure

(R3 × R, d(etα))

The Reeb field R satisfies:1 dα(R, ·) = 02 α(R) = 1

(R3 × R, d(etα))

A Morse Theory – LCH Dictionary

Morse-Witten theory:• Chain complex

generated by criticalpoints

• Differential from rigidflowlines between criticalpoints

Legendrian CH:• Associative algebra

generated by Reebchords

• Differential from rigidJ-hol ∂-punctured diskswith ∂ on L× R

– –

Transition to Combinatorics in (R3, ξ0)

For knots in (R3, ξ0), we want to compute Legendrian contacthomology combinatorially using xy diagrams. What happens when weproject R3 × R → R2

• Since the Reeb field for α0 is ∂z , Reeb chords project to doublepoints:

• Rigid J-holomorphic disks project to rigid holomorphic disks inR2 = C

(Can think of these as smoothly immersed disks byRiemann Mapping Theorem)

• Rigid J-holomorphic disks project to rigid holomorphic disks inR2 = C (Can think of these as smoothly immersed disks byRiemann Mapping Theorem)

Where Are We?

4 Extensions

Extensions

Lagrangian Cobordism

Replacing L× R with an exactLagrangian cobordism L between L andL′ allows us to define a map:

φL : LCH(L) → LCH(L′)

[Ekholm-Honda-Kálmán]

For L′ = ∅, φL is an augmentation, andthere are close connections to rulingshere.

Extensions

Lagrangian Cobordism

Replacing L× R with an exactLagrangian cobordism L between L andL′ allows us to define a map:

φL : LCH(L) → LCH(L′)

[Ekholm-Honda-Kálmán]

For L′ = ∅, φL is an augmentation, andthere are close connections to rulingshere.

Extensions

Legendrian Symplectic Field Theory

Why not consider disks in LCH with multiple positive punctures?

Difficult (but possible) to define an algebraic structure that takes new“bubbling” phenomena into account [Ng]

This might lead to holomorphic curve-based invariants of transverseknots!

Extensions

Higher Dimensions

• Geometric ideas lead to an analytic definition of LCH forhigher-dimensional Legendrians [Ekholm-Etnyre-Sullivan]; a version ofduality still holds [Ekholm-Etnyre-JMS]

• It is possible to use LCH of Legendrian tori to get invariants ofsmooth knots

• What about rulings? [Henry] Or GF homology?

Extensions

Higher Dimensions

Extensions

Higher Dimensions

Lunchtime!

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