TANGLE CONTACT HOMOLOGY JOHAN ASPLUND Abstract. Knot contact homology is an ambient isotopy invariant of knots and links in R3. The

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TANGLE CONTACT HOMOLOGY

JOHAN ASPLUND

Abstract. Knot contact homology is an ambient isotopy invariant of knots and links in R3. The

purpose of this paper is to extend this deﬁnition to an ambient isotopy invariant of tangles and

prove that gluing of tangles gives a gluing formula for knot contact homology. As a consequence of

the gluing formula we obtain that the tangle contact homology detects triviality of tangles.

Contents

1. Introduction 1

1.1. Statement of results 2

1.2. Construction and method of proof 3

1.3. Future directions 4

1.4. Related work 5

Outline 5

Acknowledgments 5

2. Simplicial decompositions for Weinstein pairs 5

2.1. Construction of simplicial decompositions for Weinstein pairs 7

2.2. Weinstein hypersurfaces and simplicial decompositions 13

2.3. Simplicial descent for Weinstein pairs 18

2.4. Relative Legendrian submanifolds 20

3. Tangle contact homology and gluing formulas 24

3.1. Unframed knot contact homology 24

3.2. Knot contact homology for links 26

3.3. Tangle contact homology 27

3.4. Gluing formula recovering knot contact homology 28

3.5. Gluing formula recovering tangle contact homology 33

4. Untangle detection 34

4.1. Unknotted case 36

4.2. General case 36

5. Examples 37

References 43

1. Introduction

Knot contact homology is an ambient isotopy invariant of knots and links in R3ﬁrst deﬁned

combinatorially by Ng [Ng05a, Ng05b]. It is known to be isomorphic to the Legendrian contact

homology of the unit conormal bundle of the link, with homology coeﬃcients [EENS13]. It is known

that Knot contact homology detects the unlink, cabled knots, composite knots and torus knots,

see [GL17], [Ng08, Proposition 5.10] and [CELN17, Corollary 1.5]. Moreover, an enhanced version

of the knot contact homology is in fact a complete knot invariant [ENS18]. We refer the reader to

[Ng14] and references therein for a complete survey of knot contact homology.

arXiv:2210.03036v4 [math.SG] 20 May 2024

2 JOHAN ASPLUND

The unit conormal bundle of a link K⊂R3is the set

ΛK:=(x, v)∈T∗R3x∈K, |v|= 1,⟨v, TxK⟩= 0,

for some metric on R3. The unit cotangent bundle of R3, denoted by ST ∗R3, is a contact man-

ifold when equipped with the one-form α:= (p1dx +p2dy +p3dz)|ST ∗R3where (x, y, z) are local

coordinates in R3and (p1, p2, p3) are local coordinates in the ﬁber directions. The unit conormal

ΛK⊂ST ∗R3is a Legendrian submanifold, meaning that α|ΛK= 0.

In this paper we are concerned with the fully non-commutative version of knot contact homology

of links in R3[EENS13, CELN17, ENS18] which for our purposes is deﬁned as the Chekanov–

Eliashberg dg-algebra of ΛKin the unit cotangent bundle of R3with loop space coeﬃcients

KCC∗

EENS(K):=CE∗(ΛK, C−∗(ΩΛK)),

see [EL23] for the deﬁnition of Chekanov–Eliashberg dg-algebras with loop space coeﬃcients.

The homology of the Chekanov–Eliashberg dg-algebra is a Legendrian isotopy invariant of Leg-

endrian submanifolds in contact manifolds [EES05, EES07, EN15, DR16, Kar20]. The diﬀerential

of the Chekanov–Eliashberg dg-algebra is deﬁned by counting punctured J-holomorphic disks in

R×ST ∗R3with boundary in R×ΛK. It was ﬁrst studied independently by Chekanov and Eliash-

berg [Che02, Eli98] and it is part of the more general symplectic ﬁeld theory package deﬁned by

Eliashberg–Givental–Hofer [EGH00].

Using methods developed in [AE22] we may understand the Chekanov–Eliashberg dg-algebra

with loop space coeﬃcients as the Chekanov–Eliashberg dg-algebra of a cotangent neighborhood

of ΛK, denoted by N(ΛK), together with a choice of handle decomposition hwhich encodes all

Weinstein handles and their attaching maps, see Deﬁnition 2.5 for details. Using the latter point

of view, we deﬁne KCC∗(K, h):=CE∗((N(ΛK), h); T∗R3) using notation as in [AE22]. For each

choice of h,KCC∗(K, h) is an ambient isotopy invariant of the link K⊂R3, and for a certain

choice of hit recovers KCC∗

EENS(K), see Theorem 3.21.

1.1. Statement of results. The main construction in this paper is that of tangle contact homology.

It is the homology of a dg-algebra associated to an oriented tangle Tin R3

x≥0, and a choice of handle

decomposition hof a Weinstein neighborhood of its unit conormal bundle which is denoted by

N(ΛT), see Deﬁnition 2.7. The unit conormal bundle of Tis a Legendrian manifold-with-boundary

ΛT⊂T∗R3

x≥0. We show that tangle contact homology is independent of choices and an ambient

isotopy invariant of T(with ﬁxed boundary).

Suppose that K⊂R3is an oriented link and H⊂R3is a smooth submanifold that is diﬀeomor-

phic to R2such that Kintersects Htransversely. The hypersurface Hsplits Kinto two tangles

T1, T2⊂R3

x≥0, and conversely we say that Kis the gluing of T1and T2.

In the following, let X→Ydenote an oriented binormal geodesic for X, Y ∈ {T1, T2, H}in the

appropriate copy of R3

x≥0. Let GTi→H→Tjdenote the set of words of oriented binormal geodesic

chords of the form

Ti→H→ · · · → H→Tj.

Let ⟨S⟩denote the free algebra on the set S,∗denote free product of algebras, and ∗Cdenote

amalgamated free product of algebras, see Remark 2.46 for details. Our main result is the following

gluing formula for KCC∗(K, h).

Theorem 1.1 (Theorem 3.32).Let T1and T2be two tangles in R3

x≥0whose gluing is the link

K⊂R3, where H:=∂R3

x≥0. Let h1and h2be choices of handle decomposition of N(ΛT1)and

N(ΛT2), respectively. The free algebra

K:=KCC∗(T1, h1)∗KCC∗(∂T,h∂)KCC∗(T2, h2)∗ ⟨GT1→H→T2⟩∗⟨GT2→H→T1⟩ ⊂ KCC∗(K, h1#h2),

is a canonical subalgebra. Furthermore, ﬁxing a generic choice of metric and almost complex struc-

ture we have the following:

TANGLE CONTACT HOMOLOGY 3

(1) Kis dg-algebra when equipped with the same diﬀerential as the one in KCC∗(K, h1#h2)

such that KCC∗(T1, h1), KCC∗(T2, h2)⊂Kare dg-subalgebras.

(2) There is a quasi-isomorphism of dg-algebras

K∼

=KCC∗(K, h1#h2).

Remark 1.2.(1) Only knowing the dg-algebras KCC∗(T1, h1) and KCC∗(T2, h2) is not suﬃ-

cient to recover KCC∗(K, h1#h2). To recover KCC∗(K, h1#h2) we also need to know the

set of oriented binormal geodesic chords between Tiand Hand between Hand Tjin the

two copies of R3

x≥0.

(2) The two algebras ⟨GT1→H→T2⟩and ⟨GT2→H→T1⟩and their free product by themselves are

not dg-algebras when equipped with the diﬀerential in KCC∗(K, h1#h2), and we do not

know any way of expressing KCC∗(K, h1#h2) as the colimit of dg-algebras in general.

(3) See Section 3.4.1 and in particular Figures 15 and 16 for a discussion which handle decom-

positions to consider in order to recover KCC∗

EENS(K).

(4) Theorem 1.1 and every construction in this paper still works if the link K⊂R3is split into

tangles by any smooth codimension 1 submanifold of R3. A particularly important case

is that of S2, which is more amenable to concrete calculations (see Section 5). We have

chosen to state results for R3

≥0as it is the most “symmetric”.

(5) Theorem 1.1 is the simplest gluing formula to state, however, the general framework is

signiﬁcantly more ﬂexible. For instance we can further decompose R3

x≥0by splitting it

further by hypersurfaces R3

x=afor a > 0 which allows us to ﬁnd gluing formulas for tangle

contact homology itself, see Theorem 3.37.

We have the following geometric characterization of tangle contact homology.

Theorem 1.3 (Theorem 3.29).Let Tbe a tangle in R3

x≥0where H:=∂R3

x≥0. Let hbe a choice

of handle decomposition of N(ΛT). Let Tdenote the algebra generated by

(1) Composable words of oriented binormal geodesic chords in R3

x≥0of the form T→Tor

T→H→ · · · → H→T.

(2) Oriented binormal geodesic chords ∂T →∂T in H.

Then T⊂KCC∗(T, h)is a canonical subalgebra. For generic choices of metrics and almost complex

structures Tbecomes a dg-algebra when equipped with the same diﬀerential as in KCC∗(T, h)and

there is a quasi-isomorphism

T∼

=KCC∗(T, h).

Remark 1.4.The construction of KCC∗(T, h) makes sense for tangles of any dimension, and a

version of Theorem 1.1 still holds in this case.

Theorem 1.5 (Theorem 4.10).Let hbe one of the two handle decompositions described in Sec-

tion 3.4.1. Then KCC∗(T, h)detects the r-component trivial tangle.

Theorem 1.5 is a consequence of Theorem 1.1, and by the property that the fully non-commutative

knot contact homology detect the unlink. In fact, we are also utilizing a gluing formula for tangle

contact homology itself described in Remark 1.2(5).

1.2. Construction and method of proof. We ﬁrst give a rough description of the deﬁnition of

tangle contact homology of a tangle T⊂R3

x≥0. Taking the unit conormal bundle of T(denoted by

ΛT) naturally yields a Legendrian submanifold in the contact boundary of the (open) Weinstein

sector T∗R3

x≥0. This sector corresponds to the Weinstein pair (B6, T ∗H) where H:=∂R3

x≥0and

ΛTis viewed as a Legendrian submanifold in ∂B6with Legendrian boundary in ∂T ∗H. After

picking a handle decomposition hTof N(ΛT) which is a Weinstein neighborhood of ΛT, the dg-

algebra KCC∗(T, hT) is deﬁned as the Chekanov–Eliashberg dg-algebra of the pair (N(ΛT), hT)

4 JOHAN ASPLUND

in (B6, T ∗H), see Section 3.3 for details. This is the natural deﬁnition of what the Chekanov–

Eliashberg dg-algebra with loop space coeﬃcients would be for a Legendrian with boundary.

In order to prove Theorem 1.1 we generalize the gluing formulas for Chekanov–Eliashberg dg-

algebras proven in [AE22, Asp23], to hold for Chekanov–Eliashberg dg-algebras with loop space

coeﬃcients. We now give a rough sketch of the gluing formula.

First consider two Weinstein pairs (X1, V ) and (X2, V ), we glue X1and X2together along their

common Weinstein hypersurface V2n−2→∂X2nto obtain a new Weinstein manifold denoted by

X1#VX2. This operation is called Weinstein connected sum [Avd21, Eli18, ´

AGEN22], and gives

gluing formulas for the Chekanov–Eliashberg dg-algebra with ﬁeld coeﬃcients of the Legendrian

attaching link [AE22, Asp23].

For the generalization of the above, assume we have two Weinstein pairs (X1, W1) and (X2, W2)

where Wi=V2n−2

i#Q2n−4(V′)2n−2for i∈ {1,2}. We glue X1and X2along the common Weinstein

subhypersurface (V′)2n−2⊂W2n−2

i. The result is a new Weinstein pair (X, V ), where X=

X1#V′X2, and V=V1#QV2, see Figure 1. We show that this type of gluing yields a gluing

formula for the Chekanov–Eliashberg dg-algebra of the Weinstein hypersurface (V, h) where his a

handle decomposition of V, which for certain choices of his the Chekanov–Eliashberg dg-algebra

with loop space coeﬃcients, see Section 2.3 for details.

X1X2

V′×DεT∗∆1

V1×(−ε, ε)V2×(−ε, ε)

Q×DεT∗∆1

Figure 1. The Weinstein pair (X1#V′X2, V1#QV2) obtained by gluing together

the Weinstein pairs (X1, W1) and (X2, W2) along their common Weinstein subhy-

persurface V′⊂W1, W2.

1.3. Future directions. Knot contact homology of (framed oriented) knots in R3is closely related

to several smooth knot invariants. Some examples include string homology, the cord algebra,

the Alexander polynomial and the augmentation polynomial which is closely related to the A-

polynomial (and also conjectured to be related to a specialization of the HOMFLY-PT polynomial)

[Ng05b, Ng08, CELN17, Ekh18]. The knot group and its peripheral subgroup can be extracted

from the enhanced version of the knot contact homology [ENS18].

In view of Theorem 1.3, it is natural to propose the following conjecture, extending the scope of

[CELN17, Theorem 1.1].

Conjecture 1.6. If Tis a framed oriented r-component tangle in R3

x≥0there exists a choice of

handle decomposition hof N(ΛT)such that we have an isomorphism of Z[µ±1

1, . . . , µ±1

r]-algebras

KCH0(T, h)∼

=Cord(T, H).

Here Cord(T, H) denotes the framed cord algebra of T, relative to H. It is deﬁned as the algebra

of cords that are allowed to have none, either, or both endpoints on H, modulo homotopy and

Skein relations (compare with the usual deﬁnition in [Ng05b, Section 4.3] and [Ng08, Section 2.1]).

TANGLE CONTACT HOMOLOGY 5

Remark 1.7.A reﬁnement of Conjecture 1.6 is that the choice of handle decomposition mentioned

should be the one depicted in Figure 16. There is a certain asymmetry in choices of handle

decompositions h1and h2to recover KCC∗

EENS(K) as described in Section 3.4.1. Thus we expect

to ﬁnd a gluing formula similar to that of Theorem 1.1, recovering Cord(K) from Cord(T1, H) and

]

Cord(T2, H) where ]

Cord(T2, H):=KCH0(T2, h2) with h2chosen as in Figure 15. We also expect

there to be a suitable topological interpretation of Cord(T1, H) and ]

Cord(T2, H).

1.4. Related work. Dattin has deﬁned a sutured Legendrian isotopy invariant of sutured Legen-

drian submanifolds in sutured contact manifolds called cylindrical sutured homology [Dat22b]. In

case the sutured contact manifold is balanced, we expect that the dg-algebra of ΛT⊂∂T ∗R3

x≥0

deﬁned by Dattin is quasi-isomorphic to KCC∗(T, h).

Let Σbe a surface without boundary and let B⊂Σ×[−1,1] be a braid. It is proven that a

certain quotient of cylindrical sutured homology of ΛB⊂∂T ∗(Σ×[−1,1]) together with a product

structure gives a complete invariant of braids [Dat22b, Dat].

Outline. In Section 2 we generalize results from [Asp23]. Namely we construct simplicial decom-

positions of Weinstein pairs, and prove that the Chekanov–Eliashberg dg-algebra of top attaching

spheres satisﬁes a gluing formula. This specializes to gluing formulas for the Chekanov–Eliashberg

dg-algebra with loop space coeﬃcients.

In Section 3 we deﬁne tangle contact homology for tangles in R3

x≥0. We apply the machinery of

Section 2 to show that gluing of tangles induces gluing formulas for the tangle contact homologies.

In Section 5 we compute tangle contact homology in some examples and end with a calculation

of the knot contact homology of the unknot via the gluing formula.

Acknowledgments. The author thanks Tobias Ekholm and Cˆome Dattin for helpful discussions,

and Lenhard Ng for his correspondence. This paper grew out as an oﬀshoot from on-going col-

laboration with William E. Olsen, to whom the author extends a special thanks to for carefully

reading earlier drafts of this paper. Finally, the author thanks the anonymous referee whose helpful

comments improved the exposition of this paper. The author was supported by the Knut and Alice

Wallenberg Foundation.

2. Simplicial decompositions for Weinstein pairs

In this section we generalize the notion of a simplicial decomposition of a Weinstein manifold

introduced in [Asp23] to Weinstein pairs. We ﬁrst review some basic deﬁnitions we use throughout

the paper.

ALiouville domain is a pair (X2n, λ) of a smooth 2n-dimensional manifold-with-boundary X

and a one-form λsuch that ω:=dλ is symplectic and the ω-dual of λ(called the Liouville vector

ﬁeld) is outwards pointing along ∂X. A Liouville manifold is an exact symplectic manifold (X, λ)

which admits an exhaustion by successively larger Liouville domains, such that the Liouville vector

ﬁeld is complete. The skeleton of a Liouville manifold, denoted by Skel X, is the subset which does

not escape every successive exhausting Liouville domain, under the ﬂow of the Liouville vector ﬁeld.

AWeinstein manifold is a Liouville manifold for which the Liouville vector ﬁeld is gradient-like for

a Morse function on X. It is well-known that any Weinstein manifold can equivalently be obtained

by successive attachments of standard Weinstein handles of index k∈ {0, . . . , n}, see [Wei91]. A

Weinstein manifold Xis called subcritical if it can be built using no Weinstein handles of index n.

Deﬁnition 2.1 (Weinstein hypersurface).Let (X2n, λ)bea2n-dimensional Weinstein manifold. A

Weinstein hypersurface consists of a (2n−2)-dimensional Weinstein manifold (V2n−2, λV) together

with a Weinstein embedding (V, λV)→(X\Skel X, λ) such that the induced map V→∂X is an

embedding. We will use the shorthand V →∂X to denote a Weinstein hypersurface.

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TANGLECONTACTHOMOLOGYJOHANASPLUNDAbstract.KnotcontacthomologyisanambientisotopyinvariantofknotsandlinksinR3.Thepurposeofthispaperistoextendthisdefinitiontoanambientisotopyinvariantoftanglesandprovethatgluingoftanglesgivesagluingformulaforknotcontacthomology.Asaconsequenceofthegluingformulaweobtainth...

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TANGLE CONTACT HOMOLOGY JOHAN ASPLUND Abstract. Knot contact homology is an ambient isotopy invariant of knots and links in R3. The

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