Legendrian contact homology of conormal lifts Tobias Ekholm - - PowerPoint PPT Presentation

Legendrian contact homology

• f conormal lifts

Tobias Ekholm

Uppsala University, Sweden

Berkeley, April 4, 2011

Plan of the talk

Background Legendrian contact homology Computations for links in R3 Application to transverse links in (R3, ξstd) Remarks about other dimensions and relations to string topology

Background – basic observations

Diff top. A smooth submanifold K ⊂ M.

Background – basic observations

Diff top. A smooth submanifold K ⊂ M. Sympl geom. The Lagrangian conormal of K in T ∗M, LK =

• p ∈ T ∗M : π(p) ∈ K, p|TK = 0
• ⊂ T ∗M,

with ideal Legendrian boundary ΛK = LK ∩ S∗M.

Background – basic observations

Diff top. A smooth submanifold K ⊂ M. Sympl geom. The Lagrangian conormal of K in T ∗M, LK =

• p ∈ T ∗M : π(p) ∈ K, p|TK = 0
• ⊂ T ∗M,

with ideal Legendrian boundary ΛK = LK ∩ S∗M.

• Isotopy. An isotopy Kt of K induces a Lagrangian (resp. a

Legendrian) isotopy of LKt (resp. ΛKt).

Background – Invariants

Symplectic deformation invariants of ΛK give isotopy invariants of K ⊂ M. We will study holomorphic curve invariants on the symplectic side.

Background – Invariants

Symplectic deformation invariants of ΛK give isotopy invariants of K ⊂ M. We will study holomorphic curve invariants on the symplectic side. For most of the talk we restrict attention to the case M = R3 and K ⊂ R3 a smooth link. In this case ΛK ⊂ S∗R3 is a union of Legendrian tori. We will compute its Legendrian contact homology (the most basic part of SFT).

Background – Survey of results

For knots, the knot contact homology of L. Ng is a combinatorial invariant that was inspired by this construction and conjectured to agree with the Legendrian contact homology of ΛK. The work surveyed in this talk proves that conjecture.

Background – Survey of results

For knots, the knot contact homology of L. Ng is a combinatorial invariant that was inspired by this construction and conjectured to agree with the Legendrian contact homology of ΛK. The work surveyed in this talk proves that conjecture. Ng derived many properties of his knot contact homology. In particular, it follows from his work that the Legendrian contact homology of ΛK is an unknot detector that contains the A-polynomial.

Background – Survey of results

For knots, the knot contact homology of L. Ng is a combinatorial invariant that was inspired by this construction and conjectured to agree with the Legendrian contact homology of ΛK. The work surveyed in this talk proves that conjecture. Ng derived many properties of his knot contact homology. In particular, it follows from his work that the Legendrian contact homology of ΛK is an unknot detector that contains the A-polynomial. In this talk we focus on describing the holomorphic curves needed to compute the Legendrian homology. This description lead to new invariants of transverse knots in R3 and also constitutes an important model for computations in other manifolds and in higher dimensions.

Background – some refs

In the case under study, the technical details of Legendrian contact homology were worked out in

Background – some refs

In the case under study, the technical details of Legendrian contact homology were worked out in Ekholm, T.; Etnyre, J.; Sullivan, M. Legendrian contact homology in P × R. Trans. Amer. Math. Soc. 359 (2007),

• no. 7, 3301–3333

Ekholm, T.; Etnyre, J.; Sullivan, M. Orientations in Legendrian contact homology and exact Lagrangian immersions, Internat. J. Math. 16 (2005), no. 5, 453–532.

Legendrian contact homology – set up

The Reeb vector field R of a contact form α on a contact manifold Y is characterized by dα(R, ·) = 0 and α(R) = 1. For Y = S∗M identified with the unit conormal bundle, and α = p dq, the flow of R is the geodesic flow. A flow line of R beginning and ending on a Legendrian submanifold ΛK ⊂ Y is a Reeb chord of Λ. For Λ = ΛK Reeb chords correspond to binormal geodesic chords on K.

Legendrian contact homology – set up

The Reeb vector field R of a contact form α on a contact manifold Y is characterized by dα(R, ·) = 0 and α(R) = 1. For Y = S∗M identified with the unit conormal bundle, and α = p dq, the flow of R is the geodesic flow. A flow line of R beginning and ending on a Legendrian submanifold ΛK ⊂ Y is a Reeb chord of Λ. For Λ = ΛK Reeb chords correspond to binormal geodesic chords on K. The symplectization of Y is X = Y × R with symplectic form d(etα) where t ∈ R. Fix an almost complex structure J on X such that J(∂t) = R, J(ker α) = ker α, and dα(v, Jv) > 0.

Legendrian contact homology – set up

The Reeb vector field R of a contact form α on a contact manifold Y is characterized by dα(R, ·) = 0 and α(R) = 1. For Y = S∗M identified with the unit conormal bundle, and α = p dq, the flow of R is the geodesic flow. A flow line of R beginning and ending on a Legendrian submanifold ΛK ⊂ Y is a Reeb chord of Λ. For Λ = ΛK Reeb chords correspond to binormal geodesic chords on K. The symplectization of Y is X = Y × R with symplectic form d(etα) where t ∈ R. Fix an almost complex structure J on X such that J(∂t) = R, J(ker α) = ker α, and dα(v, Jv) > 0. If c is a Reeb chord of Λ then c × R ⊂ X is a J-holomorphic strip with boundary on the Lagrangian submanifold Λ × R.

Legendrian contact homology – DGA

The Legendrian contact homology algbera of Λ is the free unital (non-commutative) algebra A(Λ) = Z[H1(Λ)]

• Reeb chords
• (For general Y , A(Λ) is an algebra over the contact homology

algebra Q(Y ) of Y generated by closed Reeb orbits. In this talk we work with contact forms for which there are no closed

• rbits in Y and for which c1(ker α) = 0)

Legendrian contact homology – DGA

The Legendrian contact homology algbera of Λ is the free unital (non-commutative) algebra A(Λ) = Z[H1(Λ)]

• Reeb chords
• (For general Y , A(Λ) is an algebra over the contact homology

algebra Q(Y ) of Y generated by closed Reeb orbits. In this talk we work with contact forms for which there are no closed

• rbits in Y and for which c1(ker α) = 0)

A(Λ) is a DGA. The grading |c| of a Reeb chord is defined by a Maslov index. The differential ∂ : A(Λ) → A(Λ) is linear, satisfies Leibniz rule, and is defined on generators through a holomorphic curve count. The DGA (A(λ), ∂) is invariant under deformations up to quasi-isomorphism.

The Legendrian contact homology – the differential

∂a =

• |a|−|A|−|b|=1

#MA(a, b)Ab, where MA(a; b) is the moduli space of holomorphic disks u: D → X, dim(MA(a; b)) = |a| − |A| − |b|.

Computations – strategy

To compute the Legendrian contact homology of ΛK for a link K, we braid K around the unknot U. Then ΛK ⊂ J1(ΛU) ⊂ S∗R3.

Computations – strategy

To compute the Legendrian contact homology of ΛK for a link K, we braid K around the unknot U. Then ΛK ⊂ J1(ΛU) ⊂ S∗R3. In the limit as K → U, holomorphic disks with boundary on ΛK admits a description in terms of holomorphic disks with boundary on ΛU with flow trees determined by ΛK ⊂ J1(Λ) attached along their boundaries.

Computations – strategy

To compute the Legendrian contact homology of ΛK for a link K, we braid K around the unknot U. Then ΛK ⊂ J1(ΛU) ⊂ S∗R3. In the limit as K → U, holomorphic disks with boundary on ΛK admits a description in terms of holomorphic disks with boundary on ΛU with flow trees determined by ΛK ⊂ J1(Λ) attached along their boundaries. Our computation thus has three main steps:

Describe holomorphic disks for ΛU. Describe flow trees for ΛK ⊂ J1(ΛU). Count the resulting rigid disks with flow trees.

Computations – some refs

The computation is joint work: Ekholm, T.; Etnyre, J.; Ng, L.; Sullivan, M. Knot contact homology, 105 pages, in preparation.

Computations – some refs

The computation is joint work: Ekholm, T.; Etnyre, J.; Ng, L.; Sullivan, M. Knot contact homology, 105 pages, in preparation. the techniques are based on Ekholm, T. Morse flow trees and Legendrian contact homology in 1-jet spaces, Geom. Topol. 11 (2007), 1083–1224. Ekholm, T.; Etnyre, J.; Sabloff, J. A duality exact sequence for Legendrian contact homology, Duke Math. J. 150 (2009),

• no. 1, 1–75.

Computations – the unknot

To describe holomorphic disks of (S∗R3 × R, ΛU × R) we use the contactomorphism Ψ: S∗R3 = J1(S2) = T ∗S2 × R, Ψ(x, y) =

• y, x − x, yy, x, y
• ,

and the correspondence between holomorphic disks in T ∗S2 with boundary on the projection π(ΛK) and 1-parameter families of holomorphic disks in J1(S2) × R with boundary on ΛK × R.

Computations – the unknot

The front of the unknot in J0(S2) ≈ S2 × R can be drawn as follows: Recall Ψ(x, y) =

• y, x − x, yy, x, y
• , the first and last

coordinates are depicted.

Computations – the unknot

In order for the disk/tree relation to hold, ΛU must be front generic, i.e. in general position with respect to the fibers in J1(S2). This holds outside neighborhoods of the poles and near the poles we get the following after perturbation:

Computations – the unknot

0-dim trees/disks: A(ΛU) = Z[λ±1, µ±1]c, e, |c| = 1, |e| = 2, ∂e = c − c = 0, ∂c = 1 + λ + µ + µλ

Computations – the unknot

1-dim trees/disks:

Computations – general links

We represent a general link K as a braid B on n strands in a tube U × D2. Picking a base point in U we think of U as parametrized by [0, 2π] and the braid B as given by a collection fB of functions fj : [0, 2π] → D2, j = 1, . . . , n, such that {f1(0), . . . , fn(0)} = {f1(2π), . . . , fn(2π)} (and such that fB is a smooth multi-section).

Computations – general links

We represent a general link K as a braid B on n strands in a tube U × D2. Picking a base point in U we think of U as parametrized by [0, 2π] and the braid B as given by a collection fB of functions fj : [0, 2π] → D2, j = 1, . . . , n, such that {f1(0), . . . , fn(0)} = {f1(2π), . . . , fn(2π)} (and such that fB is a smooth multi-section). Let ΛU = U × S1, S1 = {ξ ∈ R2 : |ξ| = 1}. Then there exists a 1-jet neighborhood N of ΛU such that ΛK is Legendrian isotopic to the graph Γj1(FB), where FB is the multifunction with components Fj(s, ξ) = fj(s) · ξ, j = 1, . . . , n.

Computations – general links

For simpler formulas assume K has only one component. Then A = A(ΛK) = Z[λ±1, µ±1]

• aij, bij, cst, est
• 1≤i=j≤n, 1≤s,t≤n,

|aij| = 0, |bij| = |cst| = 1, |est| = 2.

Computations – general links

The multi graph of FB looks as follows: With all strands of the trivial braid on one a line in D2, all aij and bij as well as the stable manifolds of aij lie right above each other. The differential depends on how we choose to perturb out of that situation.

Computations – general links

A twist in the braid between the kth and (k + 1)th strands affects the stable manifolds as follows

Computations – general links

In order to describe trees we subdivide them into parts lying in twist slices and introduce twist homomorphisms φσk : A0 → A0: φσk(aij) = aij i, j = k, k + 1; φσk(ak k+1) = −ak+1 k; φσk(ak+1 k) = −ak k+1; φσk(ai k+1) = aik i = k, k + 1; φσk(ak+1 i) = aki i = k, k + 1; φσk(aik) = ai k+1 − aikak k+1 i < k; φσk(aik) = ai k+1 − aikak k+1 i > k + 1; φσk(aki) = ak+1 i − ak+1 kaki i = k, k + 1. Here aij should be thought of as the stable manifold corresponding to the chord from the jth to the ith strand in the trivial slices of the braid next to the twist.

Computations – general links

A braid B then induces a homomorphism φB : A0 → A0 by composing its twist automorphisms.

Computations – general links

A braid B then induces a homomorphism φB : A0 → A0 by composing its twist automorphisms. Adding an (n + 1)th strand to a braid which does not interact with other strands we get a braid ˆ

• B. Further, φ ˆ

B(aj n+1)

(resp. φ ˆ

B(an+1 j)) is linear in the generator aj n+1

(resp. an+1 j), which occur at the beginning (resp. end) of any monomial that contributes. We thus find matrices ΦL

B and

ΦR

B with coefficients in A0 such that

φ ˆ

B(an+1 i) =

• j

an+1 j

• ΦR

B

• ji ,

φ ˆ

B(ai n+1) =

• j
• ΦL

B

• ij aj n+1,

Computations – general links

Define the n × n-matrices Aij =      aij if i < j, µaij if i > j, 1 + µ if i = j, Bij =      bij if i < j, µbij if i > j, if i = j, Cij = cij, Eij = eij, λij =      λ if i = j = 1, 1 if i = j > 1, if i = j.

Computations – general links

Define the n × n-matrices Aij =      aij if i < j, µaij if i > j, 1 + µ if i = j, Bij =      bij if i < j, µbij if i > j, if i = j, Cij = cij, Eij = eij, λij =      λ if i = j = 1, 1 if i = j > 1, if i = j. Then φB(A) = ΦL

B · A · ΦR B, where φB(A)ij = φB(aij)

(enough to check for elementary twists) and the differential on A(ΛK) is expressed through ΦL

B and ΦR B as follows.

Computations – general links

Theorem (Ekholm, Etnyre, Sullivan, Ng) The differential on A(ΛK) is determined by the following matrix equations ∂A = 0, ∂B = −λ−1 · A · λ + ΦL

B · A · ΦR B,

∂C = A · λ + A · ΦR

B,

∂E = B · (ΦR

B)−1 + B · λ−1 − ΦL B · C · λ−1 + λ−1 · C · (ΦR B)−1.

Computations – general links

∂B:

Computations – general links

∂C:

Computations – general links

B-components of ∂E:

Computations – general links

C-components of ∂E:

Computations – general links

Signs:

Computations – general links

σn,Γ(t) = sign

• w2(t) − w1(t), vcon(t)
• ,

σn,Γ(q) = sign

• w(qj), vker(qj)
• .

For a tree Γ with a normal vector n at its positive puncture, negative punctures at q1, . . . , qk, and trivalent vertices t1, . . . , tk−1, the sign input is: σ(n, Γ) = Πk

j=1σn,Γ(qj) Πk−1 j=1σn,Γ(tj).

Transverse links

Consider the (tight) contact form α = dx3 − x2dx1 + x1dx2

• n R3. Let H±

α = {(x, y) ∈ S∗R3 : y = ±tα, t > 0}.

Transverse links

Consider the (tight) contact form α = dx3 − x2dx1 + x1dx2

• n R3. Let H±

α = {(x, y) ∈ S∗R3 : y = ±tα, t > 0}.

If K is transverse to ker α then H±

α ∩ ΛK = ∅ and one can

check that Reeb chords of ΛK are disjoint from H±

α as well.

Note that a braid near U is transverse to α.

Transverse links

Consider the (tight) contact form α = dx3 − x2dx1 + x1dx2

• n R3. Let H±

α = {(x, y) ∈ S∗R3 : y = ±tα, t > 0}.

If K is transverse to ker α then H±

α ∩ ΛK = ∅ and one can

check that Reeb chords of ΛK are disjoint from H±

α as well.

Note that a braid near U is transverse to α. There is an almost complex structure J on S∗R3 × R such that H±

α × R are J-holomorphic hypersurfaces. Taking the

fiber S2 to be flat near (0, 0, 1) we have H±

α near x = 0

parametrized by ψ(x) = (x, (−x2, x1, 1)) and ∂x1ψ = ((1, 0, 0), (0, 1, 0)), ∂x2ψ = ((0, 1, 0), (−1, 0, 0)), J0∂x1ψ = ∂x2ψ.

Transverse links

Define the matrices λ :      λij = 0 if i = j, λii = λµw if i = 1, λii = 1 if i = 1, where w is the writhe (algebraic crossing number) of the braid (which we assume has only one component for simpler notation). To capture the two filtrations extend the coefficient ring by two formal variables U, V and define: AU :      AU

ij = Uaij

i < j, AU

ij = µaij

i > j, AU

ii = U + µ,

BU :      BU

ij = Ubij

i < j, BU

ij = µbij

i > j, BU

ii = 0,

AV :      AV

ij = aij

i < j, AV

ij = µV aij

i > j, AV

ii = 1 + µV,

BV :      BV

ij = bij

i < j, BV

ij = µV bij

i > j, BV

ii = 0.

Transverse links

Theorem (Ekholm, Etnyre, Ng, Sullivan) Counting intersections with H±

α × R gives a double filtration on

A(ΛK), the power of U (resp. V ) corresponds to intersections with H+

α (resp. H− α ). The filtered quasi-isomorphism type is invariant

under transverse isotopies of K. Furthermore, the filtered differential is determined by the following matrix equations ∂−A = 0, ∂−B = −λ−1 · A · λ + ΦL

B · A · ΦR B,

∂−C = AV · λ + AU · ΦR

B,

∂−E = BV · (ΦR

B)−1 + BU · λ−1

− ΦL

B · C · λ−1 + λ−1 · C · (ΦR B)−1,

Transverse links

Two transverse representatives of the mirror of the knot 76 in Rolfsen’s table with the same self-linking number, but not isotopic through transverse knots as their filtered DGAs are non-isomorphic (distinguished by a count of certain homomorphisms to Z3). They do however have the same transverse invariants coming from Heegaard-Floer theory and Khovanov homology.

Relation to string topology

We define a version of (relative) string topology for K ⊂ R3.

Relation to string topology

We define a version of (relative) string topology for K ⊂ R3. Let C∗ denote chains on the space of alternating strings.

Relation to string topology

We define a version of (relative) string topology for K ⊂ R3. Let C∗ denote chains on the space of alternating strings. With “co-products” δN : C∗ → C∗−1 and δQ : C∗ → C∗−1, ∆ = ∂sing + δQ + δN is a differential on C∗.

Relation to string topology

Let c be a Reeb chord of ΛK. The moduli space M(c) of holomorphic disks in T ∗R3 with positive puncture at c and with boundary on LK ∪ R3 gives a chain in the space of alternating strings by evaluation.

Relation to string topology

Let c be a Reeb chord of ΛK. The moduli space M(c) of holomorphic disks in T ∗R3 with positive puncture at c and with boundary on LK ∪ R3 gives a chain in the space of alternating strings by evaluation. The description of the boundary of M(c) shows that this induces a chain map A(ΛK) → (C∗, ∆).

Relation to string topology

Let c be a Reeb chord of ΛK. The moduli space M(c) of holomorphic disks in T ∗R3 with positive puncture at c and with boundary on LK ∪ R3 gives a chain in the space of alternating strings by evaluation. The description of the boundary of M(c) shows that this induces a chain map A(ΛK) → (C∗, ∆). Theorem (Cieliebak, Ekholm, Latchev, Ng; work in progress) The degree 0 Legendrian homology is isomorphic to the degree 0 string topology H0(A(ΛK)) ≈ Hstring (K).