An Introduction toInvariants of Legendrian Knots
Joshua M. Sabloff
Haverford College
AIM Workshop on Legendrian and Transverse Knots
Goals
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
Goals
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
Goals
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
Goals
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
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
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
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
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
Where Are We?
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
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
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
Classical Invariants
Resolution of Front Diagrams
We can “resolve” a front diagram into a Lagrangian diagram of anisotopic knot:
1
2
3 4 5
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 6 / 45
Classical Invariants
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
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 7 / 45
Classical Invariants
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
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 7 / 45
Classical Invariants
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
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 7 / 45
Classical Invariants
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
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 7 / 45
Classical Invariants
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
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 7 / 45
Classical Invariants
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
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 7 / 45
Classical Invariants
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 8 / 45
Classical Invariants
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 8 / 45
Classical Invariants
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 8 / 45
Classical Invariants
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]
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 9 / 45
Classical Invariants
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]
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 9 / 45
Classical Invariants
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]
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 9 / 45
Classical Invariants
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]
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 9 / 45
Classical Invariants
Geography via “Mountain Ranges”
Unknot [Eliashberg-Fraser]
0 1 2 3–1–2–3
–1
–2
–3
–4
tb
r
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 10 / 45
Classical Invariants
Geography via “Mountain Ranges”
52 Knot [Chekanov, Epstein-Fuchs-Meyer]
0 1 2 3–1–2–3
1
0
–1
–2
tb
r
?
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 10 / 45
Classical Invariants
Geography via “Mountain Ranges”
(2,3) Cable of (2,3) Torus Knot [Etnyre-Honda]
0 1 2 3–1–2–3
6
5
4
3
tb
r
–4 4
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 10 / 45
Where Are We?
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
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
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 12 / 45
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 cusps
2 Any two paths meet only at crossings and cuspsNote that fronts of stabilized knots do not have any rulings
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 12 / 45
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
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 12 / 45
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
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 12 / 45
Combinatorial Definitions of Invariants Normal Rulings
The Normality Condition
We don’t want to allow all types of “switches”:
Banning these switches gives a normal ruling of a front
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 13 / 45
Combinatorial Definitions of Invariants Normal Rulings
The Normality Condition
We don’t want to allow all types of “switches”:
Banning these switches gives a normal ruling of a front
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 13 / 45
Combinatorial Definitions of Invariants Normal Rulings
The Normality Condition
We don’t want to allow all types of “switches”:
Banning these switches gives a normal ruling of a front
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 13 / 45
Combinatorial Definitions of Invariants Normal Rulings
Two Examples
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 14 / 45
Combinatorial Definitions of Invariants Normal Rulings
Graded Rulings
We can refine the notion of a normal ruling further by assigning agrading (modulo 2r(K )) to each crossing.
–10
1
2
3
0
1
2
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 15 / 45
Combinatorial Definitions of Invariants Normal Rulings
Graded Rulings
We can refine the notion of a normal ruling further by assigning agrading (modulo 2r(K )) to each crossing.
–10
1
2
3
0
1
2
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 15 / 45
Combinatorial Definitions of Invariants Normal Rulings
Graded Rulings
We can refine the notion of a normal ruling further by assigning agrading (modulo 2r(K )) to each crossing.
–10
1
2
3
0
1
2
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 15 / 45
Combinatorial Definitions of Invariants Normal Rulings
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
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 16 / 45
Combinatorial Definitions of Invariants Normal Rulings
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
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 16 / 45
Combinatorial Definitions of Invariants Normal Rulings
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
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 17 / 45
Combinatorial Definitions of Invariants Normal Rulings
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
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 17 / 45
Combinatorial Definitions of Invariants Normal Rulings
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 18 / 45
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}.
1
2
3 4 5
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 19 / 45
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}.
1
2
3 4 5
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 19 / 45
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}.
1
2
3 4 5
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 19 / 45
Combinatorial Definitions of Invariants The Chekanov-Eliashberg DGA
The Differential
To define the differential, first decorate the crossings:
1
2
345
+ +––
+ +––
+ +––
+ +––+ +
––
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 20 / 45
Combinatorial Definitions of Invariants The Chekanov-Eliashberg DGA
The Differential
To define the differential, first decorate the crossings:
1
2
345
+ +––
+ +––
+ +––
+ +––+ +
––
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 20 / 45
Combinatorial Definitions of Invariants The Chekanov-Eliashberg DGA
The Differential
To define the differential, first decorate the crossings:
1
2
345
+ +––
+ +––
+ +––
+ +––+ +
––
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.
Record the − corners in counterclockwise order:
∂q1 = q5q4q3
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 20 / 45
Combinatorial Definitions of Invariants The Chekanov-Eliashberg DGA
The Differential
To define the differential, first decorate the crossings:
1
2
345
+ +––
+ +––
+ +––
+ +––+ +
––
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.
Record the − corners in counterclockwise order:
∂q1 = q5q4q3 + q5
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 20 / 45
Combinatorial Definitions of Invariants The Chekanov-Eliashberg DGA
The Differential
To define the differential, first decorate the crossings:
1
2
345
+ +––
+ +––
+ +––
+ +––+ +
––
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.
Record the − corners in counterclockwise order:
∂q1 = q5q4q3 + q5 + 1 + · · ·
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 20 / 45
Combinatorial Definitions of Invariants The Chekanov-Eliashberg DGA
Yes, Immersed!
The following disk would be counted (as a 1) in the differential:
1
3+
2+
+
++
+
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 21 / 45
Combinatorial Definitions of Invariants The Chekanov-Eliashberg DGA
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 22 / 45
Combinatorial Definitions of Invariants The Chekanov-Eliashberg DGA
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 22 / 45
Combinatorial Definitions of Invariants The Chekanov-Eliashberg DGA
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 22 / 45
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!
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 24 / 45
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) = 12 ε ◦ ∂ = 0
3 ε is graded if it has support on generators of grading 0There are only finitely many of these!
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 24 / 45
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) = 12 ε ◦ ∂ = 03 ε is graded if it has support on generators of grading 0
There are only finitely many of these!
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 24 / 45
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) = 12 ε ◦ ∂ = 03 ε is graded if it has support on generators of grading 0
There are only finitely many of these!
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 24 / 45
Combinatorial Definitions of Invariants Computable Invariants from the DGA
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 ))
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 25 / 45
Combinatorial Definitions of Invariants Computable Invariants from the DGA
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 ))
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 25 / 45
Combinatorial Definitions of Invariants Computable Invariants from the DGA
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 ))
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 25 / 45
Combinatorial Definitions of Invariants Computable Invariants from the DGA
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 ))
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 25 / 45
Combinatorial Definitions of Invariants Computable Invariants from the DGA
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 )
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 26 / 45
Combinatorial Definitions of Invariants Computable Invariants from the DGA
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 )
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 26 / 45
Combinatorial Definitions of Invariants Computable Invariants from the DGA
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 27 / 45
Combinatorial Definitions of Invariants Computable Invariants from the DGA
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 27 / 45
Combinatorial Definitions of Invariants Computable Invariants from the DGA
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 27 / 45
Combinatorial Definitions of Invariants Computable Invariants from the DGA
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
1)
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}
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 28 / 45
Combinatorial Definitions of Invariants Computable Invariants from the DGA
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
1)
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}
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 28 / 45
Combinatorial Definitions of Invariants Computable Invariants from the DGA
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
1)
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}
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 28 / 45
Combinatorial Definitions of Invariants Computable Invariants from the DGA
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]
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 29 / 45
Combinatorial Definitions of Invariants Computable Invariants from the DGA
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]
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 29 / 45
Combinatorial Definitions of Invariants Computable Invariants from the DGA
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]
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 29 / 45
Combinatorial Definitions of Invariants Computable Invariants from the DGA
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]
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 29 / 45
Where Are We?
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
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 31 / 45
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 31 / 45
Geometry of the Invariants Generating Families
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 32 / 45
Geometry of the Invariants Generating Families
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 32 / 45
Geometry of the Invariants Generating Families
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 32 / 45
Geometry of the Invariants Generating Families
An Example
x
e
f
e
f
e
f
e
f
e
f
z
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 33 / 45
Geometry of the Invariants Generating Families
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]).
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 34 / 45
Geometry of the Invariants Generating Families
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]).
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 34 / 45
Geometry of the Invariants Generating Families
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]).
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 34 / 45
Geometry of the Invariants Generating Families
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]).
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 34 / 45
Geometry of the Invariants Generating Families
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)
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 35 / 45
Geometry of the Invariants Generating Families
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)
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 35 / 45
Geometry of the Invariants Generating Families
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 )
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 36 / 45
Geometry of the Invariants Generating Families
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 )
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 36 / 45
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 . . .
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 37 / 45
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 . . .
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 37 / 45
Geometry of the Invariants Legendrian Contact Homology
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 38 / 45
Geometry of the Invariants Legendrian Contact Homology
Geometric preliminaries
The Reeb field R satisfies:1 dα(R, ·) = 0
2 α(R) = 1This 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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 38 / 45
Geometry of the Invariants Legendrian Contact Homology
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 38 / 45
Geometry of the Invariants Legendrian Contact Homology
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 38 / 45
Geometry of the Invariants Legendrian Contact Homology
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 38 / 45
Geometry of the Invariants Legendrian Contact Homology
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.
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 38 / 45
Geometry of the Invariants Legendrian Contact Homology
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
+
– –
ML
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 39 / 45
Geometry of the Invariants Legendrian Contact Homology
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
+
– –
ML
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 39 / 45
Geometry of the Invariants Legendrian Contact Homology
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
+
– –
ML
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 39 / 45
Geometry of the Invariants Legendrian Contact Homology
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
+
– –
ML
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 39 / 45
Geometry of the Invariants Legendrian Contact Homology
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
+
– –
ML
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 39 / 45
Geometry of the Invariants Legendrian Contact Homology
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
xy?
• Since the Reeb field for α0 is ∂z , Reeb chords project to doublepoints:
γ
πl
• Rigid J-holomorphic disks project to rigid holomorphic disks inR2 = C
(Can think of these as smoothly immersed disks byRiemann Mapping Theorem)
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 40 / 45
Geometry of the Invariants Legendrian Contact Homology
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
xy?
• Since the Reeb field for α0 is ∂z , Reeb chords project to doublepoints:
γ
πl
• Rigid J-holomorphic disks project to rigid holomorphic disks inR2 = C
(Can think of these as smoothly immersed disks byRiemann Mapping Theorem)
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 40 / 45
Geometry of the Invariants Legendrian Contact Homology
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
xy?
• Since the Reeb field for α0 is ∂z , Reeb chords project to doublepoints:
γ
πl
• Rigid J-holomorphic disks project to rigid holomorphic disks inR2 = C
(Can think of these as smoothly immersed disks byRiemann Mapping Theorem)
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 40 / 45
Geometry of the Invariants Legendrian Contact Homology
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
xy?
• Since the Reeb field for α0 is ∂z , Reeb chords project to doublepoints:
γ
πl
• Rigid J-holomorphic disks project to rigid holomorphic disks inR2 = C (Can think of these as smoothly immersed disks byRiemann Mapping Theorem)
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 40 / 45
Where Are We?
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
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.
ML
L´
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 42 / 45
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.
ML
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 42 / 45
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!
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 43 / 45
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!
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 43 / 45
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!
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 43 / 45
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?
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 44 / 45
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?
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 44 / 45
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?
Joshua M. Sabloff (Haverford College) Legendrian Invariants AIM Workshop 44 / 45
Lunchtime!