Filtrations on the knot contact homology of transverse knots

Abstract

We construct a new invariant of transverse links in the standard contact structure on ${\mathbb R }^3.$ This invariant is a doubly filtered version of the knot contact homology differential graded algebra (DGA) of the link, see (Ekholm et al., Knot contact homology, Arxiv:1109.1542, 2011; Ng, Duke Math J 141(2):365–406, 2008). Here the knot contact homology of a link in ${\mathbb R }^3$ is the Legendrian contact homology DGA of its conormal lift into the unit cotangent bundle $S^*{\mathbb R }^3$ of ${\mathbb R }^3$, and the filtrations are constructed by counting intersections of the holomorphic disks of the DGA differential with two conormal lifts of the contact structure. We also present a combinatorial formula for the filtered DGA in terms of braid representatives of transverse links and apply it to show that the new invariant is independent of previously known invariants of transverse links.

1 Introduction

Constructing effective invariants of transverse knots in contact 3-manifolds that go beyond the obvious homotopy invariants has been a notoriously difficult problem. For knots transverse to the standard contact structure on ${\mathbb R }^3$, the first such invariant proven to be effective was constructed only recently, using combinatorial knot Heegaard Floer homology [14]; a related transverse invariant in the Heegaard Floer homology of a general compact contact $3$-manifold was subsequently constructed in [10]. In this paper we define and combinatorially present a new invariant for transverse links in the standard contact ${\mathbb R }^3$. The new invariant is a filtration on the knot contact homology differential graded algebra (DGA) [4, 12] for transverse links and it appears to be quite different from the Heegaard Floer invariants. The construction of the invariant can be applied more generally to produce invariants of transverse links in any contact 3-manifold and might even say interesting things in higher dimensions, but the details are considerable more difficult, as are the computations. So we defer the discussion of the general case to a future paper.

1.1 Transverse knot contact homology

In [12], the third author constructs a combinatorial DGA associated to a framed knot $K \subset {\mathbb R }^3$ and shows that its homology, combinatorial knot contact homology, is an invariant of framed knots. This invariant detects the unknot and encodes the Alexander polynomial, among other things.

The current authors prove in [4] that the combinatorial DGA is (stable tame) isomorphic to the Legendrian contact homology DGA of the conormal lift of a knot $K$ and also generalize the combinatorial description of knot contact homology to many-component links. Here the conormal lift of a link $K$ is a union $\varLambda _K$ of Legendrian tori in the unit cotangent bundle $S^{*}{\mathbb R }^{3}$ of ${\mathbb R }^{3}$ with the contact structure given as the kernel of the canonical $1$-form. The Legendrian contact homology DGA of $\varLambda _K$ is an algebra generated by its Reeb chords with differential defined by a count of holomorphic disks in $S^{*}{\mathbb R }^{3}\times {\mathbb R }$ in the spirit of symplectic field theory [8]. Its calculation in [4] uses the contactomorphism $S^{*}{\mathbb R }^{3}\approx T^{*}S^{2}\times {\mathbb R }$ to transfer the holomorphic disk count in $S^{*}{\mathbb R }\times {\mathbb R }$ to $T^{*}S^{2}$. The disk count is then carried out using the relation between holomorphic disks and Morse flow trees [2].

We will call the Legendrian invariant of $\varLambda _K$ defined using holomorphic disks the knot contact homology of $K$. We denote the underlying DGA by $(KC\!\fancyscript{A}(K),\partial )$ and its homology by $KCH(K)$. The algebra $KC\!\fancyscript{A}(K)$ is a free graded tensor algebra over the coefficient ring $R={\mathbb Z }[H_1(\varLambda _K)]={\mathbb Z }[\lambda _1^{\pm 1}, \mu ^{\pm 1}_1,\dots ,\lambda _r^{\pm 1},\mu _r^{\pm 1}]$, where $r$ is the number of components of $K$; the differential $\partial$ depends on $K$ as well, but we suppress this dependence to simplify notation.

Let $(x_1,x_2,x_3)$ be coordinates on ${\mathbb R }^{3}$ and let $\xi _0=\ker (dx_3-x_1\,dx_2+x_2\,dx_1)$ denote the standard tight, rotationally symmetric contact structure on ${\mathbb R }^3.$ For links $K$ transverse to $\xi _0$, we extend the coefficient ring of the knot contact homology DGA to $R[U,V],$ where $U$ and $V$ are two formal variables which encode intersections of holomorphic disks with the natural lifts of $\xi _0$ that correspond to its two coorientations. The resulting DGA will be denoted $(KC\!\fancyscript{A}^-(K),\partial ^-)$ and has a double filtration: by positivity of intersections the differential $\partial ^-$ does not decrease the exponents of $U$ or $V$. Our main result is as follows.

Theorem 1.1

The filtered stable tame isomorphism type of $(KC\!\fancyscript{A}^-(K),\partial ^-),$ and hence its homology, is an invariant of the transverse knot or link $K$ in $({\mathbb R }^3,\xi _0).$ The DGA can be computed from a braid representative of $K$ as described in Theorem 1.3 below.

This invariant of transverse links in particular distinguishes transverse knots with the same self-linking number and with the same Heegaard Floer invariants, see Sect. 5.2. The explicit formula for $(KC\!\fancyscript{A}^-(K),\partial ^-)$ is somewhat involved and is therefore presented separately in Sect. 1.2. It should however be noted that our method for obtaining the combinatorial formula relies heavily on the explicit description of holomorphic disks used in the computation of knot contact homology in [4]. An alternate and purely combinatorial approach to Theorem 1.1 is worked out in [13]. It begins with the combinatorial description of Theorem 1.3 in terms of a braid representation of the knot and proves invariance under transverse isotopy, without any reference to the underlying geometry, via a study of effects of Markov moves. See [13] also for various algebraic properties of the invariant and more detailed calculations which demonstrate its effectiveness.

Those familiar with the algebraic formalism of Heegaard Floer homology will notice that we can construct several other invariants from $(KC\!\fancyscript{A}^-(K),\partial ^-).$ Two simpler invariants $(\widehat{KC\!\fancyscript{A}}(K),\widehat{\partial })$ and $\big (\widehat{\widehat{KC\!\fancyscript{A}}}(K), \widehat{\widehat{\partial }}\big )$ arise by setting $(U,V)=(0,1)$ and $(0,0)$, respectively. Setting $(U,V)=(1,1)$, on the other hand, reduces $(KC\!\fancyscript{A}^-(K),\partial ^-)$ to the original knot contact homology DGA, $(KC\!\fancyscript{A}(K),\partial ).$ Alternatively, one can tensor $(KC\!\fancyscript{A}^-(K),\partial ^-)$ with an extended coefficient ring $R[U^{\pm 1},V^{\pm 1}]$, to obtain a DGA, $(KC\!\fancyscript{A}^\infty (K),\partial ^\infty ).$ It turns out that $(KC\!\fancyscript{A}^\infty (K),\partial ^\infty )$ depends on only the topological isotopy class of $K$, rather than its transverse isotopy class.

Theorem 1.2

If $K$ is a transverse knot, then as a DGA over ${\mathbb Z }[\lambda ^{\pm 1},\mu ^{\pm 1},U^{\pm 1},V^{\pm 1}]$, $(KC\!\fancyscript{A}^\infty (K),\partial ^\infty )$ is an invariant, up to stable tame isomorphism, of the topological knot underlying $K$.

In fact, it equals the Legendrian contact homology of $\varLambda _K$ with coefficients in a relative homology group, see Sect. 3.7 and also [13]. A statement similar to Theorem 1.2 holds for general transverse links but will not be proven here.

1.2 Calculating $(KC\!\fancyscript{A}(K)^{-},\partial ^-)$ from a braid presentation

We now turn to the computation of our transverse link invariant. To this end we first need to introduce some notation. Our notation and conventions differ slightly from those of [13], in which the equivalence of conventions is discussed.

The unit circle $U$ in the plane $\{x_3=0\}$ is transverse to the contact structure defined by the $1$-form $dx_3-x_2\,dx_1+x_1\,dx_2$. By work of Bennequin [1], any transverse link $K$ can be isotoped so that it is braided around $U$, i.e., contained in a tubular neighborhood $U\times D^2$ of $U$ so that it is transverse to the disk fibers. So $K$ can be represented as the closure of a braid $B$ in $U\times D^2=S^1\times D^2$. The braid $B$ is an element of the braid group $B_n$ for some $n$ and can be expressed as a word in the standard generators of $B_n$ and their inverses; here the standard generator $\sigma _j$ ($j=1,\ldots ,n-1$) corresponds to a braid that intertwines strands $j$ and $j+1$ positively.

The Legendrian contact homology DGA of $\varLambda _K$ is a free associative non-commutative unital DGA over the group ring of $H_1(\varLambda _{K};{\mathbb Z })$. Fixing a framing on each component of $K$ (e.g., the standard topological $0$-framing) yields a distinguished basis $\{\lambda _1,\mu _1,\ldots ,\lambda _r,\mu _r\}$ of $H_1(\varLambda _{K};{\mathbb Z })$, where $r$ is the number of components of $K$. More precisely, by identifying $T{\mathbb R }^3$ with $T^*{\mathbb R }^3$, we can identify $\varLambda _K$ with the boundary of a tubular neighborhood of $K$, which consists of $r$ disjoint tori. Each component of $K$ has a natural orientation given by the positive coorientation of the contact structure $\xi _0$. Then $\mu _j,\lambda _j$ are the meridian and framed longitude of the $j$-th torus.

Now, as described above, suppose that $K$ is the closure of a braid $B\in B_n$ in a tubular neighborhood of the standard transverse unknot $U.$ Let ${\fancyscript{A}}_n$ denote the graded unital algebra over ${\mathbb Z }[\lambda _1^{\pm 1},\mu _1^{\pm 1},\dots ,\lambda _r^{\pm 1},\mu _r^{\pm 1}]$ freely generated by

$$\begin{aligned}&\{a_{ij}\}_{1\le i,j\le n;\,i\ne j} \text{ in} \text{ degree} 0,\\&\{b_{ij}\}_{1\le i,j\le n;\,i\ne j} \text{ in} \text{ degree} 1,\\&\{c_{ij}\}_{1\le i,j\le n} \text{ in} \text{ degree} 1\text{,} \text{ and}\\&\{e_{ij}\}_{1\le i,j\le n\,} \text{ in} \text{ degree} 2, \end{aligned}$$

where the degrees of $\lambda _j^{\pm 1}$ and $\mu _j^{\pm 1}$ equal $0$ for $j=1,\dots ,r$. (When there is only one link component, we drop the subscripts on $\lambda _j,\mu _j$.) Let ${\fancyscript{A}}^{(0)}_n$ denote the subalgebra of ${\fancyscript{A}}_n$ of elements of degree $0$.

We define an automorphism $\phi _B:{\fancyscript{A}}^{(0)}_n\rightarrow {\fancyscript{A}}^{(0)}_n$ as follows. Introduce auxiliary variables $\tilde{\mu }_1,\ldots ,\tilde{\mu }_n$ of degree $0$, and write $\tilde{{\fancyscript{A}}}^{(0)}_n$ for the unital algebra over ${\mathbb Z }[\tilde{\mu }_1^{\pm 1},\ldots ,\tilde{\mu }_n^{\pm 1}]$ freely generated by $\{a_{ij}\}_{1\le i,j\le n;i\ne j}$. For $k=1,\ldots ,n-1$, let $\phi _{\sigma _k} : \tilde{{\fancyscript{A}}}^{(0)}_n \rightarrow \tilde{{\fancyscript{A}}}^{(0)}_n$ be given by

$$\begin{aligned} \begin{array}{ll@{\quad }l} \phi _{\sigma _k}(a_{ij})&= a_{ij} \quad&i,j\ne k, k+1\\ \phi _{\sigma _k}(a_{k\,k+1})&= -a_{k+1\,k} \quad&\\ \phi _{\sigma _k}(a_{k+1\,k})&= -\tilde{\mu }_k\tilde{\mu }_{k+1}^{-1}a_{k\,k+1}\quad&\\ \phi _{\sigma _k}(a_{i\,k+1})&= a_{ik}\quad&i\not = k, k+1\\ \phi _{\sigma _k}(a_{k+1\, i})&= a_{ki}\quad&i\not =k, k+1\\ \phi _{\sigma _k}(a_{ik})&= a_{i\,k+1} - a_{ik}a_{k\, k+1}\quad&i < k\\ \phi _{\sigma _k}(a_{ik})&= a_{i\,k+1} - \tilde{\mu }_{k}\tilde{\mu }_{k+1}^{-1}a_{ik}a_{k\, k+1}\quad&i > k+1\\ \phi _{\sigma _k}(a_{ki})&= a_{k+1\,i} - a_{k+1\,k}a_{ki}\quad&i \ne k, k+1\\ \phi _{\sigma _k}\big (\tilde{\mu }_k^{\pm 1}\big )&= \tilde{\mu }_{k+1}^{\pm 1} \\ \phi _{\sigma _k}\big (\tilde{\mu }_{k+1}^{\pm 1}\big )&= \tilde{\mu }_k^{\pm 1} \\ \phi _{\sigma _k}\big (\tilde{\mu }_i^{\pm 1}\big )&= \tilde{\mu }_i^{\pm 1} \quad&i\ne k,k+1. \end{array} \end{aligned}$$

Write $B$ in terms of braid group generators, $B=\sigma _{i_0}^{m_0}\ldots \sigma _{i_l}^{m_l}\in B_n,$ and let $\phi _B = (\phi _{\sigma _{i_0}})^{m_0}\circ \cdots \circ (\phi _{\sigma _{i_l}})^{m_l}$. Then $\phi$ descends to a homomorphism from $B_n$ to the automorphism group of $\tilde{{\fancyscript{A}}}^{(0)}_n$; in particular, $\phi$ satisfies the braid relations.

For $j=i,\ldots ,n$, let $\alpha (i)\in \{1,\ldots ,r\}$ be the number of the link component of $K$ corresponding to the $i$-th strand of $B$. Then $\phi _B$ can be viewed as an automorphism of ${\fancyscript{A}}^{(0)}_n$ by setting $\tilde{\mu }_i = \mu _{\alpha (i)}$ for all $i$ and having $\phi _B$ act as the identity on $\lambda _i$ for all $i$. As an automorphism of ${\fancyscript{A}}^{(0)}_n$, $\phi _B$ acts as the identity on $\mu _i$ as well.

For convenient notation we assemble the generators of ${\fancyscript{A}}_n$ into $(n\times n)$-matrices. Writing ${\mathbf M }_{ij}$ for the element in position $ij$ in the $(n\times n)$-matrix ${\mathbf M }$, we define the $(n\times n)$-matrices

$$\begin{aligned} \begin{aligned} {\mathbf A }:\quad&{\left\{ \begin{array}{ll} {\mathbf A }_{ij}=a_{ij}&\text{ if} i<j,\\ {\mathbf A }_{ij}=\mu _{\alpha (j)} a_{ij}&\text{ if} i>j,\\ {\mathbf A }_{ii}=1+\mu _{\alpha (i)}, \end{array}\right.}&\qquad \qquad {\mathbf B }:\quad&{\left\{ \begin{array}{ll} {\mathbf B }_{ij}=b_{ij}&\text{ if} i<j,\\ {\mathbf B }_{ij}=\mu _{\alpha (j)} b_{ij}&\text{ if} i>j,\\ {\mathbf B }_{ii}=0, \end{array}\right.}\\ {\mathbf C }:\quad&{\left\{ \begin{array}{ll} {\mathbf C }_{ij}=c_{ij}, \end{array}\right.}&{\mathbf E }:\quad&{\left\{ \begin{array}{ll} {\mathbf E }_{ij}=e_{ij}. \end{array}\right.} \end{aligned} \end{aligned}$$

We also associate $(n\times n)$-matrices with coefficients in ${\fancyscript{A}}_n^{(0)}$ to the braid $B$ as follows. Let $\phi _B({\mathbf A })$ be the matrix defined by $\bigl (\phi _B({\mathbf A })\bigr )_{ij}=\phi _B({\mathbf A }_{ij})$. Then there are invertible matrices $\varPhi ^L_B$ and $\varPhi ^R_B$ so that

$$\begin{aligned} \phi _B({\mathbf A })=\varPhi ^{L}_{B}\cdot {\mathbf A }\cdot \varPhi ^{R}_{B}. \end{aligned}$$

More specifically, we define these matrices by setting $B^{\prime }$ to be the $(n+1)$-braid obtained by adding an extra strand labeled $0$ to $B$ (that is, viewing the word defining $B$ as a word in the $(n+1)$-strand braid group generated by $\sigma _0,\ldots ,\sigma _{n-1}$). Let $\phi ^{\prime }_B$ be the corresponding automorphism of the $\{a_{ij}\}_{0\le i,j\le n;i\ne j}.$ Then define $\varPhi _B^L$ and $\varPhi _B^R$ by

$$\begin{aligned} \phi ^{\prime }_B (a_{i0})=\sum _{j=1}^n \left( \varPhi _B^L\right)_{ij} a_{j0}\quad \text{ and}\quad \phi ^{\prime }_B(a_{0j})=\sum _{i=1}^na_{0i}\left( \varPhi ^R_B\right)_{ij}; \end{aligned}$$

see also [4, 11, 12]. (Note that since the $0$-th strand does not interact with the others, $\tilde{\mu }_0$ does not appear anywhere in the expressions for $\phi ^{\prime }_B(a_{i0})$ and $\phi ^{\prime }_B(a_{0j})$, and so $\varPhi ^L_B,\varPhi ^R_B$ have coefficients in ${\fancyscript{A}}^{(0)}_n$.)

Also, define an $(n\times n)$ coefficient matrix ${\varvec{\lambda }}$ as follows. Among the strands $1,\ldots ,n$ of the braid, call a strand leading if it is the first strand belonging to its component. Then let ${\varvec{\lambda }}$ be the diagonal matrix defined by

$$\begin{aligned} {\varvec{\lambda }}:\quad {\left\{ \begin{array}{ll} {\varvec{\lambda }}_{ij}=0&\text{ if} i\ne j,\\ {\varvec{\lambda }}_{ii}=\lambda _{\alpha (i)}\mu _{\gamma (i)}^{w(i)}&\text{ if} \text{ the} i\mathrm{th} \text{ strand} \text{ is} \text{ leading},\\ {\varvec{\lambda }}_{ii}= 1&\text{ otherwise}, \end{array}\right.} \end{aligned}$$

where $w(i)$ is the writhe (algebraic crossing number) of the $i$-th component of the braid, considered by itself. Finally, in order to capture the two filtrations, we define the following additional matrices:

$$\begin{aligned} \begin{aligned} {\mathbf A }^{U}:\quad&{\left\{ \begin{array}{ll} {\mathbf A }_{ij}^{U}= U a_{ij}&\text{ if} i<j,\\ {\mathbf A }_{ij}^{U}=\mu _{\alpha (j)} a_{ij}&\text{ if} i>j,\\ {\mathbf A }_{ii}^{U}=U+\mu _{\alpha (i)}, \end{array}\right.}\qquad \qquad&{\mathbf B }^{U}:\quad&{\left\{ \begin{array}{ll} {\mathbf B }_{ij}^{U}= U b_{ij}&\text{ if} i<j,\\ {\mathbf B }_{ij}^{U}=\mu _{\alpha (j)} b_{ij}&\text{ if} i>j,\\ {\mathbf B }_{ii}^{U}=0, \end{array}\right.}\\ {\mathbf A }^{V}:\quad&{\left\{ \begin{array}{ll} {\mathbf A }_{ij}^{V}= a_{ij}&\text{ if} i<j,\\ {\mathbf A }_{ij}^{V}=\mu _{\alpha (j)} V a_{ij}&\text{ if} i>j,\\ {\mathbf A }_{ii}^{V}=1+\mu _{\alpha (i)} V, \end{array}\right.}&{\mathbf B }^{V}:\quad&{\left\{ \begin{array}{ll} {\mathbf B }_{ij}^{V}=b_{ij}&\! \text{ if} i\!<\!j,\\ {\mathbf B }_{ij}^{V}=\mu _{\alpha (j)} V b_{ij}&\! \text{ if} i\!>\!j,\\ {\mathbf B }_{ii}^{V}=0. \end{array}\right.} \end{aligned} \end{aligned}$$

Theorem 1.3

The filtered knot contact homology DGA, $(KC\!\fancyscript{A}(K)^{-},\partial ^{-})$ of a transverse link $K$ represented as a braid $B$ on $n$ strands is given by the DGA, $({\fancyscript{A}}_n,\partial ^-)$, with the differential $\partial ^-:{\fancyscript{A}}_n\rightarrow {\fancyscript{A}}_n$ defined by the following matrix equations:

$$\begin{aligned} \partial ^- \mathbf A&= 0, \\ \partial ^- \mathbf B&= -{\varvec{\lambda }}^{-1}\cdot {\mathbf A }\cdot {\varvec{\lambda }}\,\, +\,\, \varPhi ^L_B \cdot {\mathbf A }\cdot \varPhi ^R_B, \\ \partial ^- \mathbf C&= {\mathbf A }^{V}\cdot {\varvec{\lambda }}\,\, +\,\, {\mathbf A }^{U}\cdot \varPhi ^{R}_B, \\ \partial ^- \mathbf E&= {\mathbf B }^{V}\cdot (\varPhi _B^{R})^{-1}\,\, +\,\, {\mathbf B }^{U}\cdot {\varvec{\lambda }}^{-1}\,\, -\,\, \varPhi ^L_B\cdot {\mathbf C }\cdot {\varvec{\lambda }}^{-1}\,\, +\,\, {\varvec{\lambda }}^{-1}\cdot {\mathbf C }\cdot (\varPhi ^R_B)^{-1}, \end{aligned}$$

where if ${\mathbf M }$ is an $(n\times n)$-matrix, the matrix $\partial ^- {\mathbf M }$ is defined by $(\partial ^- {\mathbf M })_{ij}=\partial ^- ({\mathbf M }_{ij})$.

The rest of the paper is organized as follows. We provide some general geometric background in Sect. 2 before defining the transverse invariant $(KC\!\fancyscript{A}^-(K),\partial ^-)$ and proving invariance in Sect. 3. In Sect. 4, we derive the combinatorial formula for $(KC\!\fancyscript{A}^-(K),\partial ^-)$ by proving Theorem 1.3, and present examples in Sect. 5.

2 Geometric constructions

In this section we begin by recalling the definition of Legendrian contact homology and then discuss filtrations on the Legendrian contact homology DGA induced by complex hypersurfaces with certain properties. In the next subsection we recall the conormal construction and see how it can be used to construct invariants of smooth embeddings using Legendrian contact homology. In the last subsection we show how to construct an appropriate complex hypersurface in the case of the Legendrian contact homology of the conormal lift of a transverse knot in the standard contact structure on ${\mathbb R }^3$ and that it gives an invariant of transverse knots. We note that this section gives a geometric construction of transverse knots invariants that can be generalized to other situations, but to actually compute the invariant for transverse knots in the standard contact structure on ${\mathbb R }^3$ we will need to slightly alter the complex hypersurfaces so that they interact well with the constructions in [4]. This is done explicitly in Sect. 3.

2.1 Legendrian contact homology

In [8], the Legendrian contact homology $LCH(\varLambda )$ of a Legendrian submanifold $\varLambda$ of a contact manifold $(V,\xi )$ was introduced. The analytic underpinnings were worked out in detail in [7] for a fairly general and useful case (but under the simplifying assumption that the chosen Reeb field of $\xi$ has no closed orbits, see below). In this case the Legendrian contact homology $LCH(\varLambda )$ is the homology of a DGA, $(LC\!\fancyscript{A}(\varLambda ),\partial )$, over a fixed ring, which changes by a particular type of quasi-isomorphism, called a stable tame isomorphism, as $\varLambda$ changes by Legendrian isotopy. Thus, the stable tame isomorphism class of $(LC\!\fancyscript{A}(\varLambda ),\partial )$ might be considered to be the actual Legendrian invariant underlying $LCH(\varLambda )$.

We briefly sketch the definition of this DGA in the case handled in [7] for the convenience of the reader and to establish some notation; for a more complete definition, see [7] and for generalizations see [8].

Let $P$ be a manifold with exact symplectic form $d\lambda .$ The manifold $P\times {\mathbb R }$ has a natural contact structure $\xi =\ker (dz-\lambda )$ where $z$ is the coordinate on ${\mathbb R }.$ The Reeb vector field of this form is $\partial _z$ and consequently there are no closed Reeb orbits. Consider the projection

$$\begin{aligned} \pi _{\mathbb C }:P\times {\mathbb R }\rightarrow P. \end{aligned}$$

The algebra $LC\!\fancyscript{A}(\varLambda )$ is the free tensor algebra generated over ${\mathbb Z }[H_1(\varLambda )]$ by the double points of $\pi _{\mathbb C }(\varLambda ).$ Notice that the double points are in one to one correspondence with “Reeb chords”, that is, flow lines of the Reeb vector field that begin and end on $\varLambda .$ Thus we will frequently refer to double points as Reeb chords. For the double points we choose “capping paths” in $\varLambda$: that is, paths in $\varLambda$ that connect any Reeb chord endpoint to a fixed base point in its connected component, and fixed paths connecting the base points of distinct components; together, these give paths that connect the two points in $\varLambda$ which project to a double point in $\pi _{\mathbb C }(\varLambda ).$ At a double point $p$ there are two points in $\varLambda$ that project to it. We label the one with larger $z$ coordinate $z^+$ and the other $z^-.$ The projection of a neighborhood of $z^+$ in $\varLambda$ to $P$ will be called the upper sheet at z and the projection of a neighborhood of $z^-$ will be called the lower sheet at z. We then can define a Maslov type index $|c|$ and $|A|$ of Reeb chords $c$ and homology classes $A\in H_1(\varLambda )$ to provide a grading on $LC\!\fancyscript{A}(\varLambda ).$

To define the differential we fix an almost complex structure $J$ on $P$ (which can be thought of as an almost complex structure on $\xi$ using the isomorphism $d\pi _{\mathbb C }|_\xi$). For this almost complex structure the differential is determined by counting (pseudo-)holomorphic disks mapped into $P$ with boundary on $\pi _{\mathbb C }({\varLambda }).$ Given a Reeb chord $a,$ a (noncommutative) word of Reeb chords ${\mathbf b } = b_1\cdots b_m,$ and a homology class $A \in H_1(\varLambda ),$ we define a moduli space

$$\begin{aligned} {{\fancyscript{M}}}_A(a;\mathbf b ) \end{aligned}$$

(2.1)

of holomorphic disks $u:D\rightarrow P$ with: boundary on $\varLambda ;$ one positive¹ puncture at $a$ and negative punctures at $b_1,\dots ,b_m$ in the order given by the boundary orientation; and the homology class $A$ given by the lift of $u(\partial D)$ to $\varLambda$ together with the chosen capping paths. For a generic almost complex structure, this moduli space is a manifold of dimension $|a| - |\mathbf b | - |A|-1$, where $|\mathbf b |=\sum _{i=1}^{m} |b_i|$. Furthermore, the space has a natural compactification which consists of (several level) broken curves and which admits the structure of a manifold with boundary with corners. The moduli spaces can be coherently oriented provided the Legendrian submanifold $\varLambda$ is spin.

Define the differential on the generators of $LC\!\fancyscript{A}(\varLambda )$ by

$$\begin{aligned} \partial a = \sum _{\{u \in {{\fancyscript{M}}}_A(a;{\mathbf b }) \,\,|\,\, |a|-|\mathbf b |-|A| -1=0\}} (-1)^{|a|+1} \sigma (u) e^A {\mathbf b }, \end{aligned}$$

(2.2)

where $\sigma (u) \in \{\pm 1\}$ is determined by the moduli space orientation. The differential is then extended to all of $LC\!\fancyscript{A}(L)$ by the graded product rule and linearity.

2.2 Almost complex hypersurfaces and filtration on Legendrian contact homology

We discuss how to use a complex hypersurface to add a “filtration” to the Legendrian contact homology DGA. With the notation above, suppose that $\overline{H}$ is a submanifold of $P$ such that

• 1. $\overline{H}$ is an almost complex hypersurface ($J(T\overline{H})=T\overline{H}$ and the (real) codimension of $\overline{H}$ equals $2$) and

• 2. $\varLambda \cap \overline{H}=\varnothing .$

Given such $\varLambda$ and $\overline{H}$, we can extend the base ring for the contact homology DGA of $\varLambda$ from ${\mathbb Z }[H_1(\varLambda )]$ to ${\mathbb Z }[H_1(\varLambda )][U_{\overline{H}}]$, by adjoining a formal variable $U_{\overline{H}}$ and changing the definition of the boundary map, using powers of $U_{\overline{H}}$ to keep track of the number of times the holomorphic disks in the definition of the boundary map intersect $U_{\overline{H}}.$ Specifically, given $u\in \fancyscript{M}_A(a;{\mathbf b }),$ positivity of intersection shows that the intersection number of the image of $u$ with ${\overline{H}}$ is a well-defined and nonnegative integer which we denote $n_{\overline{H}}(u).$ We can now modify the differential in Eq. (2.2) and define instead:

$$\begin{aligned} \partial _f a = \sum _{\{u \in \fancyscript{M}_A(a;{\mathbf b }) \,\,|\,\, |a|-|\mathbf b |-|A| -1=0\}} (-1)^{|a|+1} \sigma (u) U_{\overline{H}}^{n_{\overline{H}}(u)}e^A {\mathbf b }. \end{aligned}$$

(2.3)

Conditions $(1)$ and $(2)$ above ensure that $\partial _f$ is a filtered differential; that is, it respects the filtration

$$\begin{aligned} LC\!\fancyscript{A}\supset U_{\overline{H}} \cdot LC\!\fancyscript{A}\supset U_{\overline{H}}^2 \cdot LC\!\fancyscript{A}\supset \cdots . \end{aligned}$$

The proof that $(LC\!\fancyscript{A}(\varLambda ),\partial )$ is invariant up to stable tame isomorphism carries over to show that the stable tame isomorphism type of $(LC\!\fancyscript{A}(\varLambda ),\partial _f)$ over ${\mathbb Z }[H_1(\varLambda )][U_{\overline{H}}]$ is an invariant of $\varLambda$ under isotopies $\varLambda _t$, $0\le t\le 1$, such that $\varLambda _t$ satisfies condition $(2)$ for all $t$, see Theorem 3.11 below.

2.3 The conormal construction and knot contact homology

Given any $n$-manifold $M,$ the cotangent bundle $T^*M$ has a canonical symplectic structure $d\lambda$ where $\lambda$ is the Liouville 1-form. If we choose a metric $g$ on $M$ then we can consider the unit cotangent bundle $S^*M.$ The restriction of $\lambda$ to $S^*M$ is a contact form which we denote by $\alpha$, and $\xi =\ker \alpha$ is a contact structure on $S^*M.$

Let $\mathrm pr :T^{*} M\rightarrow M$ denote the natural projection. If $K$ is a submanifold of $M$ (of any dimension) then the unit conormal bundle

$$\begin{aligned} \varLambda _K= \{\beta \in S^*M: \mathrm pr (\beta )=p\in K \mathrm {and} \beta (v)=0 \mathrm {for all} v\in T_pK \} \end{aligned}$$

is a Legendrian submanifold of $(S^*M,\xi ).$ If we smoothly isotop $K$ in $M$, it is clear that $\varLambda _K$ will undergo a Legendrian isotopy in $(S^*M,\xi ).$ Thus any Legendrian isotopy invariant of $\varLambda _K$ is a smooth isotopy invariant of $K.$

In this paper, we consider conormal lifts of links $K\subset {\mathbb R }^{3}$ which are Legendrian submanifolds $\varLambda _K\subset S^{*}{\mathbb R }^{3}$. There is a well known contactomorphism $S^{*}{\mathbb R }^{3}\cong J^{1}(S^{2})=T^{*}S^{2}\times {\mathbb R }$ and we will thus consider $\varLambda _K\subset T^{*} S^{2}\times {\mathbb R }$ and use the version of Legendrian contact homology of $\varLambda _K$ defined in [7]. In particular, we define the knot contact homology algebra of a link $K$ in ${\mathbb R }^3$ to be the Legendrian contact homology algebra of $\varLambda _K \subset J^1(S^2)$. We denote it $(KC\!\fancyscript{A}(K),\partial )$ and note that the stable tame isomorphism class of $(KC\!\fancyscript{A}(K),\partial )$ is an isotopy invariant of $K$. In [4] the authors show how to compute $(KC\!\fancyscript{A}(K),\partial )$ and demonstrate that it is equivalent to the combinatorial knot DGA introduced by the third author in [12].

We recall for later use that the projection $\pi _F:J^1(S^2)\rightarrow S^2\times {\mathbb R }$ is called the front projection and that a generic Legendrian submanifold $\varLambda$ in $J^1(S^2)$ can be recovered uniquely from $\pi _F(\varLambda )\subset S^2\times {\mathbb R }.$ Since $S^2\times {\mathbb R }$ can be visualized as ${\mathbb R }^3\setminus \{(0,0,0)\},$ it will frequently be useful to study a Legendrian submanifold $\varLambda$ via its front projection.

2.4 Transverse link invariants

Here we describe the geometry underlying the claimed filtration on knot contact homology, in the case when $K \subset {\mathbb R }^3$ is a transverse link and not simply a topological link. Consider a contact structure $\xi$ on ${\mathbb R }^3.$ If there is a Reeb vector field $R_\xi$ for $\xi$ such that the flow lines of the vector field trace out geodesics (in the flat Euclidean metric on ${\mathbb R }^3$) then we can consider the submanifolds

$$\begin{aligned} H_\xi ^\pm =\{\eta \in S^*{\mathbb R }^3: \pm \eta (R_\xi )>0 \text{ and} \eta (v)=0 \text{ for} \text{ all} v\in \xi \}. \end{aligned}$$

Since the Reeb flow lines of $R_\xi$ are geodesics, $H_\xi ^\pm$ are foliated by Reeb flow lines in $S^*{\mathbb R }^3.$ In other words, identifying $S^*{\mathbb R }^3$ with $J^1(S^2)=T^*S^2\times {\mathbb R }$, the projected submanifold $\overline{H}_\xi ^\pm =\pi _{\mathbb C }(H_\xi ^\pm )$ in $T^*S^2$ is an embedded codimension 2 submanifold. One may also choose the almost complex structure on $T^*S^2$ so that $\overline{H}_\xi ^\pm$ is a holomorphic submanifold. Moreover, if $K$ is a knot in ${\mathbb R }^3$ that is transverse to $\xi$ then its conormal lift $\varLambda _K$ projects to an exact Lagrangian submanifold in $T^*S^2$ that is disjoint from $\overline{H}_\xi ^\pm .$ Thus, as discussed above, we can construct an associated filtered contact homology DGA that will be an invariant of the transverse isotopy class of $K.$

To carry out the above construction one must choose a contact form $\alpha$ with $\xi = \ker (\alpha )$ so that its Reeb flow traces out geodesics. The standard contact structure does have such representatives, for example $\xi _0=\ker (\sin x_1\, dx_2 + \cos x_2 \, dx_3),$ but it is quite difficult to actually compute the filtered contact homology DGA for this contact structure. To take advantage of the computations in [4] we would prefer to work with the contact form $\ker (dx_3+r^2\, d\theta ),$ but this contact form does not have a Reeb vector field with the requisite properties. In the remainder of the paper we overcome this problem by considering a different $S^2$-subbundle $B$ of $T^*{\mathbb R }^3$ instead of the unit cotangent bundle. As long as each fiber in this subbundle bounds a convex region containing the origin, we can still identify $B$ with $J^1(S^2).$ By a judicious choice of $B$ we will see that the projection of $H_\xi ^\pm$ to $T^*S^2$ retains enough of the properties discussed above to allow us to explicitly calculate a filtered invariant for transverse knots in ${\mathbb R }^3.$

3 The filtered DGA of transverse links in $({\mathbb R }^{3},\xi _0)$

In this section we show how to construct from the standard contact structure $({\mathbb R }^{3}, \xi _{0})$ a pair of complex hypersurfaces $\overline{H}_{\pm }$ in $T^{*} S^{2}$ satisfying the following: if $K$ is any link in a sufficiently small ball around the origin which is transverse to $\xi _0$, then $\overline{H}_{\pm }\cap \pi _{\mathbb C }(\varLambda _K)=\varnothing$. In order to get a computable invariant we adapt the geometry and use a slightly non-standard version of $S^{*}{\mathbb R }^{3}$.

In Sect. 3.1 we describe our geometric model of $S^{*}{\mathbb R }^{3}$ and its relation to $T^{*} S^{2}\times {\mathbb R }$. In Sect. 3.2 we show that we can control the image of the holomorphic disks used to compute the knot contact homology so that they lie near the zero section in $T^{*} S^2$ if the original link is sufficiently small. We then discuss the conormal lift of the standard contact structure on ${\mathbb R }^3$ in Sect. 3.3 and show in Sect. 3.4 that it is no restriction to assume that all transverse links and isotopies lie in a small ball around the origin. In Sects. 3.5 and 3.6, we define a suitable almost complex structure on $T^{*} S^2$ and then prove the filtered DGA of a transverse link is well-defined and invariant up to stable tame isomorphisms under isotopies through transverse links. Finally, in Sect. 3.7, we explain why the infinity theory for knots in ${\mathbb R }^3$ is a topological invariant.

3.1 The spherical conormal bundle and the $1$-jet space of $S$

Let $y=(y_1,y_2,y_3)$ be standard Euclidean coordinates on ${\mathbb R }^{3}$. Let $S$ denote the smooth boundary of a (not necessarily strictly) convex subset of ${\mathbb R }^{3}$ which is symmetric with respect to reflection in the $y_1y_2$-plane and with respect to rotations about the $y_3$-axis. For $y\in S$, let $\nu (y)$ denote the outward unit normal to $S$ at $y$. Note that the symmetries of $S$ imply that $\nu (y_1,y_2,y_3)$ lies in the subspace spanned by the vectors $(y_1,y_2,0)$ and $(0,0,y_3)$. In particular, $\nu (0,0,y_3)=(0,0,\pm 1)$ and $\nu (y_1,y_2,0)=\frac{1}{\sqrt{y_1^{2}+y_2^{2}}}(y_1,y_2,0)$.

We represent the spherical cotangent bundle of ${\mathbb R }^{3}$ as

$$\begin{aligned} S^*{\mathbb R }^{3}={\mathbb R }^{3}\times S\subset T^{*}{\mathbb R }^3={\mathbb R }^3\times {\mathbb R }^{3} \end{aligned}$$

and use coordinates $(x,y)=(x_1,x_2,x_3,y_1,y_2,y_3)$ on $T^{*}{\mathbb R }^{3}$. The contact form on $S^*{\mathbb R }^{3}$ is the restriction of the Liouville $1$-form $y\cdot dx=\sum _{j=1}^{3} y_j\,dx_j$ on $T^{*}{\mathbb R }^{3}$ to $S^*{\mathbb R }^{3}$. We compute the Reeb vector field as follows.

Lemma 3.1

The Reeb vector field on $S^*{\mathbb R }^{3}$ is

$$\begin{aligned} R(x,y)=|y|^{-1}(\nu (y)\cdot \partial _x)=|y|^{-1}\sum \nu _j(y)\partial _{x_j} \end{aligned}$$

(3.1)

and the time $t$ flow starting at $(x,y)$ is

$$\begin{aligned} \varPhi _R^{t}(x,y)=(x+t(|y|^{-1}\nu (y))\,,\,y). \end{aligned}$$

(3.2)

Proof

If $i$ denotes the standard complex structure on ${\mathbb C }^{3}={\mathbb R }^{3}+i{\mathbb R }^{3}=T^{*}{\mathbb R }^{3}$ then the Reeb field lies in the intersection of $T(S^*{\mathbb R }^3)$ and the complex tangent line at $y$ containing $\nu (y)$. Thus up to normalization the Reeb field equals $-i\nu (y)=\nu (y)\cdot \partial _{x}$. The lemma follows.

The spherical cotangent bundle $S^* {\mathbb R }^3$ can be identified with the 1–jet space $J^1(S)=T^*S\times {\mathbb R }$ as follows, where we use the flat metric on ${\mathbb R }^{3}$ to identify vectors and covectors.

Lemma 3.2

The map $\phi :S^*{\mathbb R }^{3}\rightarrow J^{1}(S)=T^{*}S\times {\mathbb R }$ given by

$$\begin{aligned} \phi (x,y)=(y,x-(x\cdot \nu )\nu ,x\cdot y). \end{aligned}$$

is a contactomorphism

$$\begin{aligned} \phi :(S^*{\mathbb R }^{3},y\cdot dx)\rightarrow (J^{1}(S),dz-p\cdot dq), \end{aligned}$$

where $q=(q_1,q_2)$ are local coordinates on $S,$ $p=(p_1,p_2)$ give the coordinates on the fiber of the cotangent bundle and $z$ is the coordinate on ${\mathbb R }.$

Proof

Note that

$$\begin{aligned} \phi ^{*}(dq)&= dy,\\ \phi ^{*}(dz)&= x\cdot dy + y\cdot dx, \end{aligned}$$

and thus

$$\begin{aligned} \phi ^{*}(dz-p\cdot dq)=x\cdot dy +y\cdot dx - (x-(x\cdot \nu )\nu )\cdot dy= y\cdot dx, \end{aligned}$$

where we use $\nu \cdot dy=0$, which holds since $y\in S$ and $\nu$ is the normal of $S$ at $y$.

3.2 Confining holomorphic curves

Fix $\delta >0,$ let $S\subset {\mathbb R }^{3}$ be as above and write $\rho _0=\max \{|y|:y\in S\}$. Below we will measure lengths of cotangent vectors in $T^{*}S$ using the metric coming from the one induced on $T^{*}{\mathbb R }^{3}$ by the flat metric on ${\mathbb R }^{3}$. If $(q,p)\in T^{*}S$ then we write $|p|$ for the length of the cotangent vector measured with respect to this metric.

Lemma 3.3

If $K$ is any link contained in $B(\delta ),$ the ball of radius $\delta$ about the origin in ${\mathbb R }^3,$ then

$$\begin{aligned} \varLambda _K\subset \{(q,p,z)\in T^{*}S\times {\mathbb R }:|p|\le \delta \}. \end{aligned}$$

Moreover, if $c$ is a Reeb chord of $\varLambda _K$ then

$$\begin{aligned} \int _c (dz-p\,dq)\le 2\rho _0\delta . \end{aligned}$$

Proof

By Lemma 3.2 if $x\in K$ and $(x,y)\in \varLambda _K\subset S^*{\mathbb R }^{3}$ then

$$\begin{aligned} |p|=|x-(\nu (y)\cdot x)\nu (y)|\le |x|\le \delta . \end{aligned}$$

For the second statement we note that Lemma 3.1 implies that a Reeb chord $c$ of $\varLambda _K$ is a lift $(l_c,y)$ into $S^*{\mathbb R }^{3}$ of a line segment $l_c$ in ${\mathbb R }^3$ with endpoints on $K$ and that the action of the chord is

$$\begin{aligned} \int _{l_c} y\cdot dx\le \int _{l_c}\rho _0|dx|\le 2\rho _0\delta . \end{aligned}$$

As a consequence of Lemma 3.3 we can confine holomorphic curves with boundary on $\varLambda _K.$ As in Sect. 2.3, $\pi _{\mathbb C }:J^1(S)\rightarrow T^*S$ will denote the projection map and $\overline{\varLambda }_K=\pi _{\mathbb C }(\varLambda _K)$.

Lemma 3.4

Let $J$ be an almost complex structure on $T^*S$ that is compatible with the symplectic form on $T^*S.$ Fix $\delta _0>0$ a constant. Then there exists $0<\delta <\delta _0$ such that if $K$ is a link in $B(\delta )$ then any $J$-holomorphic disk with boundary on $\overline{\varLambda }_K$ and one positive puncture lies in $\{(q,p)\in T^{*}S:|p|<2\delta _0\}$.

Proof

Consider $0<\delta <\delta _0$. By Stokes’ theorem and Lemma 3.3, the area of a disk $u$ as described is bounded above by $2\rho _0\delta$ and its boundary is contained in the region where $|p|\le \delta$. By monotonicity (see for example Proposition 4.3.1 [9]), there exists a constant $C_0$ (depending only on $J$) such that if $u$ leaves the region where $|p|\le 2\delta _0$ then the area of $u$ is at least $C_0\delta _0^{2}$. If we now take $\delta < C_0\delta ^{2}_0/2\rho _0$, the lemma follows.

3.3 A contact form on ${\mathbb R }^{3}$ and its spherical cotangent lifts

Fix the contact form $\alpha _0=dx_3-x_2\,dx_1+x_1\,dx_2$ on ${\mathbb R }^{3}$ and write, as in Sect. 1, $\xi _0=\ker (\alpha _0)$ for the corresponding contact structure. Note that $\alpha _0$ is invariant under rotations in the $x_1x_2$-plane and that the diffeomorphism

$$\begin{aligned} (x_1,x_2,x_3)\mapsto \left(\frac{1}{2}x_1\,,\,\frac{1}{2}x_2\,,\,x_3-\frac{1}{2} x_1x_2\right) \end{aligned}$$

gives a contactomorphism between $\alpha _0$ and the standard contact form on ${\mathbb R }^{3}$, $dx_3-x_2\,dx_1$.

If $v\in {\mathbb R }^{3}$ is a non-zero vector, we denote the two open half rays determined by $v$ as follows:

$$\begin{aligned} {\mathbb R }_\pm \cdot v=\{x\in {\mathbb R }^{3}:x=tv,\,\,\pm t>0\}. \end{aligned}$$

The positive and negative spherical lifts of $\alpha _0$ are

$$\begin{aligned} \begin{aligned} H_{+}&=\{(x,y)\in S^*{\mathbb R }^{3}:y\in {\mathbb R }_+\cdot \alpha _0(x)\} \text{ and} \\ H_{-}&=\{(x,y)\in S^*{\mathbb R }^{3}:y\in {\mathbb R }_-\cdot \alpha _0(x)\}, \end{aligned} \end{aligned}$$

(3.3)

respectively. At all points $x$ on a transverse link $K,$ $\alpha _0$ fails to annihilate the tangent space $T_xK$. Thus we have the following immediate result.

Lemma 3.5

If $K$ is transverse to $\xi _0$ then the conormal lift $\varLambda _K$ of $K$ is disjoint from $H_{\pm }.$

3.4 Shrinking transverse links

The following straightforward lemma reduces the study of transverse links in ${\mathbb R }^{3}$ with its standard contact structure to the study of transverse links lying in an arbitrary small fixed neighborhood of the origin. Let $B^{d}(r)$ denote the closed $d$-dimensional ball of radius $r$ around the origin and let $B(r) = B^3(r).$

Lemma 3.6

Fix $\delta >0.$ Let $K(s)$, $s\in B^{d}(1),$ be a continuous family of transverse links in $({\mathbb R }^{3},\xi _0)$ such that $K(s)\subset B(\delta )$ for $s\in \partial B^{d}(1).$ Then there is a homotopy $K(s,t)$, $0\le t\le 1$, with $K(s,t)=K(s)$ if $s\in \partial B^{d}(1)$, and such that $K(s,0)=K(s)$ and $K(s,1)\subset B(\delta )$ for each $s\in B^{d}(1)$. In particular, the space of transverse links in ${\mathbb R }^{3}$ is weakly homotopy equivalent to the space of transverse links in $B(\delta )$.

Proof

Note that if $K$ is a transverse link in $({\mathbb R }^{3},\alpha )$ then so is $F_c(K)$ where

$$\begin{aligned} F_c(x_1,x_2,x_3)=(\sqrt{c}\,x_1,\sqrt{c}\,x_2,c x_3) \end{aligned}$$

for $c>0.$ Choose $\varepsilon _1, \varepsilon _2 >0$ sufficiently small so that $K(s) \subset B(\varepsilon _1^{-1} \delta )$ for all $s \in B^d(1)$ and $K(s) \subset B(\delta )$ for all $s \in (B^d(1) \setminus B^d(1-\varepsilon _2)).$ Choose any smooth $\varepsilon : B^d(1) \rightarrow [\varepsilon _1, 1]$ such that

$$\begin{aligned} \varepsilon (s) = 1 \,\,\, \text{ if} \,\, s \in \partial B^d(1), \quad \text{ and} \,\,\, \varepsilon (s) = \varepsilon _1 \,\,\, \text{ if} \,\, s \in B^d(1 - \varepsilon _2). \end{aligned}$$

Then $K(s,t)=F_{(1-t)+t\varepsilon (s)}(K(s))$ is a homotopy with the required properties.

Using Lemma 3.6 in conjunction with Lemma 3.4, we can restrict our attention to a small neighborhood of the zero section in $T^*S$ when counting holomorphic curves with boundary on the conormal lift of transverse knots.

3.5 Almost complex structures

We choose $S\subset {\mathbb R }^{3}$ as in Sect. 3.1 with the additional requirement that $S$ is flat near the north and south poles. More precisely, for some fixed $\delta _0>0$ we require that

$$\begin{aligned} S \cap \left\{ y\in {\mathbb R }^{3}:\sqrt{y_1^{2}+y_2^{2}}\le 3\delta _0\right\} =\left\{ (y_1,y_2,\pm 1)\in {\mathbb R }^{3}:\sqrt{y_1^{2}+y_2^{2}}\le 3\delta _0\right\} .\nonumber \\ \end{aligned}$$

(3.4)

For $0 < k <3$, write

$$\begin{aligned} E_{k\delta _0}\!=\! S \cap \left\{ y\in {\mathbb R }^{3}:\sqrt{y_1^{2}+y_2^{2}}\le k\delta _0\right\} =\left\{ (y_1,y_2,\pm 1)\in {\mathbb R }^{3}:\sqrt{y_1^{2}+y_2^{2}}\le k\delta _0\right\} , \end{aligned}$$

and let $H_{\pm }(k\delta _0)$ denote the intersection $H_{\pm }\cap (T^{*}E_{k\delta _0} \times {\mathbb R }).$ Note that the metric on $S$ is flat in $E_{3\delta _0}$ and that the almost complex structure induced by the metric agrees with the standard integrable complex structure $J_0$ on $T^{*}E_{3\delta _0}\subset {\mathbb C }^{2}$.

Let $K$ be a transverse link in the ball $B(\delta )$ and $\varLambda _K$ its conormal lift. As usual, let $\pi _{\mathbb C }:J^{1}(S)\rightarrow T^{*}S$ denote the projection, and write $\overline{H}_\pm = \pi _{\mathbb C }({H}_\pm )$ and $\overline{H}_\pm (k\delta _0) = \pi _{\mathbb C }(H_{\pm }(k\delta _0)).$

Lemma 3.7

The spherical lifts $H_{\pm }$ of $\alpha _0$ intersect the subset $\partial (T^{*}E_{2\delta _0}\times {\mathbb R })\subset J^{1}(S)$ transversely. Moreover, $H_{\pm }(2\delta _0)$ is invariant under the Reeb flow and its projection ${\overline{H}}_{\pm }(2\delta _0) \subset T^{*}E_{2\delta _0}$ is a smooth $J_0$-complex subvariety.

Proof

By Formula (3.4), the normal vector to $E_{2\delta _0}$ is $\nu = (0,0,1).$ By the definition of the contact form $\alpha _0$ and Lemma 3.2,

$$\begin{aligned}&\displaystyle H_{\pm }(2 \delta _0) = \bigl \{ (q,p,z)\in T^{*}S\times {\mathbb R }:&(q,p,z)=\left(\pm (-x_2,x_1,1)\,,\,(x_1,x_2,0)\,,\,x_3\right)\nonumber \\&\displaystyle \qquad \qquad \qquad \qquad \text{ and} x_1^{2}+x_2^{2}\le 4\delta _0^2\, \bigr \} \end{aligned}$$

(3.5)

(see the Proof of Lemma 3.8 for a parameterized version). This is clearly transverse to $\partial (T^{*}E_{2\delta _0}\times {\mathbb R })$ and invariant under the Reeb flow (which is just translation in the $z$-direction). Furthermore, under the identification $((u_1,u_2,\pm 1),(v_1,v_2,0))\mapsto (u_1+iv_1,u_2+iv_2)\in {\mathbb C }^{2}$, $\overline{H}_{\pm }(2 \delta _0)$ corresponds to the complex line

$$\begin{aligned} (u_1+iv_1,u_2+iv_2)=(i\zeta ,\zeta ), \end{aligned}$$

where $\zeta =x_1+ix_2$.

Fix $\delta \in (0,\delta _0)$ so that Lemma 3.4 holds. As pointed out in Sect. 3.4, Lemma 3.6 implies that when studying the isotopy classification of links in ${\mathbb R }^{3}$ which are transverse to $\alpha$, it is no restriction to assume that all such links are contained in $B(\delta )$ and that all isotopies are through links inside $B(\delta )$. We thus make this assumption throughout the rest of the paper.

Lemma 3.8

Let $K$ be a transverse link (by our standing assumption $K\subset B(\delta )$). Then the sets $\overline{\varLambda }_K$ and $\overline{H}_{\pm }$ are disjoint in $T^{*}S$. In addition, $\overline{H}_{\pm } \setminus \overline{H}_{\pm }(2 \delta _0)$ does not intersect any $J$-holomorphic disk with boundary on $\overline{\varLambda }_K$ and one positive puncture.

Proof

Lemmas 3.5 and 3.7 imply that $\overline{H}_\pm (2 \delta _0)$ and $\overline{\varLambda }_K$ are disjoint, for if $H_\pm (2 \delta _0)$ intersected a Reeb flow line emanating from $\varLambda _K,$ then it would intersect $\varLambda _K$ itself since ${H}^\pm (2 \delta _0)$ is foliated by Reeb flow lines.

Lemmas 3.3 and 3.4 imply that any holomorphic curve with boundary on $\overline{\varLambda }_K$ will be contained in

$$\begin{aligned} \{(q,p)\in T^{*}S:|p|<2\delta _0\}. \end{aligned}$$

Write $x=(x_1,x_2,x_3).$ The contactomorphism of Lemma 3.2 and the properties of $S$, see Sect. 3.1, imply that $\overline{H}_\pm$ consists of the points $(p(x),q(x))\in T^{*} S$ whose coordinates satisfy the following:

$$\begin{aligned} q(x)=b(-x_2,x_1,1), \end{aligned}$$

where $b>0$ is such that $q(x)\in S;$ and

$$\begin{aligned}&p(x)\\&\quad \!=\!\left(x_1-x_2x_3\sqrt{1-a^2(x_1^2+x_2^2)},\,\, x_2\!-\!vx_1x_3\sqrt{1-a^2(x_1^2+x_2^2)},\,\, a^2x_3(x_1^2\!+\!x_2^2)\right), \end{aligned}$$

where $a\ge 0$ is chosen so that $\nu (q(x))= (-ax_2, ax_1, \sqrt{1-a^2(x_1^2+x_2^2)})$. It follows that

$$\begin{aligned} |p(x)|^{2}=\big (x_1^2+x_2^2\big )\big (1+a^2x_3^2\big ). \end{aligned}$$

Since the second factor is bigger than or equal to 1, the lemma clearly follows.

Lemma 3.9

There exists an almost complex structure $J$ on $T^{*} S$ which agrees with $J_0$ in a neighborhood of $\overline{ H}_{\pm }$ and outside the region where $|p|\le \delta _0,$ and which is regular for $\overline{\varLambda }_K$ in the sense that $0$- and $1$-dimensional moduli spaces of holomorphic disks with boundary on $\varLambda _K$ and one positive puncture are transversely cut out.

Proof

Proposition 2.3(1) in [7] shows that the asserted regularity can be achieved by perturbing $J$ in an arbitrary small neighborhood of the double points of $\overline{\varLambda }_K$. Since these double points lie neither on $\overline{H}_{\pm }$ nor in the region where $|p|\ge \delta _0$ the lemma follows.

3.6 A filtered DGA

Following the discussion in Sect. 2, we now define a filtered version of the Legendrian contact homology DGA of $\varLambda _K$ when $K$ is a transverse link. See [7] for further background details on the unfiltered DGA.

Let $KC\!\fancyscript{A}^-(K) = LC\!\fancyscript{A}(\varLambda _K)$ be the graded free associative non-commutative unital algebra over the ring ${\mathbb Z }[H_1(\varLambda _K)][U, V]$ generated by the Reeb chords of $\varLambda _K$. Here $U,V$ are two (formal) variables of grading 0. Other generators and coefficients have grading exactly as in the usual Legendrian contact homology DGA determined via a Maslov index. We denote the grading $|\cdot |$.

Consider a Reeb chord $a,$ a monomial of Reeb chords ${\mathbf b } = b_1\cdots b_m,$ and a homology class $A \in H_1(\varLambda _K).$ Recall from (2.1) the moduli space ${{\fancyscript{M}}}_A(a;{\mathbf b })$ of holomorphic disks with boundary on $\varLambda _K$, which has dimension $|a|-|\mathbf b |-|A|-1$. For $u \in {{\fancyscript{M}}}_A(a;{\mathbf b }),$ let $n_U(u)$ and $n_V(u)$ denote the algebraic intersection of $u$ and $\overline{H}_+$ and $u$ and $\overline{H}_-,$ respectively. Lemmas 3.7 and 3.8 imply that these counts are well-defined and non-negative for the $J$ and $K$ we consider.

Define the differential ${\partial }^- :KC\!\fancyscript{A}^-(K) \rightarrow KC\!\fancyscript{A}^-(K)$ by

$$\begin{aligned} {\partial }^- a = \sum _{\{u \in {{\fancyscript{M}}}_A(a;{\mathbf b }) \,\,|\,\, |a|-|\mathbf b |-|A|=1\}} (-1)^{|a|+1} \sigma (u) \,U^{n_U(u)} V^{n_V(u)}\, e^A\, {\mathbf b }, \end{aligned}$$

(3.6)

where $\sigma (u) \in \{\pm 1\}$ is determined by the moduli space orientation; compare to Eq. (2.3). Setting $U = V=1$ we recover the differential used in the Legendrian contact homology DGA defined in [7].

Theorem 3.10

The above definition gives a filtered differential: $\partial ^{-}$ does not decrease the exponents of $U$ or $V,$ and $(\partial ^-)^2=0$.

Proof

The unfiltered version of the differential squares to zero due to the usual transversality, gluing and compactness arguments [7]. The filtered version follows from the fact that $n_U$ and $n_V$ are nonnegative, the fact that gluing and compactness respects the filtration, and Lemmas 3.8 and 3.9.

We call the above filtered DGA the transverse knot DGA of $K$ and denote it $(KC\!\fancyscript{A}^-(K),\partial ^-)$. We next show that the filtered DGA of $K$ is invariant under transverse isotopies of $K$ up to filtered stable tame isomorphism. (For the definition of stable tame isomorphism, which extends to our situation, see [7] for example.) In particular, the homology of the filtered DGA $(KC\!\fancyscript{A}^-(K),\partial ^-)$ is a transverse link invariant.

Theorem 3.11

The filtered DGA, $(KC\!\fancyscript{A}^-(K),\partial ^-),$ is invariant under transverse isotopies of $K$ up to filtered stable tame isomorphism.

Proof

By Lemma 3.6 we may assume that $K_t$, $0\le t\le 1$ is an isotopy of transverse links inside $B(\delta )$ connecting two given transverse links. Then $\varLambda _{K_t}$ is an isotopy of Legendrian submanifolds confined to the region in $J^{1}(S)$ where $|p|\le \delta$. To prove the invariance statement we generalize the invariance proof in [6].

We study parameterized moduli spaces and first note that, as in [6, Lemma 2.11], when there are no disks of index $-1$ and no births/deaths of intersection points, the moduli spaces change by cobordisms and the filtered differential is unchanged.

Suppose at some critical time $t^{\prime }$ an index $-1$ disk exists. Like in [6, Section 10], we use a small perturbation of the trace of the fronts of the isotopies near the critical instance to create a Legendrian submanifold $\varLambda _{K_{t^{\prime }}}\times {\mathbb R }\subset J^{1}(S\times {\mathbb R })$ and study a compact part of this manifold corresponding to $[-1,1]\subset {\mathbb R }$. Straightforward modifications of Lemmas 3.4, 3.7, and 3.8 show that $\overline{H}_{\pm }\times {\mathbb C }\subset T^{*}(S\times {\mathbb R })$ (where we think of $T^{*}{\mathbb R }$ as ${\mathbb C }$) are disjoint from $\pi _{\mathbb C }(\varLambda _{K_{t^{\prime }}}\times {\mathbb R })$ and is complex in regions where holomorphic disks with one positive puncture might exist. As above it then follows that $\overline{H}_{\pm }\times {\mathbb C }$ give filtrations on $LC\!\fancyscript{A}^-(\varLambda _{K_{t^{\prime }}}\times {\mathbb R })$ compatible with those induced on $LC\!\fancyscript{A}^{-}(\varLambda _{K_{t^{\prime }}})$. From this filtered differential we construct a filtered tame isomorphism by repeating, essentially verbatim, the construction of the tame isomorphism in the unfiltered case given in [6, Lemma 2.12].

When a birth/death of intersection points occurs, we use the same analytical “degenerate” gluing results [6, Proposition 2.16 and 2.17] and a filtered version of the algebraic arguments in [6, Lemma 2.15] to construct an explicit filtered stable tame isomorphism.

3.7 Topological invariance of the infinity version in ${\mathbb R }^3$

Before delving in depth into the technical details of the computation of the filtered DGA in ${\mathbb R }^3$ in Sect. 4, we give a geometric explanation of Theorem 1.2, which says that the infinity version of the transverse invariant in ${\mathbb R }^3$ is actually a topological link invariant. To simplify notation, we treat only the single-component knot case in this subsection. An alternative discussion in the algebraic setting can be found in [13], though the presentation here has the advantage that it explains the underlying geometric reason for this phenomenon.

Let $K$ be an oriented transverse knot in ${\mathbb R }^3$. The homology $H_1(\varLambda _K) \cong {\mathbb Z }^2$ has a distinguished set of generators corresponding to the meridian and ($0$-framed) longitude of $K$, allowing us to identify ${\mathbb Z }[H_1(\varLambda _K)]$ with $R={\mathbb Z }[\lambda ^{\pm 1},\mu ^{\pm 1}]$. We can then rewrite Eq. (3.6) as

$$\begin{aligned} \partial ^- a = \sum _{\{u \in (\fancyscript{M}_A(a;{\mathbf b })/{\mathbb R }) \,\,|\,\, \dim \fancyscript{M} = 1\}} (-1)^{|a|+1} \sigma (u) U^{n_U(u)}V^{n_V(u)} \lambda ^\mathrm{long (A)}\mu ^\mathrm{mer (A)} {\mathbf b },\nonumber \\ \end{aligned}$$

(3.7)

where $A$ is the linear combination of $\mathrm long (A)$ longitudes and $\mathrm mer (A)$ meridians.

As in [13], define the infinity DGA, $(KC\!\fancyscript{A}^\infty (K),\partial ^\infty ),$ by tensoring $(KC\!\fancyscript{A}^-(K),\partial ^-)$ with $R[U^{\pm 1},V^{\pm 1}]$ and replacing $\lambda$ by $\lambda (U/V)^{-(\mathrm{sl }(K)+1)/2}$, where $\mathrm{sl }(K)$ is the self-linking number of $K$. We are now ready to prove Theorem 1.2 from the introduction.

Proof of Theorem 1.2

Let $u$ denote a holomorphic disk contributing to the Legendrian contact homology of $\varLambda _K \subset S^*{\mathbb R }^3$. Just as we viewed the boundary of $u$ in Sect. 2.3 as an element of $H_1(\varLambda _K)$ by appending capping paths at Reeb chords, we can view the entirety of $u$ as an element $[u]$ of the relative homology $H_2(S^*{\mathbb R }^3,\varLambda _K)$ by appending capping surfaces at Reeb chords.

Note that $S^*{\mathbb R }^3 \cong J^1(S^2)$ is topologically $S^2 \times {\mathbb R }^3$. The exact sequence

$$\begin{aligned} \cdots \!\longrightarrow \! 0 \!\longrightarrow \! H_2(S^*{\mathbb R }^3)={\mathbb Z }\longrightarrow H_2(S^*{\mathbb R }^3,\varLambda _K) \!\longrightarrow \! H_1(\varLambda _K) \!=\! {\mathbb Z }^2 \!\longrightarrow \! 0 \!\longrightarrow \! \cdots \end{aligned}$$

implies that $H_2(S^*{\mathbb R }^3,\varLambda _K) \cong {\mathbb Z }^3$. Pick a basis $s,l,m$ of $H_2(S^*{\mathbb R }^3,\varLambda _K)$ such that the following holds:

• $s = [S^2]$, the class of $S^2$ in $S^*{\mathbb R }^3 \cong {\mathbb R }^3\times S^2$;

• $l$ is the homology class of the conormal lift of a cooriented Seifert surface of $K$;

• $m$ is the hemisphere of the $S^2$ fiber of $S^*{\mathbb R }^3$ over some point $p\in K$, bounded by the intersection of $\varLambda _K$ with this fiber and containing the positive lift of $\xi$ over $p$.

Note that under the boundary map $H_2(S^*{\mathbb R }^3,\varLambda _K) \longrightarrow H_1(\varLambda _K)$, the classes $s,l,m$ map to $0$, the longitude, and the meridian, respectively. Now intersecting $H_\pm$ or projecting to $T^*S^2$ and intersecting with with $\overline{H}_{\pm }$ defines linear maps

$$\begin{aligned} n_U,n_V : H_2(S^*{\mathbb R }^3,\varLambda _K) \rightarrow {\mathbb Z }\end{aligned}$$

such that $n_U(s) = n_V(s) = 1$, $n_U(m)=1$, and $n_V(m)=0$. By the definition of self-linking number, the difference $n_U(l)-n_V(l)$ is $\mathrm{sl }(K)$; by adding the appropriate multiple of $s$ to $l$, we may assume that $n_U(l) = \frac{\mathrm{sl }(K)-1}{2}$ and $n_V(l) = -\frac{\mathrm{sl }(K)+1}{2}$.

If we write $[u] = s(u)s+l(u) l+m(u)m$ for $s(u),l(u),m(u) \in {\mathbb Z }$, then $n_U[u] = s(u)+\frac{\mathrm{sl }(K)-1}{2}l(u)+m(u)$ and $n_V[u] = s(u)-\frac{\mathrm{sl }(K)+1}{2}l(u)$. From Eq. (3.7), the contribution of $u$ to the differential $\partial ^\infty$ has coefficient

$$\begin{aligned}&(-1)^{|a|+1} \sigma (u) U^{n_U[u]} V^{n_V[u]} (\lambda (U/V)^{-(\mathrm{sl }(K)+1)/2})^{l(u)} \mu ^{m(u)}\\&\quad =(-1)^{|a|+1} \sigma (u) (\lambda /U)^{l(u)} (\mu U)^{m(u)} (UV)^{s(u)}. \end{aligned}$$

Now in the definition of the unfiltered Legendrian contact homology of $\varLambda _K$, which is a topological knot invariant, one could use the coefficient ring ${\mathbb Z }[H_2(S^*{\mathbb R }^3,\varLambda _K)]$ rather than ${\mathbb Z }[H_1(\varLambda _K)]$. If one writes $\tilde{\lambda },\tilde{\mu },\tilde{\sigma }$ for the multiplicative generators of ${\mathbb Z }[H_2(S^*{\mathbb R }^3,\varLambda _K)]$ corresponding to $l,m,s$, then the contribution of $u$ to Legendrian contact homology with this enhanced coefficient ring is

$$\begin{aligned} (-1)^{|a|+1} \sigma (u) \tilde{\lambda }^{l(u)} \tilde{\mu }^{m(u)} \tilde{\sigma }^{s(u)}. \end{aligned}$$

But this is precisely the coefficient of the contribution of $u$ to $\partial ^{\infty }$, once we make the global substitutions $\tilde{\lambda } = \lambda /U$, $\tilde{\mu } = \mu U$, $\tilde{\sigma } = UV$.

It follows that this global substitution turns $(KC\!\fancyscript{A}^\infty (K),\partial ^\infty )$ into the unfiltered Legendrian contact homology DGA of $\varLambda _K$ with coefficients in ${\mathbb Z }[H_2(S^*{\mathbb R }^3,\varLambda _K)]$. Since the latter is a topological invariant, the result follows.

We remark that our choice of basis $(l,m,s)$ for $H_2(S^*{\mathbb R }^3,\varLambda _K)$ in the proof of Theorem 1.2 is canonical, depending only on the topological type of $K$; as a consequence, the (stable tame) isomorphisms on Theorem 1.2 act as the identity, not just an isomorphism, on the base ring ${\mathbb Z }[\lambda ^{\pm 1},\mu ^{\pm 1},U^{\pm 1},V^{\pm 1}]$. This is clear for $m$, which can be defined using the orientation on $K$ rather than the contact structure, and for $s$. For $l$, suppose that there is a topological isotopy $K_t$ between two transverse knots $K_0$ and $K_1$, with a corresponding isotopy $\Sigma _t$ of Seifert surfaces, such that $K_t$ fails to be a transverse knot at finitely many moments. At these moments, the self-linking number may jump, generically by $\pm 2$, but $n_U([\Sigma _t])$ and $n_V([\Sigma _t])$ also each jump by $\pm 1$. Thus the quantity $n_U([\Sigma _t]) - \frac{\mathrm{sl }(K_t)-1}{2}$ remains unchanged during the isotopy. Since $l$ (which is $[\Sigma _t]$ plus some multiple of $s$) is chosen so that $n_U(l) = \frac{\mathrm{sl }(K_t)-1}{2}$, the isotopy $K_t$ preserves $l$.

4 Computing the filtered DGA of a transverse link

In this section we compute the filtered DGA of a transverse link to prove Theorem 1.3. We begin by describing the main strategy used in [4] for calculating knot contact homology. This description leads to a sufficient understanding of the behavior of all holomorphic disks needed for the calculation of the filtered differential.

4.1 Scheme for calculating knot contact homology

The calculation of knot contact homology in [4] proceeds as follows. Consider a link $K$ braided around the unknot $U$ with $n$ strands and view it as a multisection of a fibration $U\times D^{2}$, where $D^{2}$ is the $2$-disk, corresponding to a tubular neighborhood of $U$. As we degenerate the multisection toward $U$ the conormal lift $\varLambda _K$ approaches the conormal lift of the unknot $\varLambda _U$ (with multiplicity $n$). More precisely, a neighborhood of $\varLambda _U\subset S^{*}{\mathbb R }^{3}$ is contactomorphic to the $1$-jet space $J^{1}(\varLambda _U)$ of $\varLambda _U$ and for $K$ sufficiently close to $U$, $\varLambda _K$ is a multisection of $J^{1}(\varLambda _U)\rightarrow \varLambda _U$ with $n$ sheets. In other words, over any open disk $W\subset \varLambda _U$, $\varLambda _K$ is given by the $1$-jet extension of $n$ functions $F_j:W\rightarrow {\mathbb R }$, $j=1,\dots ,n$.

It will be useful to recall that when a Legendrian submanifold $\varLambda$ in $J^1(S^2)$ is given locally as the $1$-jet of $n$ functions $F_i$ then its Reeb chords (that is double points of $\pi _{\mathbb C }(\varLambda )$) correspond to critical points with positive critical values of the difference of these local functions $F_i-F_j.$

By [4, §3], we know that close to the limit as $\varLambda _K$ approaches $\varLambda _U,$ the Reeb chords of $\varLambda _K$ are of two types:

I :

Near each Reeb chord $ch$ of $\varLambda _U$ there are $n^{2}$ Reeb chords $ch_{ij}$, $1\le i,j\le n$ of $\varLambda _K$, where we write $ch_{ij}$ for the chord near $ch$ that starts on the $j\mathrm{th}$ local sheet of $\varLambda _K$ near the start point of $ch$ and ends at the $i\mathrm{th}$ local sheet near the endpoint of $ch$.

II :

There are $n(n-1)$ small Reeb chords corresponding to critical points of positive local function differences of the form $F_i-F_j$. These critical points are either maxima or saddle points. We denote the former by $b_{ij}$ and the latter by $a_{ij}$, where $1\le i,j\le n$, $i\ne j$, and we use the same notational conventions for subscripts as above.

Thus Reeb chords of $\varLambda _K$ are either Reeb chords of $\varLambda _U$ with a small chord added or subtracted or small Reeb chords entirely inside the neighborhood of $\varLambda _U.$

One of the main technical results of [4] (see [5] for a similar result) shows that holomorphic disks admit a similar description. Near the limit as $\varLambda _K$ approaches $\varLambda _U$, rigid holomorphic disks in $T^{*} S$ with boundary on $\overline{\varLambda }_K = \pi _{\mathbb C }(\varLambda _K)$ and one positive puncture are of two types:

I :

They can lie in an arbitrarily small neighborhood of the union of the following: a disk with one positive puncture and boundary on $\varLambda _U;$ and, certain flow trees of the function differences $F_i-F_j$ attached along the disk’s boundary.

II :

They can lie entirely inside a small neighborhood of $\varLambda _K$ and are given locally as flow trees of the functional differences $F_i-F_j$.

Furthermore, any disk with its positive puncture at a chord of type $\mathbf I$ (resp. $\mathbf II$) is of type $\mathbf I$ (resp. $\mathbf II$). For the notion of flow trees we refer to [2, Section 2]. We do not give a complete definition of flow trees here, but merely note that they are made from pieces of flow lines of the functional differences $F_i-F_j$ and that as $\varLambda _K$ collapses onto $\varLambda _U$ the corresponding holomorphic disks stay in smaller and smaller neighborhoods of $\varLambda _U.$

In conclusion, in order to compute the differential $\partial :KC\!\fancyscript{A}(K)\rightarrow KC\!\fancyscript{A}(K)$ we need to understand holomorphic disks with boundary on $\overline{\varLambda }_U$ and flow trees determined by $F_i-F_j$. The latter can be understood using finite dimensional Morse theory. In order to understand the former we use the correspondence between holomorphic disks and flow trees on fronts from [2].

4.2 The conormal lift of the unknot

One may easily compute (or see [3, 4] and Lemma 4.1 below) the conormal lift $\varLambda _U$ of the unknot $U$ represented as the circle of radius $r<\delta$ in the $x_1x_2$-plane. This lift can be slightly perturbed in $J^1(S)$ so that it has two Reeb chords denoted $c$ and $e,$ see Fig. 1.

Fig. 1

On the left is ${\mathbb R }^3$ with the unknot in the $xy$-plane and a normal circle at one point is shown. The middle figure shows the normal circle seen in $S^2\times {\mathbb R }\cong {\mathbb R }^3-\{(0,0,0)\}$ which is thought of as the (front) projection of $S^*{\mathbb R }^3\cong J^1(S^2)$ to $S^2\times {\mathbb R }.$ On the right is the front projection of the entire conormal lift of the unknot which is obtained by rotating the circle shown in the middle figure about the line through the north and south poles. We have also slightly perturbed the picture on the right so that there are only two Reeb chords, labeled $c$ and $e$ in the figure 

For purposes of finding holomorphic disks via flow trees, $\varLambda _U$ must be perturbed to be in general position with respect to the front projection into $S^2\times {\mathbb R }.$ Notice that in Fig. 1 there is a circle in $\varLambda _U$ that is mapped to the north pole and a circle mapped to the south pole. Since $\varLambda _U$ is already in general position outside the fibers over the north and south pole of $S$ we concentrate our attention there. Near the poles, the projection $\overline{\varLambda }_U$ looks as described in the following lemma.

Lemma 4.1

If $U$ is the circle of radius $r<\delta$ in the $x_1x_2$-plane, then

$$\begin{aligned} \overline{\varLambda }_U\cap T^{*}E_{2\delta _0}\!=\!\{(q,p):q=(\xi _1,\xi _2,\pm 1),\,p=(x_1,x_2,0)\}\cong S^{1}\!\times \!(-1,1) \times \{\pm 1\}, \end{aligned}$$

where $\sqrt{x_1^{2}+x_2^{2}}=r$, $-\xi _1x_2+\xi _2x_1=0$, and $\cong$ denotes diffeomorphism.

Proof

This is immediate from the definition.

In Fig. 2 we see the front projection of $\varLambda _U$ over the region where $S$ is flat. The left pictures show $\varLambda _U$ in a neighborhood of the circle over the north (or south) pole, as described in Lemma 4.1, and the corresponding cone in the front projection of $\varLambda _U$. The right pictures show $\varLambda _U$ after small perturbation near the cone point that makes the front projection generic. Using the right representation we get the following result describing holomorphic disks of $\varLambda _U$, taken from [4, §3]. See Fig. 5 for a description of the flow trees on the front (which are close to the projection of the disk boundaries) and Fig. 3 for a description of their lifts into $\varLambda _U$.

Fig. 2

Along the top of the figure is an annular neighborhood of the circle in $\varLambda _U$ that maps to the fiber above the north pole. On the middle left, we see the image of this annulus near the north pole in the front projection, a cone whose boundary is two circles. On the bottom left is the image of this annulus near the north pole in $S^2$ (that is, the top view of the cone where we have slightly offset the circles so that they are both visible). On the middle right, we see the top view of the cone after it has been perturbed to have a generic front projection. More specifically, the lighter outer curve is the image of the cusp curves, the dotted lines are the image of double points in the front projection and the darkest inner curve is the image of the circle that mapped to the cone point before the perturbation. On the bottom right, we see the image in $S^2$ of the cusp curve and the two boundary circles on $\varLambda _U$

Fig. 3

The rigid disks from Lemma 4.2 in the northern hemisphere of $S^2$

Fig. 4

The one dimensional families of trees from Lemma 4.2 in the northern hemisphere of $S^2$

Lemma 4.2

(Ekholm et al. [4]) There are exactly six rigid holomorphic disks with boundary on $\overline{\varLambda }_U$: four ($I_N$, $Y_N$, $I_S$, and $Y_S$) with positive puncture at $c$ and no negative puncture, and two ($E_1$,$E_2$) with positive puncture at $e$ and negative puncture at $c$. If $q$ is any point in $\varLambda _{U}$ lying over a point where the front of $\varLambda _U$ has 2 sheets then there are exactly two constrained rigid holomorphic disks with positive puncture at $e$ and boundary constrained to pass through $q$. These two disks correspond to two constrained rigid flows which lie in two of the $1$-parameter families $\widetilde{I}_N$, $\widetilde{Y}_N$, $\widetilde{I}_S$, and $\widetilde{Y}_S$ of flow trees with positive puncture at $e$, and are rigidified by the condition that their $1$-jet lift passes through $q$. The boundaries of these $1$-parameter families are as follows:

$$\begin{aligned} \begin{aligned} \partial \widetilde{I}_N&= (E_1\,\#\, I_N) \cup (E_2\,\#\, I_N),\quad \partial \widetilde{Y}_N = (E_1\,\#\, Y_N) \cup (E_2\,\#\, Y_N),\\ \partial \widetilde{I}_S&= (E_1\,\#\, I_S) \cup (E_2\,\#\, I_S),\quad \partial \widetilde{Y}_S= (E_1\,\#\, Y_S) \cup (E_2\,\#\, Y_S). \end{aligned} \end{aligned}$$

Here $E_1\,\#\, I_N$ denotes the broken flow tree obtained by adjoining $I_N$ to $E_1$ etc., see Fig. 4.

Using the capping path convention and isomorphism $H_1(\varLambda _U) = {\mathbb Z }\langle \mu , \lambda \rangle$ specified in [4], the disks $I_N, Y_N, I_S,Y_S$ contribute $1, \lambda , \lambda \mu , \mu ,$ respectively, to $\partial c.$ Similarly $E_1$ and $E_2$ contribute $-\lambda ^{-1}c$ and $\lambda ^{-1}c,$ respectively, to $\partial e.$ This gives the differential

$$\begin{aligned} \partial c&= 1 + \lambda + \lambda \mu + \mu \\ \partial e&= (-\lambda ^{-1} + \lambda ^{-1}) c = 0. \end{aligned}$$

As in [4, §3], it suffices to consider constrained rigid trees when studying the filtered contributions of the curves. (See the third result of Theorem 4.6 below.) Although not needed for the purposes of calculating the filtered differential in this paper, it is possible, as mentioned in [4, §3], to prove that 1-parameter families of holomorphic disks with positive puncture at $e$ are in natural one-to-one correspondence with the $1$-parameter families of flow trees mentioned in Lemma 4.2.

4.3 Intersection numbers for disks with boundary on $\overline{\varLambda }_U$

Now that we have recalled the computation of the knot contact homology of the standard unknot in the $x_1x_2$-plane, we turn to computing the filtration on this knot thought of as a transverse knot.

Theorem 4.3

Let $U$ be the transverse unknot with self-linking number $-1.$ The filtered DGA $(KC\!\fancyscript{A}^-(U),\partial ^-)$ is filtered stable tame isomorphic to the algebra over $R[U,V]$ generated by $c$ and $e,$ where $|c|=1$ and $|e|=2,$ and

$$\begin{aligned} \partial ^- c= U+\lambda +\lambda \mu V+\mu \quad \text{ and} \quad \partial ^- e=0. \end{aligned}$$

Proof

From the explicit description of the knot contact homology differential in Lemma 4.2, it suffices to compute the intersection numbers of the four holomorphic disks $I_N,Y_N,I_S,Y_S$ with $\overline{H}_\pm .$ To this end, we will compute the intersection numbers of $\overline{H}_\pm$ with a disk $D_+$ homotopic to “$Y_N - I_N$” to determine the relative intersection numbers of $Y_N$ and $I_N$ with $\overline{H}_\pm .$ The absolute intersection numbers will follow from positivity of intersections. A similar argument will apply to $I_S$ and $Y_S.$

We first note that Lemma 3.8 implies that any point in the intersection between $\overline{H}_\pm$ and a holomorphic disk with boundary on $\varLambda _U$ must lie in $T^{*}E_{2\delta _0}$ so it will be sufficient to consider the parts of the holomorphic disks which lie in this region. As we shrink the unknot $U$ towards the $x_3$ axis in ${\mathbb R }^3$, its conormal lift $\varLambda _U$ approaches the $0$-section in $J^{1}(S).$ We can also see that the perturbed front generic version of $\varLambda _U$ collapses onto the $0$-section. As we degenerate $\varLambda _U$ onto the $0$-section, the boundaries of the holomorphic disks converge to the curves on the torus $\varLambda _U$ depicted in Fig. 5.

Fig. 5

Boundaries of holomorphic disks on $\varLambda _U$

The Legendrian torus $\varLambda _U$ can be described as the $1$-jet of a multifunction from $S^2$ to ${\mathbb R }$ and we notice that the curves in Fig. 5 are given by gradient flow lines for the functional differences of the multifunction. Furthermore, for an appropriate almost complex structure, the holomorphic disks $C^{1}$-converge to the strips corresponding to these flow lines outside any fixed neighborhood of its vertices, see [2, Lemma 5.13, Remark 5.14, and Subsection 6.4], and inside neighborhoods of its vertices they converge to other local models, see [2, Subsection 6.1]. Here the strip of a flow line consists of the line segments in the cotangent fibers between its cotangent lift. In particular, if we choose a perturbation of $\varLambda _U$ so that its projection to $T^{*} S$ consists of affine subspaces in a neighborhood of $(0,0,1)\in S$ then the standard complex structure for which $\overline{H}_{\pm }$ is a complex hypersurface is appropriate in the above sense, see [2, Subsection 4.1, 2nd bullet from the end].

Fig. 6

Two holomorphic disks in distinct homotopy classes. We have chosen coordinates on $S^2$ near the north pole so that the north pole corresponds to $(0,0)$

Consider the flow-line disks depicted in Fig. 6. These flow lines lie in the two distinct homotopy classes of the disks $I_N$ and $Y_N$. (Which of these disks looks like the letter $I$ or $Y$ depends on the perturbation of $\varLambda _U$ which makes its front generic, see also Fig. 4.) The disks $I_1$ and $I_2$ correspond to flow lines of the difference of two functions that locally describe part of $\varLambda _U$ near the north pole. With notation and coordinates near the north pole of $S^2$ indicated in Fig. 6, the gradient of the function describing the upper sheet of $I_1$ equals $-\partial _{u_1}$ and that of its lower sheet is $\partial _{u_2},$ and the flow line corresponding to $I_1$ in $S$ follows the curve $C_1=(t,t)$, $-T\le t\le \varepsilon$ where $\varepsilon >0$ is arbitrarily small and not yet fixed. Thus the flow line strip of $I_1$ is

$$\begin{aligned} (t,s)\mapsto ((t,t), (s(-\partial _{u_1})+(1-s)\partial _{u_2})), \quad -T\le t\le \varepsilon ,\,\,0\le s\le 1, \end{aligned}$$

near the origin. Similarly, the flow line strip of $I_2$ is

$$\begin{aligned} (t,s)\mapsto ((-t,t), (s\partial _{u_1}+(1-s)\partial _{u_2})), \quad -T\le t\le \varepsilon ,\,\,0\le s\le 1. \end{aligned}$$

On the other hand the intersection of $\overline{H}_+$ with the fibers of $T^{*} S$ over the point $(t,t)\in C_1$ is $a( t\partial _{u_1}-t\partial _{u_2})$ for some $a>0$, see Eq. (3.5). That is,

$$\begin{aligned} \overline{H}_+\cap T^{*}C_1=\{((t,t), a(t\partial _{u_1}-t\partial _{u_2}))\}, \end{aligned}$$

and we see that the strip and the hypersurface intersect once (over the point $(-\frac{1}{2a},-\frac{1}{2a})$). Similarly, over the flow line of $I_2$ we have

$$\begin{aligned} \overline{H}_+\cap T^{*}C_1=\{((-t,t), a(t\partial _{u_1}+t\partial _{u_2}))\}, \end{aligned}$$

and we see that the strip and the hypersurface do not intersect: the solution of the equation which sets the fiber coordinates equal would lie over the point $(-t,t)$ where $t=\frac{1}{2a}$ but if we set $\varepsilon < \frac{1}{2a},$ this point does not lie on the flow line of $I_2$.

In order to relate the above calculation to Lemma 4.2, consider the disk $D_\pm$ in the fiber of $T^*S$ bounded by the circle $((0,0,\pm 1),p)\in \overline{\varLambda }_U\cap T^{*}E_{2\delta _0}$ and oriented according to the induced orientation on the fiber. We notice that this circle is a lift of a longitude for $U.$ Using the description of $\overline{H}_{\pm }$ in Lemma 3.7 it is easy to see that $D_\pm$ intersects $\overline{H}_{\pm }$ with intersection number $\mp 1.$ (Note here that the orientation of the base followed by the orientation of the fiber gives the orientation opposite to the complex orientation on $T^{*}S$ and that the orientation induced on the normal bundle to the fiber by $\overline{H}_{+},$ respectively $\overline{H}_{-},$ agrees, respectively disagrees, with the orientation on the base.) Consulting Fig. 5 and considering the algebraic topology of $\varLambda _U\subset T^*S^2$ one may easily see that the difference cycle between the disk corresponding to $I_N$ and $Y_N$ is homologous to $D_+$. Since the intersection number of these disks with $\overline{H}_+$ equals $1$ or $0$ (by the above calculation) it follows that the disk corresponding to $I_N$ has intersection number $1$ with $\overline{H}_+$ and the disk corresponding to $Y_N$ has intersection number $0$. Similarly, the disks corresponding to $I_S$ and $Y_S$ intersect $\overline{H}_-$ with intersection number $1$ and $0$, respectively. In addition, since $D_{\mp }$ does not intersect $\overline{H}_{\pm }$, we find that the $I_S$ and $Y_S$ (respectively $I_N$ and $Y_N$) disks do not intersect $\overline{H}_+$ (respectively $\overline{H}_-$).

4.4 The filtered DGA of a transverse link

We begin by recalling the computation of the knot contact homology from [4, Theorem 1.1]. Using the notation in the introduction we have the following result.

Theorem 4.4

(Ekholm et al. [4]) The differential in the Legendrian DGA associated to the conormal lift $\varLambda _K$ of a framed knot $K$ is $(KC\!\fancyscript{A}(K),\partial )$ where $KC\!\fancyscript{A}(K)$ is generated over ${\mathbb Z }[\mu ^{\pm }, \lambda ^{\pm }]$ by the $a_{ij},b_{ij}, c_{ij},$ and $e_{ij}$ described in Sect. 4.1 and 4.2 and the map $\partial :KC\!\fancyscript{A}(K)\rightarrow KC\!\fancyscript{A}(K)$ is determined by the following matrix equations:

$$\begin{aligned} \partial \mathbf A&= 0, \\ \partial \mathbf B&= -{\varvec{\lambda }}^{-1}\cdot {\mathbf A }\cdot {\varvec{\lambda }}\,\, +\,\, \varPhi ^L_B \cdot {\mathbf A }\cdot \varPhi ^R_B, \\ \partial \mathbf C&= {\mathbf A }\cdot {\varvec{\lambda }}\,\, +\,\, {\mathbf A }\cdot \varPhi ^{R}_B, \\ \partial \mathbf E&= {\mathbf B }\cdot (\varPhi _B^{R})^{-1}\,\, +\,\, {\mathbf B }\cdot {\varvec{\lambda }}^{-1}\,\, -\,\, \varPhi ^L_B\cdot {\mathbf C }\cdot {\varvec{\lambda }}^{-1}\,\, +\,\, {\varvec{\lambda }}^{-1}\cdot {\mathbf C }\cdot (\varPhi ^R_B)^{-1}, \end{aligned}$$

where $\mathbf A ,\mathbf B ,\mathbf C ,\mathbf E ,{\varvec{\lambda }},\varvec{\varPhi }^\mathbf L _{\mathbf B }, \varvec{\varPhi }^{\mathbf R }_{\mathbf B }$ are as in Sect. 1.2, and if ${\mathbf M }$ is an $(n\times n)$-matrix, the matrix $\partial {\mathbf M }$ is defined by $(\partial {\mathbf M })_{ij}=\partial {\mathbf M }_{ij}$.

In Sect. 4.1 above we discussed the holomorphic curves involved in the computation of the differential. In particular, one may easily conclude the following result.

Theorem 4.5

(Ekholm et al. [4]) Given any $\delta ^{\prime }>0,$ $\varLambda _K$ can be Legendrian isotoped to be close enough to $\varLambda _U$ so that any holomorphic disk with boundary on $\overline{\varLambda }_K,$ one positive puncture, and involving only the chords $a_{ij}$ and $b_{ij},$ has its image contained within a $\delta ^{\prime }$-neighborhood of $\overline{\varLambda }_U.$

Proof

While this lemma follows from the results in [4, §3.4], we comment that it can also be seen by observing that the Reeb chords $a_{ij}$ and $b_{ij}$ have small action. This action can be made arbitrarily small as we isotop $K$ to be close to $U.$ Now a monotonicity argument will confine the holomorphic curves to stay close to $\overline{\varLambda }_U.$ $\square$

To compute the filtration on a transverse knot we will need to explicitly describe the holomorphic disks used in the computation of the differential. More specifically we need to understand disks of Type I, discussed in Sect. 4.1. To this end we summarize the computations from [4, §4.4].

Theorem 4.6

(Ekholm et al. [4]) Given any $\delta ^{\prime }>0,$ $\varLambda _K$ can be Legendrian isotoped to be close enough to $\varLambda _U$ so that any rigid holomorphic disk with boundary on $\overline{\varLambda }_K$ either has its image contained within a $\delta ^{\prime }$-neighborhood of $\overline{\varLambda }_U$ and one of the rigid disks described in Lemma 4.2, or has its image contained within a $\delta ^{\prime }$-neighborhood of $\overline{\varLambda }_U$ and one of the constrained rigid disks described in Lemma 4.2 constrained by the projection $q \in \varLambda _{U}$ of one of the endpoints of a Reeb chord $b_{ij}$.

• 1. The holomorphic disks that contribute to the terms in $\partial c_{ij}$ satisfy

  • the terms in ${\mathbf A }{\varvec{\lambda }}$ with $\mu$ coefficients are contained in neighborhoods of $I_S$ (and $\overline{\varLambda }_U$),

  • the terms in ${\mathbf A }{\varvec{\lambda }}$ without $\mu$ coefficients are contained in neighborhoods of $Y_N$ (and $\overline{\varLambda }_U$),

  • the terms in ${\mathbf A }\varPhi ^R_B$ with $\mu$ coefficients are contained in neighborhoods of $Y_S$ (and $\overline{\varLambda }_U$), and

  • the terms in ${\mathbf A }\varPhi ^R_B$ without $\mu$ coefficients are contained in neighborhoods of $I_N$ (and $\overline{\varLambda }_U$).

• 2. The holomorphic disks that contribute to the $\varPhi ^L_B\cdot {\mathbf C }\cdot {\varvec{\lambda }}^{-1}\,\, +\,\, {\varvec{\lambda }}^{-1}\cdot {\mathbf C }\cdot \mathrm{(}\varPhi ^R_B\mathrm{)}^{-1}$ terms in $\partial e_{ij}$ are contained in neighborhoods of either $E_0$ or $E_1$ (and $\overline{\varLambda }_U$).

• 3. The holomorphic disks that contribute to the ${\mathbf B }\cdot (\varPhi _B^{R})^{-1}\,\, +\,\, {\mathbf B }\cdot {\varvec{\lambda }}^{-1}$ terms of $\partial e_{ij}$ are contained in neighborhoods of one of the disks in the 1-parameter families $\widetilde{I}_N$, $\widetilde{Y}_N$, $\widetilde{I}_S$, and $\widetilde{Y}_S$ (and $\overline{\varLambda }_U$). That is, they are close to the union of $\overline{\varLambda }_U\cup E_1\cup E_2$ and one of the disks $I_N, Y_N, I_S, Y_S,$ shifted in the $\lambda$-direction some fixed distance (the same distance for all disks). More precisely,

  • disks associated to those terms in ${\mathbf B }{\varvec{\lambda }}^{-1}$ with $\mu$ coefficients correspond to $Y_S,$

  • disks associated to those terms in ${\mathbf B }{\varvec{\lambda }}^{-1}$ without $\mu$ coefficients correspond to $I_N,$

  • disks associated to those terms in ${\mathbf B }\left(\varPhi ^R_B\right)^{-1}$ with $\mu$ coefficients correspond to $I_S,$ and

  • disks associated to those terms in ${\mathbf B }\left(\varPhi ^R_B\right)^{-1}$ without $\mu$ coefficients correspond to $Y_N.$

We are now in a position to prove the combinatorial expression for the filtrations given in Theorem 1.3.

Proof of Theorem 1.3

The first equation in the theorem follows for grading reasons.

The second equation in the theorem follows from Theorem 4.4 and the fact that none of the holomorphic disk contributing to the differential of a $b_{ij}$ can intersect $\overline{H}_\pm$ by Lemma 4.5.

Since the disks $E_0$ and $E_1$ lie close to the equator, Lemma 3.8 and the third paragraph of Theorem 4.6 imply that there are no intersections of $\overline{H}_{\pm }$ with the disks with positive puncture at an $e_{ij}$-chord and one negative puncture at an $c_{ij}$-chord. Thus, $\varPhi ^L_B\cdot {\mathbf C }\cdot {\varvec{\lambda }}^{-1}\,\, +\,\, {\varvec{\lambda }}^{-1}\cdot {\mathbf C }\cdot (\varPhi ^R_B)^{-1}$ appears in both $\partial \mathbf E$ and ${\partial }^- \mathbf E .$

The remainder of the differential involves disks that intersect $\overline{H}_{\pm }$. The formula for the filtered differential that derives from these intersections is easily derived by the intersections of $\overline{H}_\pm$ with the disks from Lemma 4.2 which were worked out in Sect. 4.3. Some simple bookkeeping yields the desired computation.

5 Some examples

In this section, we present some computations of the filtered transverse knot DGA, $(KC\!\fancyscript{A}^-,\partial ^-)$. The first shows that the additional filtration structure is nontrivial; the second is an outline of a computation that shows that the filtered DGA is an effective invariant of transverse knots.

5.1 The unknot

We compute the transverse knot DGA for three versions of the unknot: the closure of the trivial $1$-braid and the closure of the $2$-braids $\sigma _1$ and $\sigma _1^{-1}$. The first two both represent the standard transverse unknot in ${\mathbb R }^3$ with self-linking number $-1$, while the third represents the transverse unknot with self-linking number $-3$. We will show that the filtered DGAs for the first two are stable tame isomorphic and are distinct from the filtered DGA for the third.

For the trivial $1$-braid, Theorem 1.3 or 4.3 yields a DGA with two generators $c,e$ and differential

$$\begin{aligned} \partial ^- c = U+\lambda +\lambda \mu V+\mu , \quad \partial ^- e = 0. \end{aligned}$$

For the $2$-braid $\sigma _1$, the relevant matrices are

$$\begin{aligned} \varPhi ^L_{\sigma _1}&= \begin{pmatrix} -a_{21}&1 \\ 1&0 \end{pmatrix} ,\quad \varPhi ^R_{\sigma _1} = \begin{pmatrix} -a_{12}&1 \\ 1&0 \end{pmatrix} ,\quad {\varvec{\lambda }}= \begin{pmatrix} \lambda \mu&0 \\ 0&1 \end{pmatrix}\\ {\mathbf A }&= \begin{pmatrix} 1+\mu&a_{12} \\ \mu a_{21}&1+\mu \end{pmatrix} , \quad {\mathbf A }^U = \begin{pmatrix} U+\mu&U a_{12} \\ \mu a_{21}&U +\mu \end{pmatrix} , \quad {\mathbf A }^V = \begin{pmatrix} 1+\mu V&a_{12} \\ \mu V a_{21}&1+\mu V \end{pmatrix},\\ {\mathbf B }^U&= \begin{pmatrix} 0&U b_{12} \\ \mu b_{21}&0 \end{pmatrix} , \quad {\mathbf B }^V = \begin{pmatrix} 0&b_{12} \\ \mu V b_{21}&0 \end{pmatrix}. \end{aligned}$$

We then calculate from Theorem 1.3 that the DGA for $\sigma _1$ has generators $a_{12}$, $a_{21}$, $b_{12}$, $b_{21}$, $c_{11}$, $c_{12}$, $c_{21}$, $c_{22}$, $e_{11}$, $e_{12}$, $e_{21}$, $e_{22}$, and differential given by $\partial ^-(a_{12})=\partial ^-(a_{21})=0$,

$$\begin{aligned}&\partial ^- \begin{pmatrix} 0&b_{12} \\ b_{21}&0 \end{pmatrix} = \begin{pmatrix} 0&-\frac{1}{\lambda \mu } a_{12}-a_{21} \\ -\lambda \mu a_{21}-a_{12}&0 \end{pmatrix}, \\&\partial ^- \begin{pmatrix} c_{11}&c_{12} \\ c_{21}&c_{22} \end{pmatrix} = \begin{pmatrix} \lambda \mu +\lambda \mu ^2 V-\mu a_{12}&\mu +U+a_{12} \\ \mu +U+\lambda \mu ^2 V a_{21}-\mu a_{21}a_{12}&1+\mu V+\mu a_{21} \end{pmatrix}, \\&\partial ^- \begin{pmatrix} e_{11}&e_{12} \\ e_{21}&e_{22} \end{pmatrix} \\&\quad =\begin{pmatrix} b_{12}+\frac{1}{\lambda \mu }(c_{12}-c_{21}+a_{21}c_{11})&U b_{12}-c_{22}+a_{21}c_{12}+b_{12}a_{12}+\frac{1}{\lambda \mu }(c_{11}+c_{12}a_{12}) \\ c_{22}+\frac{1}{\lambda }b_{21}-\frac{1}{\lambda \mu }c_{11}&\mu V b_{21}-c_{12}+c_{21}+c_{22}a_{12} \end{pmatrix}. \end{aligned}$$

We want to find a tame automorphism of this algebra that sends $\partial ^-$ to a stabilization of the differential for the trivial braid. We do this by replacing generators one by one so that the new differential on all generators but $c_{12}$ and $e_{11}$ is trivial or nearly trivial, and the new differential on $c_{12}$ and $e_{11}$ is as in the case of the trivial 1-braid; for instance, since $\textstyle {\partial ^-(c_{12}+\frac{1}{\mu }c_{11})} = U+\lambda +\lambda \mu V+\mu$, we can replace $c_{12}$ by $\textstyle {c_{12}-\frac{1}{\mu }c_{11}}$ to get a generator with differential $U+\lambda +\lambda \mu V+\mu$. In full, if we apply successively the eight tame automorphisms

$$\begin{aligned} \begin{aligned} e_{12}&\mapsto \textstyle {e_{12}-b_{12}c_{12}-\frac{1}{\lambda \mu } c_{12}^2} \\ c_{21}&\mapsto c_{21}-\mu Vb_{21}+c_{12}-c_{22}a_{12} \\ b_{12}&\mapsto \textstyle { b_{12}-\frac{1}{\mu }c_{22}+\frac{1}{\lambda \mu ^2} c_{11}} \\ b_{21}&\mapsto \textstyle { b_{21}+\frac{1}{\mu }c_{11}-\lambda c_{22}} \\ e_{11}&\mapsto \textstyle {e_{11}+\frac{1}{\lambda \mu ^2}c_{22}c_{11}-\frac{1}{\mu } e_{12}-\frac{1}{\lambda \mu } e_{22}+Ve_{21}} \\ c_{12}&\mapsto \textstyle {c_{12}-\frac{1}{\mu } c_{11}} \\ a_{21}&\mapsto \textstyle {a_{21}-V-\frac{1}{\mu }} \\ a_{12}&\mapsto a_{12}+\lambda +\lambda \mu V, \end{aligned} \end{aligned}$$

then we obtain a DGA with the same generators but differential

$$\begin{aligned}&\partial ^-(c_{11})=-\mu a_{12}, \qquad \partial ^-(c_{12}) = U+\lambda +\lambda \mu V+\mu , \qquad \partial ^-(c_{22})= \mu a_{21}, \\&\partial ^-(e_{12}) = -\mu b_{12}, \qquad \partial ^-(e_{21}) = \textstyle {\frac{1}{\lambda } b_{21}}, \qquad \partial ^-(e_{22}) = c_{21},\\&\partial ^-(a_{12}) = \partial ^-(a_{21}) = \partial ^-(b_{12}) = \partial ^-(b_{21}) = \partial ^-(c_{21}) = \partial ^-(e_{11}) = 0. \end{aligned}$$

Destabilizing yields a DGA generated by $c_{12},e_{11}$ with differential $\partial ^-(c_{12}) = U+\lambda +\lambda \mu V+\mu$, $\partial ^-(e_{11})=0$, which agrees with the DGA for the trivial $1$-braid above.

For the $2$-braid $\sigma _1^{-1}$, we use the matrices

$$\begin{aligned} \varPhi ^R_{\sigma _1^{-1}} = \begin{pmatrix} 0&1 \\ 1&-a_{21} \end{pmatrix} ,\quad {\varvec{\lambda }}= \begin{pmatrix} \lambda \mu ^{-1}&0 \\ 0&1 \end{pmatrix}. \end{aligned}$$

The expression for ${\partial }^- \mathbf C$ from Theorem 1.3 now yields in particular

$$\begin{aligned} \partial ^-(c_{11}) = \lambda \mu ^{-1}+\lambda V+U a_{12}. \end{aligned}$$

We claim that the filtered DGA for $\sigma _1^{-1}$ differs from the filtered DGA for the trivial $1$-braid by setting $(U,V)=(0,0)$ and comparing $(\widehat{\widehat{KC\!\fancyscript{A}}}, \widehat{\widehat{\partial }}).$ For $\sigma _1^{-1}$ we find that $\widehat{\widehat{\partial }}(c_{11}) = \lambda \mu ^{-1}$, a unit in ${\mathbb Z }[\lambda ^{\pm 1},\mu ^{\pm 1}]$, and so the homology of $(\widehat{\widehat{KC\!\fancyscript{A}}}, \widehat{\widehat{\partial }})$ is trivial. For the trivial $1$-braid, we obtain a DGA generated by $c,e$ with $\widehat{\widehat{\partial }}(c) = \lambda +\mu$ and $\widehat{\widehat{\partial }}(e) = 0$, and it is clear that this DGA has nontrivial homology.

5.2 The knot $m(7_6)$

Let $K_1,K_2$ be the transverse knots given by the closures of the $4$-braids

$$\begin{aligned} \sigma _1\sigma _2^{-1}\sigma _1\sigma _2^{-1}\sigma _3^{-1}\sigma _2\sigma _3^3, \qquad \sigma _1\sigma _2^{-1}\sigma _1\sigma _2^{-1}\sigma _3^3\sigma _2\sigma _3^{-1}, \end{aligned}$$

respectively. These are both transverse representatives of the mirror of the knot $7_6$, with self-linking number $-1$. The following result demonstrates that the filtered DGA is an effective invariant of transverse knots.

Theorem 5.1

The filtered DGAs for $K_1$ and $K_2$ are not filtered stable tame isomorphic, and thus $K_1$ and $K_2$ are not transversely isotopic.

Proof

Consider the DGAs $(\widehat{KC\!\fancyscript{A}}(K_i),\widehat{\partial })$ over ${\mathbb Z }[\lambda ^{\pm 1},\mu ^{\pm 1}]$ obtained from the filtered DGAs by setting $(U,V)=(0,1)$. One can count by computer the number of DGA maps (augmentations) from $(\widehat{KC\!\fancyscript{A}}(K_i),\widehat{\partial })$ to $({\mathbb Z }/3,0)$ that send $\lambda$ to $-1$ and $\mu$ to $+1$; there are $5$ for $K_1$ and $0$ for $K_2$. The result follows. (See [13] for more details.)

As noted in [13], the hat version of knot Floer homology for $m(7_6)$ is $0$ in the relevant bidegree $(0,0)$, and so the transverse invariants in knot Floer homology [10, 14] do not distinguish $K_1$ and $K_2$. One similarly finds that other previously developed transverse invariants (in Khovanov or Khovanov–Rozansky homology, for instance) do not distinguish $K_1$ and $K_2$. We conclude that the transverse invariant from knot contact homology is independent of previously known transverse invariants.