Knot Contact Homology, Chern-Simons Theory, and ... · Knot contact homology Knot contact...
Transcript of Knot Contact Homology, Chern-Simons Theory, and ... · Knot contact homology Knot contact...
Knot Contact Homology, Chern-Simons Theory,and Topological Strings
Tobias Ekholm
Uppsala University and Institute Mittag-Leffler, Sweden
Symplectic Topology, Oxford, Fri Oct 3, 2014
Plan
Reports on joint work with Aganagic, Ng, and Vafa, arXiv:1304.5778
Knot Contact Homology
Legendrian DGAsAugmentation varieties
Physics Results
Chern-Simons theory and topological stringThe conifold and large N transition
Relations
GW disk potentials and augmentationsLegendrian SFT and recursion relations for the coloredHOMFLY
Knot contact homology
Let K ⊂ S3 be a knot. The Lagrangian conormal of K is
LK = {p ∈ T ∗KS3 : p|TK = 0} ≈ S1 × R2.
The Legendrian conormal is the ideal boundary of LK :
ΛK = LK ∩ U∗S3,
where U∗S3 is the unit conormal
{(q, p) ∈ T ∗S3 : |p| = 1} ≈ S3 × S2,
with the contact form α = pdq,
ΛK ≈ S1x × S1
p ≈ T 2,
where S1x is the longitude and S1
p the meridian.
Knot contact homology
Let K ⊂ S3 be a knot. The Lagrangian conormal of K is
LK = {p ∈ T ∗KS3 : p|TK = 0} ≈ S1 × R2.
The Legendrian conormal is the ideal boundary of LK :
ΛK = LK ∩ U∗S3,
where U∗S3 is the unit conormal
{(q, p) ∈ T ∗S3 : |p| = 1} ≈ S3 × S2,
with the contact form α = pdq,
ΛK ≈ S1x × S1
p ≈ T 2,
where S1x is the longitude and S1
p the meridian.
Knot contact homology
Knot contact homology is the Legendrian contact homologyof the Legendrian conormal of a knot (or link), which wedescribe next.
The Reeb vector field R of α: dα(R, ·) = 0 and α(R) = 1.For α = p dq, the flow of R is the geodesic flow. A flow lineof R beginning and ending on a Legendrian submanifold is aReeb chord of Λ. For ΛK Reeb chords correspond to binormalgeodesic chords on K , and are the critical points of the actionfunctional γ 7→
∫γ α for curves γ with endpoints on ΛK .
The grading |c | of a Reeb chord c is defined by a Maslovindex. For binormal geodesics in knot contact homology thegrading is the Morse index: min = 0, sad = 1, max = 2.
Knot contact homology
Knot contact homology is the Legendrian contact homologyof the Legendrian conormal of a knot (or link), which wedescribe next.
The Reeb vector field R of α: dα(R, ·) = 0 and α(R) = 1.For α = p dq, the flow of R is the geodesic flow. A flow lineof R beginning and ending on a Legendrian submanifold is aReeb chord of Λ. For ΛK Reeb chords correspond to binormalgeodesic chords on K , and are the critical points of the actionfunctional γ 7→
∫γ α for curves γ with endpoints on ΛK .
The grading |c | of a Reeb chord c is defined by a Maslovindex. For binormal geodesics in knot contact homology thegrading is the Morse index: min = 0, sad = 1, max = 2.
Knot contact homology
Knot contact homology is the Legendrian contact homologyof the Legendrian conormal of a knot (or link), which wedescribe next.
The Reeb vector field R of α: dα(R, ·) = 0 and α(R) = 1.For α = p dq, the flow of R is the geodesic flow. A flow lineof R beginning and ending on a Legendrian submanifold is aReeb chord of Λ. For ΛK Reeb chords correspond to binormalgeodesic chords on K , and are the critical points of the actionfunctional γ 7→
∫γ α for curves γ with endpoints on ΛK .
The grading |c | of a Reeb chord c is defined by a Maslovindex. For binormal geodesics in knot contact homology thegrading is the Morse index: min = 0, sad = 1, max = 2.
Knot contact homology
Legendrian contact homology is in a sense the “Floerhomology” of the action functional
∫γ α. Because of certain
bubbling phenomena this theory becomes more non-linearthan in the ordinary case.
The symplectization of U∗S3 is U∗S3 × R with symplecticform d(etα) where t ∈ R. Fix an almost complex structure Jsuch that J(∂t) = R, J(kerα) = kerα, and dα(v , Jv) > 0.
If c is a Reeb chord of ΛK then c ×R is a J-holomorphic stripwith boundary on the Lagrangian submanifold ΛK × R.
Knot contact homology
Legendrian contact homology is in a sense the “Floerhomology” of the action functional
∫γ α. Because of certain
bubbling phenomena this theory becomes more non-linearthan in the ordinary case.
The symplectization of U∗S3 is U∗S3 × R with symplecticform d(etα) where t ∈ R. Fix an almost complex structure Jsuch that J(∂t) = R, J(kerα) = kerα, and dα(v , Jv) > 0.
If c is a Reeb chord of ΛK then c ×R is a J-holomorphic stripwith boundary on the Lagrangian submanifold ΛK × R.
Knot contact homology
Legendrian contact homology is in a sense the “Floerhomology” of the action functional
∫γ α. Because of certain
bubbling phenomena this theory becomes more non-linearthan in the ordinary case.
The symplectization of U∗S3 is U∗S3 × R with symplecticform d(etα) where t ∈ R. Fix an almost complex structure Jsuch that J(∂t) = R, J(kerα) = kerα, and dα(v , Jv) > 0.
If c is a Reeb chord of ΛK then c ×R is a J-holomorphic stripwith boundary on the Lagrangian submanifold ΛK × R.
Knot contact homology
The Legendrian contact homology algbera of ΛK is the freeunital (non-commutative) algebra
A(ΛK ) = C[H2(U∗S3,ΛK )]⟨
Reeb chords⟩
= C[e±x , e±p,Q±1]⟨
Reeb chords⟩
A(ΛK ) is a DGA. The differential ∂ : A(ΛK )→ A(ΛK ) islinear, satisfies Leibniz rule, decreases grading by 1, and isdefined on generators through a holomorphic curve count.The DGA (A(ΛK ), ∂) is invariant under deformations up tohomotopy and in particular up to quasi-isomorphism.
Knot contact homology
The Legendrian contact homology algbera of ΛK is the freeunital (non-commutative) algebra
A(ΛK ) = C[H2(U∗S3,ΛK )]⟨
Reeb chords⟩
= C[e±x , e±p,Q±1]⟨
Reeb chords⟩
A(ΛK ) is a DGA. The differential ∂ : A(ΛK )→ A(ΛK ) islinear, satisfies Leibniz rule, decreases grading by 1, and isdefined on generators through a holomorphic curve count.The DGA (A(ΛK ), ∂) is invariant under deformations up tohomotopy and in particular up to quasi-isomorphism.
Knot contact homology
Knot contact homology
Knot contact homology
In general, the knot contact homology can be explicitlycomputed, via Morse flow trees as an adiabatic limit ofholomorphic disks, from a braid presentation of a knot. For abraid on n strands the algebra has n(n − 1) generators indegree 0, n(2n − 1) in degree 1, and n2 in degree 2.
The unknot
A(ΛU) = C[e±x , e±p,Q±1]〈c, e〉, |c | = 1, |e| = 2,
∂e = c − c = 0, ∂c = 1− ex − ep + Qexep
Knot contact homology
In general, the knot contact homology can be explicitlycomputed, via Morse flow trees as an adiabatic limit ofholomorphic disks, from a braid presentation of a knot. For abraid on n strands the algebra has n(n − 1) generators indegree 0, n(2n − 1) in degree 1, and n2 in degree 2.
The unknot
A(ΛU) = C[e±x , e±p,Q±1]〈c, e〉, |c | = 1, |e| = 2,
∂e = c − c = 0, ∂c = 1− ex − ep + Qexep
Knot contact homology
The right handed trefoil (differential in degree 1):
A(ΛT ) = C[e±x , e±p,Q±1]〈a12, a21, b12, b21, cij , eij〉i ,j∈{1,2},|aij | = 0, |bij | = |cij | = 1, |eij | = 2,
∂b12 = e−xa12 − a21,
∂b21 = exa21 − a12,
∂c11 = epex − ex − (2Q − ep)a12 − Qa212a21,
∂c12 = Q − ep + epa12 + Qa12a21,
∂c21 = Q − ep + epexa21 + Qa12a21,
∂c22 = ep − 1− Qa21 + epa12a21,
Augmentation variety
Consider A = A(ΛK ) as a family over (C∗)3 of C-algebras,where points in (C∗)3 correspond to values of coefficients(ex , ep,Q).
An augmentation of A is a chain map ε : A → C, ε ◦ ∂ = 1, ofDGAs, where we consider C as a DGA with trivial differentialconcentrated in degree 0.
The augmentation variety VK is the algebraic closure of{(ex , ep,Q) ∈ (C∗)3 : A has augmentation
}.
Augmentation variety
Consider A = A(ΛK ) as a family over (C∗)3 of C-algebras,where points in (C∗)3 correspond to values of coefficients(ex , ep,Q).
An augmentation of A is a chain map ε : A → C, ε ◦ ∂ = 1, ofDGAs, where we consider C as a DGA with trivial differentialconcentrated in degree 0.
The augmentation variety VK is the algebraic closure of{(ex , ep,Q) ∈ (C∗)3 : A has augmentation
}.
Augmentation variety
Consider A = A(ΛK ) as a family over (C∗)3 of C-algebras,where points in (C∗)3 correspond to values of coefficients(ex , ep,Q).
An augmentation of A is a chain map ε : A → C, ε ◦ ∂ = 1, ofDGAs, where we consider C as a DGA with trivial differentialconcentrated in degree 0.
The augmentation variety VK is the algebraic closure of{(ex , ep,Q) ∈ (C∗)3 : A has augmentation
}.
Augmentation variety
The augmentation polynomial AK is the polynomialcorresponding to the hypersurface VK . It can be computedfrom the formulas for the differential by elimination theory.
The unknot U:
AU(ex , ep,Q) = 1− ex − ep + Qexep.
The trefoil T :
AT (ex , ep,Q) = (e4p − e3p)e2x
+ (e4p − Qe3p + 2Q2e2p − 2Qe2p − Q2ep + Q2)ex
+ (−Q3ep + Q4).
Augmentation variety
The augmentation polynomial AK is the polynomialcorresponding to the hypersurface VK . It can be computedfrom the formulas for the differential by elimination theory.
The unknot U:
AU(ex , ep,Q) = 1− ex − ep + Qexep.
The trefoil T :
AT (ex , ep,Q) = (e4p − e3p)e2x
+ (e4p − Qe3p + 2Q2e2p − 2Qe2p − Q2ep + Q2)ex
+ (−Q3ep + Q4).
Augmentation variety
The augmentation polynomial AK is the polynomialcorresponding to the hypersurface VK . It can be computedfrom the formulas for the differential by elimination theory.
The unknot U:
AU(ex , ep,Q) = 1− ex − ep + Qexep.
The trefoil T :
AT (ex , ep,Q) = (e4p − e3p)e2x
+ (e4p − Qe3p + 2Q2e2p − 2Qe2p − Q2ep + Q2)ex
+ (−Q3ep + Q4).
Physics results
We apply Witten’s argument, equating partition functions inU(N) Chern-Simons gauge theory on a 3-manifold M and inopen topological strings in T ∗M with N Lagrangian branes onthe 0-section M, to the case of a knot K in M = S3.
Add an additional Lagrangian brane along LK . Integrating thestrings in LK ∩ S3 = K out corresponds to inserting
det(1− Hol(K )e−x)−1
in the Chern-Simons path integral, where Hol(K ) is theholonomy of the U(N)-connection around K and e−x theholonomy of the U(1)-connectionon LK .
Physics results
We apply Witten’s argument, equating partition functions inU(N) Chern-Simons gauge theory on a 3-manifold M and inopen topological strings in T ∗M with N Lagrangian branes onthe 0-section M, to the case of a knot K in M = S3.
Add an additional Lagrangian brane along LK . Integrating thestrings in LK ∩ S3 = K out corresponds to inserting
det(1− Hol(K )e−x)−1
in the Chern-Simons path integral, where Hol(K ) is theholonomy of the U(N)-connection around K and e−x theholonomy of the U(1)-connectionon LK .
Physics results
Expanding
det(1− e−xHol(K ))−1 =∑k
trSk Hol(K ) e−kx
then gives
ΨK (ex , egs ,Q) : = ZGW (LK )/ZGW
=∑k
〈trSk Hol(K )〉e−kx =∑k
Hk(K ) e−kx ,
where Hk(K ) is a HOMFLY-invariant which is polynomial inq = egs and Q = qN = egsN . Here the string couplingconstant is gs = 2πi
N+k , where k is the level in Chern-Simonspath integral.
Physics results
X = T ∗S3 is a quadric in C4, a resolution of a cone. There isanother resolution Y , total space of O(−1)⊕O(−1)→ CP1.
Physics results
Gopakumar-Vafa show: if area(CP1) = t = Ngs , andQ = et = qN , then
ZGW (X ;N, gs) = ZGW (Y ; gs ,Q),
relating open curve counts in X to closed curve counts in Y .
Physics results
For a knot K ⊂ S3, LK can be shifted off the 0-section by anon-exact Lagrangian isotopy. Then LK ⊂ Y andGopakumar-Vafa’s argument gives
ΨK (ex , egs ,Q) = ZGW (Y ; LK )/ZGW (Y )
Physics results
Witten’s argument relates constant curves on LK toGL(1)-gauge theory on the solid torus which corresponds toordinary QM in the periods of the connection. This gives
pΨK (x) = gs∂
∂xΨK (x),
and the usual asymptotics:
ΨK (x) = exp
(1
gs
∫pdx + . . .
).
From the GW -perspective
ΨK (x) = exp
(1
gsWK (x) + . . .
),
where WK (x) is the disk potential of LK and where . . .counts curves with 2g − 2 + h > −1.
Physics results
Witten’s argument relates constant curves on LK toGL(1)-gauge theory on the solid torus which corresponds toordinary QM in the periods of the connection. This gives
pΨK (x) = gs∂
∂xΨK (x),
and the usual asymptotics:
ΨK (x) = exp
(1
gs
∫pdx + . . .
).
From the GW -perspective
ΨK (x) = exp
(1
gsWK (x) + . . .
),
where WK (x) is the disk potential of LK and where . . .counts curves with 2g − 2 + h > −1.
Physics results
We conclude that as we vary LK
p =∂WK
∂x.
This equation gives a local parameterization of an algebraiccurve that in all examples agree with the augmentation variety(and which is important from a mirror symmetry perspective).
Physics results
We conclude that as we vary LK
p =∂WK
∂x.
This equation gives a local parameterization of an algebraiccurve that in all examples agree with the augmentation variety(and which is important from a mirror symmetry perspective).
Augmentations and exact Lagrangian fillings
Exact Lagrangian fillings L of ΛK in T ∗S3 inducesaugmentations by
ε(a) =∑|a|=0
|MA(a)|A.
The map on coefficients are just the induced map onhomology.
Augmentations and exact Lagrangian fillings
There are two natural exact fillings of ΛK : LK andMK ≈ S3 − K . Thus, ep = 1 and ex = 1 belong to VK |Q=1
for any K .
For the unknot AU(ex , ep,Q = 1) = (1− ex)(1− ep).
For the trefoil AT (ex , ep,Q = 1) = (1− ex)(1− ep)(e3p − 1).
Augmentations and exact Lagrangian fillings
There are two natural exact fillings of ΛK : LK andMK ≈ S3 − K . Thus, ep = 1 and ex = 1 belong to VK |Q=1
for any K .
For the unknot AU(ex , ep,Q = 1) = (1− ex)(1− ep).
For the trefoil AT (ex , ep,Q = 1) = (1− ex)(1− ep)(e3p − 1).
Augmentations and exact Lagrangian fillings
There are two natural exact fillings of ΛK : LK andMK ≈ S3 − K . Thus, ep = 1 and ex = 1 belong to VK |Q=1
for any K .
For the unknot AU(ex , ep,Q = 1) = (1− ex)(1− ep).
For the trefoil AT (ex , ep,Q = 1) = (1− ex)(1− ep)(e3p − 1).
Augmentations and non-exact Lagrangian fillings
The Lagrangian conormal can be shifted off of the 0-section.It becomes non-exact but a similar version of compactnessstill holds: closed disks lie in a compact, and disks at infinityare as for ΛK × R.
Augmentations and non-exact Lagrangian fillings
The previous definition of a chain map does not work becauseof new boundary phenomena.
Compare the family of real curves in C2, xy = ε, ε→ 0.
Augmentations and non-exact Lagrangian fillings
We resolve this problem by using bounding chains: fix a chainσD for each rigid disk D that connects its boundary in LK to amultiple of a standard homology generator at infinity.
Augmentations and non-exact Lagrangian fillings
We introduce quantum corrected holomorphic disks withpunctures: these are ordinary holomorphic disks with allpossible insertions of σ along the boundary. (CompareFukaya’s unforgettable marked points.) In the moduli spaceMA(a;σ) of quantum corrected disks, boundary bubblingbecome interior points.
Augmentations and non-exact Lagrangian fillings
Analyzing the boundary then shows that
ε(a) =∑|a|=1
MA(a;σ)eA
is a chain map provided p = ∂WK∂x .
Augmentations and non-exact Lagrangian fillings
We find that p = ∂WK∂x parameterizes a branch of the
augmentation variety.
Here WK is the Gromov-Witten disk potential which is a sumover weighted trees, where there is a holomorphic disk at eachvertex of the tree and where each edge is weighted by thelinking number of the boundaries of the disks at the verticesof its endpoints, as measured by the bounding chains.
Augmentations and non-exact Lagrangian fillings
We find that p = ∂WK∂x parameterizes a branch of the
augmentation variety.
Here WK is the Gromov-Witten disk potential which is a sumover weighted trees, where there is a holomorphic disk at eachvertex of the tree and where each edge is weighted by thelinking number of the boundaries of the disks at the verticesof its endpoints, as measured by the bounding chains.
Many component links
For an m-component link, the GW-disk potential of theconormal filling sees only the individual knot components. Wethus need other Lagrangian fillings of the conormal tori. Suchfillings indeed exist in special cases. In the general case thereare candidates. The GW-disk potential WK of such a fillinggives a Lagrangian (locally xj = ∂W
∂pj) augmentation variety
VK in (C∗)m × (C∗)m.
Physical quantization of VK
To quantize VK , consider the topological A-model on T ∗T 2m
with a co-isotropic A-brane B on the whole space and aLagrangian brane L on VK . The algebra of open string statesfrom B to B gives the Weyl algebra. Strings with one end onB and the other on L has a unique ground state which gives afermion ψK (x) on VK . The theory of this fermion is theD-model on VK . Conjecture: the expectation value of thisfermion living on VK is the (HOMFLY) function,
〈ψK (x)〉 = ΨK (x) = ΨK (x1, . . . , xm).
The action of B − B strings on B − VK strings gives aD-module with characteristic variety VK , this is in line withthe intersection behavior of various components of VK .
This is in line with Garoufalidis’ result that the coloredHOMFLY polynomial is q-holonomic.
Physical quantization of VK
To quantize VK , consider the topological A-model on T ∗T 2m
with a co-isotropic A-brane B on the whole space and aLagrangian brane L on VK . The algebra of open string statesfrom B to B gives the Weyl algebra. Strings with one end onB and the other on L has a unique ground state which gives afermion ψK (x) on VK . The theory of this fermion is theD-model on VK . Conjecture: the expectation value of thisfermion living on VK is the (HOMFLY) function,
〈ψK (x)〉 = ΨK (x) = ΨK (x1, . . . , xm).
The action of B − B strings on B − VK strings gives aD-module with characteristic variety VK , this is in line withthe intersection behavior of various components of VK .
This is in line with Garoufalidis’ result that the coloredHOMFLY polynomial is q-holonomic.
Mathematical quantization of VK – Legendrian SFT
It is possible to define a version of Legendrian SFT in thecurrent set up which we conjecture gives the D-moduledefined by the colored HOMFLY polynomial.
Let the degree 1 Reeb chords be denoted b1, . . . bm and thedegree 0 Reeb chords a1, . . . , an
Additional data: a Morse function f on LK which givesobstruction chains. A 4-chain CK for LK with ∂CK = 2[LK ]which looks like ±J∇f near the boundary.
Mathematical quantization of VK – Legendrian SFT
It is possible to define a version of Legendrian SFT in thecurrent set up which we conjecture gives the D-moduledefined by the colored HOMFLY polynomial.
Let the degree 1 Reeb chords be denoted b1, . . . bm and thedegree 0 Reeb chords a1, . . . , an
Additional data: a Morse function f on LK which givesobstruction chains. A 4-chain CK for LK with ∂CK = 2[LK ]which looks like ±J∇f near the boundary.
Mathematical quantization of VK – Legendrian SFT
It is possible to define a version of Legendrian SFT in thecurrent set up which we conjecture gives the D-moduledefined by the colored HOMFLY polynomial.
Let the degree 1 Reeb chords be denoted b1, . . . bm and thedegree 0 Reeb chords a1, . . . , an
Additional data: a Morse function f on LK which givesobstruction chains. A 4-chain CK for LK with ∂CK = 2[LK ]which looks like ±J∇f near the boundary.
Legendrian SFT
Legendrian SFT
We define the GW-potential eF as the generating function oforiented graphs with holomorphic curves at vertices, andintersections with chains at the edges weighted by ±1
2 :
F =∑
Cχ,m,I g−χs emxai1 . . . air
Legendrian SFT
Let H(bj) denote the count of rigid holomorphic curves in thesymplectization with a positive puncture at bj :
H(bj) =∑
Cχ,m,n,I ,J g−χs emxenp ai1 . . . air g
ls
∂
∂aj1. . .
∂
∂ajl.
Then if p = gs∂∂x , e−FH(bj)e
F counts ends of a1-dimensional moduli space and in particular:
H(bj)eF = 0.
We conjecture that the above system give a generator AK
with AKeF0 = 0 for the D-module ideal corresponding to the
colored HOMFLY of K (generators AjK if K is a link).
Legendrian SFT and HOMFLY
For the unknot there are no (formal) higher genus curves andthe operator equation is
AU(ex , ep,Q) = (1− ex − ep − Qexep)ΨU = 0.
For the Hopf link L Reeb chord generators are as for thetrefoil. The relevant parts for the operator H is as follows:
H(c11) = (1− ex1 − ep1 + Qex1ep1) + g2s ∂a12∂a21 +O(a),
H(c22) = (1− ex2 − ep2 + Qex2ep2) + Qex2ep2g2s ∂a12∂a21 +O(a),
H(c12) = (ep2e−p1 − Qex2ep2)gs∂a12
+ g−1s (e−gs − 1)(1− ex2)a21 +O(a2),
H(c21) = (Qex1ep1 − egs ep1e−p2)gs∂a21
+ g−1s
((egs (egs − 1)− e2gs (egs − 1)ex1)ep1e−p2
+(egs − 1)Qex1ep1g2s ∂a12∂a21
)a12 +O(a2).
Legendrian SFT and HOMFLY
For the unknot there are no (formal) higher genus curves andthe operator equation is
AU(ex , ep,Q) = (1− ex − ep − Qexep)ΨU = 0.
For the Hopf link L Reeb chord generators are as for thetrefoil. The relevant parts for the operator H is as follows:
H(c11) = (1− ex1 − ep1 + Qex1ep1) + g2s ∂a12∂a21 +O(a),
H(c22) = (1− ex2 − ep2 + Qex2ep2) + Qex2ep2g2s ∂a12∂a21 +O(a),
H(c12) = (ep2e−p1 − Qex2ep2)gs∂a12
+ g−1s (e−gs − 1)(1− ex2)a21 +O(a2),
H(c21) = (Qex1ep1 − egs ep1e−p2)gs∂a21
+ g−1s
((egs (egs − 1)− e2gs (egs − 1)ex1)ep1e−p2
+(egs − 1)Qex1ep1g2s ∂a12∂a21
)a12 +O(a2).
Legendrian SFT and HOMFLY
After the change of variables,
ex′1 = egs ex1 , ep
′1 = egs ep1 ;
ex′2 = Q−1e−x2 , ep
′2 = e−gsQ−1e−p2 ;
Q ′ = egsQ, g ′s = −gs ,
we find the D-module ideal generators
A1L = (ex1 − ex2) + (ep1 − ep2)− Q(ex1ep1 − ex2ep2)
A2L = (1− e−gs ex1 − ep1 + Qex1ep1)(ex1 − ep2)
A3L = (1− e−gs ex2 − ep2 + Qex2ep2)(ex2 − ep1),
in agreement with HOMFLY.
Legendrian SFT and HOMFLY
Similarly, for the trefoil we get the D-module ideal generator
AT = (Q2 − Qep)egs (Q − egs e2p)Q2
+((Q2 − Qep)e5gs (Q − egs e2p)e2p
)ex
+((qe2p − ep)e3gs (Q − e3gs e2p)Q2
)ex
+(egs (Q − egs e2p)(Q − e2gse2p)(Q − e3gs e2p)
)ex
+((egs e2p − ep)e7gs (Q − e3gs e2p)e2p
)e2x .
which after a change of variables is in agreement withHOMFLY.
Note that the classical limit gs → 0 of this is
(Q − e2p)AT (ex , ep,Q).
Legendrian SFT and HOMFLY
Similarly, for the trefoil we get the D-module ideal generator
AT = (Q2 − Qep)egs (Q − egs e2p)Q2
+((Q2 − Qep)e5gs (Q − egs e2p)e2p
)ex
+((qe2p − ep)e3gs (Q − e3gs e2p)Q2
)ex
+(egs (Q − egs e2p)(Q − e2gse2p)(Q − e3gs e2p)
)ex
+((egs e2p − ep)e7gs (Q − e3gs e2p)e2p
)e2x .
which after a change of variables is in agreement withHOMFLY.
Note that the classical limit gs → 0 of this is
(Q − e2p)AT (ex , ep,Q).