# Persistent Legendrian Contact Homology in $\mathbb{R}^{3}$

Maya Basu and Austin Christian and Ethan Clayton and Daniel Irvine and
Fredrick Mooers and Weizhe Shen

###### Abstract.

This work applies the ideas of persistent homology to the problem of
distinguishing Legendrian knots. We develop a persistent version of Legendrian
contact homology by filtering the Chekanov-Eliashberg DGA using the action
(height) functional. We present an algorithm for assigning heights to a
Lagrangian diagram of a Legendrian knot, and we explain how each Legendrian
Reidemeister move changes the height of generators of the DGA in a way that is
predictable on the level of homology. More precisely, a Reidemeister move that
changes an area patch of a Lagrangian diagram by $\delta$ will induce a
$2\delta$-interleaving on the persistent Legendrian contact homology, computed
before and after the Reidemeister move. Finally, we develop strong Morse
inequalities for our persistent Legendrian contact homology.

##  1\. Introduction

This work presents a new lens through which we view the problem of classifying
Legendrian knots.

A Legendrian knot is a smooth embedding $S^{1}\to\mathbb{R}^{3}$ that is
everywhere tangent to the contact distribution given by the standard contact
form,

$\alpha_{std}=dz-y\,dx.$

More precisely, if a knot $\Lambda$ is parametrized as
$\Lambda(t)=(x(t),y(t),z(t))$ for $t\in[0,2\pi]$, then the Legendrian
condition asks that

(1) $z^{\prime}(t)-y(t)\,x^{\prime}(t)=0.$

By imposing this geometrical constraint on a knot, we are led to consider a
rich problem of classifying knots up to isotopies which preserve their
topological and geometrical properties. This, broadly, is the classification
problem in Legendrian knot theory.

A Legendrian knot properly exists in $\mathbb{R}^{3}$, but it is often
convenient to work with a two-dimensional projection of the knot. There are
two useful ways of projecting a Legendrian knot into the plane. The Lagrangian
projection is

$\Pi\colon\mathbb{R}^{3}\to\mathbb{R}^{2},\qquad\text{ given by
}\qquad(x,y,z)\mapsto(x,y).$

The Lagrangian projection operator, when restricted to a Legendrian knot
$\Lambda$, gives a particular Lagrangian diagram of $\Lambda$. More precisely,
a Lagrangian diagram is a knot diagram that is a Lagrangian projection of a
Legendrian knot, modulo all possible planar isotopies. (See Section 2.2.1.) An
example Lagrangian diagram is shown in Figure 2(a). The other important
projection is the front projection operator.

$F\colon\mathbb{R}^{3}\to\mathbb{R}^{2},\qquad\text{ given by
}\qquad(x,y,z)\mapsto(x,z).$

Legendrian contact homology (abbreviated LCH) is a powerful tool to use on the
classification problem. The survey [EN20] explains how to compute the
Legendrian contact homology, $\operatorname{LCH}_{*}(\Lambda)$, given a
Lagrangian diagram for $\Lambda$. In this introduction, we elide the details
of how to use _augmentations_ of $\operatorname{LCH}_{*}(\Lambda)$ to extract
useful information, but an explanation is given in Section 2.2.3.

The main utility of LCH comes in distinguishing Legendrian knots. We describe
here an example application, in the broadest possible terms. Given two
Legendrian knots $\Lambda_{1}$ and $\Lambda_{2}$, one could compute the so-
called Poincaré-Chekanov polynomials $PC^{\epsilon}$ for $\Lambda_{1}$ and
$\Lambda_{2}$ with respect to all possible augmentations of each (see Theorem
2.4 and Formula 5). If one arrives at two distinct collections of polynomials,
then one can conclude that $\Lambda_{1}$ and $\Lambda_{2}$ are distinct as
Legendrian knots.

The substance of this work is to introduce ideas of persistent homology
(abbreviated persistence) into the computation of the Legendrian contact
homology of a knot. True to its name, persistent homology measures the
geometro-topological features that persist throughout the many spaces that
make up a filtered knot. Given a knot $\Lambda$, a filtration is an increasing
family of subsets, $\\{\Lambda^{a}\\}_{a\in\mathbb{R}}$, such that
$\Lambda^{a}\subseteq\Lambda^{b}$, whenever $a\leq b$, and
$\Lambda^{a}=\Lambda$ for sufficiently large $a$. While this definition is
useful for intuition, in this paper we will not examine a filtration on a
Legendrian knot; instead, we will define a filtration on the chain level. A
persistence module of chain complexes is a functor from the partially-ordered
set of real numbers, $(\mathbb{R},\leq)$ treated as a poset category, to the
category of chain complexes of modules over a fixed ground ring. A more
detailed definition will be given in Section 2.1.1.

Legendrian contact homology is the homology of a differential graded algebra
referred to in the literature as the _Chekanov-Eliashberg DGA_. The Chekanov-
Eliashberg DGA is naturally filtered by the Reeb action of each generator.
This will be the filtration for our persistence module structure. We adopt the
following sign convention. Each generator $q$ of the Chekanov-Eliashberg DGA
has a geometric realization as a crossing in a Lagrangian projection. Since
the Legendrian knot $\Lambda$ exists in contact $\mathbb{R}^{3}$, we may lift
the crossing from the Lagrangian projection $\Pi(\Lambda)$ to a pair of points
$\tilde{c}\in\Lambda$ and define

$\text{action}(q)=\int_{\tilde{c}}dz.$

A chain level filtration will turn the Chekanov-Eliashberg DGA into a
persistence module with a differential operator, as defined in [PSV17]. The
Chekanov-Eliashberg DGA already has a $\mathbb{Z}$-grading, called the Maslov
grading, given by the degree of a word. Introducing a filtration adds an
$\mathbb{R}$-grading. As a means of distinguishing these two gradings, we call
$\text{action}(q)$ the height of the element $q$. Henceforth we denote the
height by $h(q)$. The evaluation of heights agrees with the algebra operations
of the Chekanov-Eliashberg DGA;

$h(q_{1}q_{2})=h(q_{1})+h(q_{2})\qquad\text{ and }\qquad
h(q_{1}+q_{2})=\max\\{h(q_{1}),h(q_{2})\\}.$

We now see that the problem of introducing persistence into Legendrian contact
homology hinges on the assignment of a height to each generator of the DGA. In
this paper we will explain why it is natural to express the LCH of a
Legendrian knot as a persistence module and demonstrate a powerful tool for
visualizing persistent LCH, called a barcode.

######  Lemma 1.1.

In this work, we are interested in the following two algebraic structures,
both of which are persistence modules.

  1. (1)

The Chekanov-Eliashberg DGA with a height filtration is an example of a
persistence module of DGAs.

  2. (2)

The (linearized) homology of (i) is a persistence module with a real
parameter, denoted $F^{\bullet}\operatorname{LCH}^{\epsilon}_{\ast}(\Lambda)$.

All persistence modules of interest to us can be decomposed in the following
way.

######  Theorem 1.2.

_(A consequence of the structure theorem of Gabriel, Auslander, Ringel–
Tachikawa, Webb, cf.[CdSGO16])_ The persistence modules in Lemma 1.1 can be
decomposed into a direct sum of interval modules.

An interval module is the simplest example of a persistence module, and this
is explained in Example 2.1, below. We use this structure theorem every time
that we compute persistent homology.

There is a pseudo-metric on the class (category) of persistence modules of
Theorem 1.2, called the _interleaving distance_. The interleaving distance
will be defined in Section 2.1.3, but for now, let us denote the distance
function by

$d\colon(\text{persistence modules})^{\times 2}\to\mathbb{R}.$

The ultimate goal of this investigation is to explain the extent to which the
persistent Legendrian contact homology is an invariant of a Lagrangian
projection diagram of a Legendrian knot $\Lambda$. In order to do this, we
must show that applying any reasonable Legendrian Reidemeister move to a
Lagrangian diagram for $\Lambda$ will change the persistent LCH by a small
amount, with respect to this interleaving distance.

######  Theorem 1.3.

Let $\Lambda_{0}$ and $\Lambda_{1}$ be Legendrian knots whose Lagrangian
projections $\Pi(\Lambda_{0})$ and $\Pi(\Lambda_{1})$ are related by a single
Legendrian Reidemeister move, and fix $\delta>0$. There exist Legendrian knots
$\tilde{\Lambda_{0}}$ and $\tilde{\Lambda_{1}}$ whose Lagrangian projections
are planar isotopic to those of $\Lambda_{0}$ and $\Lambda_{1}$, respectively,
such that

$d(F^{\bullet}\operatorname{LCH}^{\epsilon}_{\ast}(\tilde{\Lambda_{0}})\,,\,F^{\bullet}\operatorname{LCH}^{\epsilon}_{\ast}(\tilde{\Lambda_{1}}))<\delta.$

Let us discuss a bit of motivation for Theorem 1.3. Recall that
$F^{\bullet}\operatorname{LCH}^{\epsilon}_{\ast}(\Lambda)$ is a persistence
module of a real parameter. After applying Theorem 1.2, we write
$F^{\bullet}\operatorname{LCH}^{\epsilon}_{\ast}(\Lambda)$ as a direct sum of
interval modules, which we call a barcode. The interleaving distance measures
the infimum over all $\delta$ for which a $2\delta$ interleaving exists
between two barcodes. A working definition for a $2\delta$-interleaving of a
barcode is as follows. One is allowed to move the endpoints of any bar left or
right by a distance of at most $\delta$. Consequently, one may delete any bar
of finite length less than $2\delta$.

Using this working definition, one might draw a barcode of a Legendrian knot
diagram and look for a bar (interval) of length $<2\delta$, for some pre-
determined noise parameter $\delta$. By identifying the crossings that
generate the endpoints of the bar of short length, one may then look for a
Legendrian Reidemeister move to perform near those crossings which can
simplify the knot and the barcode. This is a way in which a barcode could hope
to distinguish essential information from noise.

### Future directions

The present article establishes the definition of persistent Legendrian
contact homology and offers some basic tools for computing persistent LCH in
practice. But several interesting questions remain, and we plan to address
some of these in future work. Our hope is that persistent LCH will have much
to tell us about the classification of Legendrian knots in $\mathbb{R}^{3}$.

Our first and most fundamental question is one of invariance. While a fixed
(generic) Legendrian knot in $\mathbb{R}^{3}$ has a well-defined persistent
LCH, Legendrian knots are typically studied according to their Legendrian
isotopy classes. The invariance question thus becomes

######  Question 1.1.

What collection of persistence modules can be realized as the persistent LCH
of a fixed Legendrian isotopy type?

Because Legendrian contact homology is computed combinatorially from a
Lagrangian diagram, we find it natural to decompose this question into two
finer questions:

  1. (1)

How is persistent LCH affected by Legendrian isotopies which do not change the
Lagrangian diagram?

  2. (2)

How is persistent LCH affected by Legendrian isotopies which induce
Reidemeister moves on the Lagrangian diagram?

Theorem 1.3 provides one answer to 2: a Legendrian isotopy $\Lambda_{t}$,
$t\in[t_{0}-\epsilon,t_{0}+\epsilon]\subsetneq[0,1]$, corresponding to a
single Lagrangian Reidemeister move at $t=t_{0}$ may be extended to a
Legendrian isotopy $\Lambda_{t}$, $t\in[0,1]$, whose Lagrangian projection is
constant on each of $[0,t_{0})$ and $(t_{0},1]$ and whose endpoints
$\Lambda_{0},\Lambda_{1}$ have arbitrarily small interleaving distance between
their persistent Legendrian contact homologies.

Our understanding of 1 is also quite strong. Namely, a planar isotopy of a
Lagrangian projection cannot change LCH, and thus persistent LCH will change
only by variations in the filtration — that is to say, the endpoints of a bar
in the persistent LCH barcode may change, but the number of bars in the
barcode will not. For a fixed Lagrangian diagram with $n$ crossings, the
collection of valid filtrations forms an open cone in the first orthant of
$\mathbb{R}^{n}$, and Legendrian isotopies which leave the Lagrangian diagram
unchanged correspond to paths in this cone. It is desirable to have a
preferred height assignment within this cone, so that persistent LCH is a
well-defined invariant of Lagrangian diagrams. Note that we cannot choose the
vertex of our cone, since this assigns a height of zero to every crossing. The
lack of obvious choice for a canonical height assignment leaves us with the
following question:

######  Question 1.2.

How should we upgrade persistent LCH to a planar isotopy invariant of
Lagrangian diagrams?

Before leaving this question, we outline a strategy which the authors find
promising. Each crossing of a Lagrangian diagram $D$ corresponds to a
coordinate $h_{i}$ in $\mathbb{R}^{n}$, and this coordinate must satsify
$h_{i}>0$. The Legendrian condition then leads to $n+1$ inequalities in these
coordinates, each of which has a constant term of 0. To obtain a closed set of
feasible height assignments in $\mathbb{R}^{n}$, we may fix parameters
$0<\epsilon_{A}\ll\epsilon_{H}$ and require that $h_{i}\geq\epsilon_{H}$,
while the constant term of each area inequality coming from $D$ is replaced by
$\epsilon_{A}$. The solutions to the resulting collection of inequalities form
a closed, convex, unbounded subset of the cone of height assignments described
above, with a unique vertex minimizing the sum $h_{1}+\cdots+h_{n}$. We
suspect that defining the persistent LCH of $D$ to be computed using the
height assignment corresponding to this vertex will allow us to define a
Legendrian isotopy invariant with desirable features, and these suspicions are
the subject of ongoing investigation.

In addition to our goal of realizing persistent LCH as a Legendrian isotopy
invariant, we hope to replicate some of the additional structure enjoyed by
LCH in the persistent setting. For instance, Sabloff constructed in [Sab06] an
expanded Chekanov-Eliashberg DGA from which one may derive a type of Poincaré
duality for Legendrian contact homology. In persistent LCH, this gives a
duality for infinite bars — if $r_{k}\geq 0$ is the number of infinite bars in
grading $k$ for a persistent LCH barcode, then $r_{1}=r_{-1}+1$ and
$r_{k}=r_{-k}$ for $k\neq\pm 1$. It is natural to wonder how our filtration of
the Chekanov-Eliashberg DGA might extend to the expanded algebra, and whether
this might lead us to a Poincaré duality for persistent LCH. A concrete
version of this question is as follows.

######  Question 1.3.

Does persistent LCH admit a notion of Sabloff duality for finite bars?

Our optimism for persistent LCH lies in the fact that it retains the
computability of linearized LCH while also capturing chain-level data in the
form of finite bars. However, some nonlinear aspects of LCH can be computed,
and have proven to be quite useful: in [CKE+11], cup and Massey products are
constructed on linearized LCH and used to recover higher-order linearizations
of LCH. The authors find it quite plausible that these innovations could be
duplicated in the persistent setting, and thus pose the following question.

######  Question 1.4.

Does persistent Legendrian contact homology admit a Massey product structure?

Since its introduction, Legendrian contact homology has been utilized and
generalized in a great variety of manners, and we are hopeful that some of
these successes will be replicated in persistent Legendrian contact homology.

### Organization

This paper is organized as follows. In Section 2, we provide relevant
background information on persistent homology, barcodes, the interleaving
distance, Legendrian knots, Legendrian contact homology, and augmentations. We
then describe how these ideas will fit together in this work to accomplish the
goal of using persistent homology to understand Legendrian knots. In Section
3, we provide an outline of the procedure for computing the Legendrian contact
homology barcode from a Lagrangian diagram. Two worked examples are given, of
the standard unknot and the standard trefoil knot.

In Section 4, we set about proving the main theorem of this work, which is
Theorem 1.3 above. We need to show that any Legendrian Reidemeister move
performed on a Lagrangian diagram will perturb in an inappreciable way the
area that is contained by the bounded regions of the diagram. This
perturbation of areas will, in turn, perturb the heights that can and must be
assigned to each Lagrangian diagram double point. These double points will
serve as the chain complex generators of Legendrian contact homology, and the
height of each generator will be important for the filtration that gives us
persistent homology, as explained in Section 2. Hence, to complete the proof
of the main theorem we must also check that each Legendrian Reidemeister move
perturbs generators of the aforementioned filtered chain complex in a way that
will be predictable on the level of homology.

In Section 5, we present our flooding algorithm for assigning heights to a
Lagrangian diagram. In Section 6, we associate to a Legendrian knot two
polynomials, called the Morse-Chekanov polynomial and the Poincaré-Chekanov
polynomial. We develop the so-called strong Morse inequalities in Theorem 6.1,
which says that the Morse-Chekanov polynomial majorizes the Poincaré-Chekanov
polynomial.

##  2\. Background

In this section, we gather background material for persistent homology in
Section 2.1 and for Legendrian contact homology in Section 2.2.

###  2.1. Persistent homology

Persistent homology is a popular tool in topological data analysis. In this
section, we recall the notation and results necessary to build a _barcode_ for
a filtered DGA, and describe a pseudo-metric on the category of persistence
modules.

####  2.1.1. Persistence modules

######  Definition 1.

Let $R$ be a fixed ground ring. A _persistence module over a real parameter_
is a functor from the poset category $(\mathbb{R},\leq)$ to the category of
$R$-modules. Similarly, a _persistence module of DGAs_ is a functor from
$(\mathbb{R},\leq)$ to the category of DGAs over $R$.

In practice, our ground ring $R$ will always be $\mathbb{Z}_{2}$, and the
unmodified phrase _persistence module_ will refer to a persistence module over
a real parameter. We will be interested in persistence modules of DGAs when
working at the chain level, and persistence modules over $\mathbb{Z}_{2}$ once
we have passed to homology. The data we will track at the chain level consists
of an $\mathbb{R}$-graded direct sum

$\mathcal{A}^{\bullet}:=\bigoplus_{t\in\mathbb{R}}\mathcal{A}^{t}$

of DGAs $\mathcal{A}^{t}$ over $\mathbb{Z}_{2}$, along with a DGA morphism
$\psi_{s}^{t}\colon\mathcal{A}^{s}\to\mathcal{A}^{t}$, for every pair of real
numbers $s\leq t$, satisfying

  1. (1)

$\psi_{t}^{t}=\mathrm{id}$, for every $t\in\mathbb{R}$;

  2. (2)

$\psi_{s}^{t}\circ\psi_{r}^{s}=\psi_{r}^{t}$, for every triple $r\leq s\leq
t$.

We refer to $\psi_{s}^{t}$ as a _transfer map_. At the homology level we will
consider an $\mathbb{R}$-graded direct sum of $\mathbb{Z}_{2}$-vector spaces
$V^{t}$ along with linear transfer maps $v_{s}^{t}\colon V^{s}\to V^{t}$
satisfying conditions analogous to those above. We will use the notation
$\mathcal{A}^{\bullet}$ for a persistence module of DGAs and $V^{\bullet}$ for
a persistence module over a real parameter.

######  Example 2.1.

For our purposes, the fundamental example of a persistence module is the
_interval module_. Given any interval $I\subset\mathbb{R}$, we may define a
family of $\mathbb{Z}_{2}$-vector spaces

$V^{t}:=\left\\{\begin{matrix}\mathbb{Z}_{2},&t\in I\\\
0,&\text{otherwise}\end{matrix}\right.,$

along with linear transformations $v_{s}^{t}\colon V^{s}\to V^{t}$ defined for
$s\leq t$ by

$v_{s}^{t}:=\left\\{\begin{matrix}\mathrm{id},&s,t\in I\\\
0,&\text{otherwise}\end{matrix}\right..$

This collection of vector spaces and linear maps determines a persistence
module $V^{\bullet}$; if $I=[a,b)$, we will denote this persistence module by
$\mathbb{Z}_{2}[a,b)$.

We will see that persistent homology is expressed as a direct sum of interval
modules, and this decomposition is called a barcode.

####  2.1.2. Persistent homology

The homology of a DGA $(\mathcal{A},\partial)$ over $\mathbb{Z}_{2}$ is a
direct sum of vector spaces:

$H_{*}(\mathcal{A},\partial):=\bigoplus_{k\in\mathbb{Z}}\dfrac{\ker\partial_{k}}{\operatorname{im}\partial_{k+1}}.$

Analogously, the persistent homology of a persistence module of DGAs
$(\mathcal{A}^{\bullet},\partial^{\bullet})$ over $\mathbb{Z}_{2}$ is a
$\mathbb{Z}$-graded direct sum of persistence modules. For each integer
$k\in\mathbb{Z}$, we have the persistence module

$H_{k}(\mathcal{A}^{\bullet},\partial^{\bullet}):=\bigoplus_{t\in\mathbb{R}}H_{k}(\mathcal{A}^{t},\partial^{t}),$

with transfer maps $H_{k}(\mathcal{A}^{s},\partial^{s})\to
H_{k}(\mathcal{A}^{t},\partial^{t})$ induced by the chain-level transfer maps
$\psi_{s}^{t}\colon\mathcal{A}^{s}\to\mathcal{A}^{t}$. From these we obtain
the persistent homology of $(\mathcal{A}^{\bullet},\partial^{\bullet})$:

$H_{*}(\mathcal{A}^{\bullet},\partial^{\bullet}):=\bigoplus_{k\in\mathbb{Z}}H_{k}(\mathcal{A}^{\bullet},\partial^{\bullet}).$

Thus, for instance, the vector space of
$H_{*}(\mathcal{A}^{\bullet},\partial^{\bullet})$ at filtration level
$t\in\mathbb{R}$ is given by

$H^{t}_{*}(\mathcal{A}^{\bullet},\partial^{\bullet})=\bigoplus_{k\in\mathbb{Z}}H_{k}(\mathcal{A}^{t},\partial^{t}),$

and the transfer map between the vector spaces at filtration levels $s,t$,
with $s\leq t$, is given as a direct sum of transfer maps.

####  2.1.3. Nearby persistence modules

We explain here how two objects in the category of persistence modules might
be considered nearby one another. For concreteness, let us consider two
interval modules $V^{\bullet}=\mathbb{Z}_{2}[a_{1},b_{1})$ and
$U^{\bullet}=\mathbb{Z}_{2}[a_{2},b_{2})$. Recall that an interval module is a
special kind of persistence module whose target (here in the category of
vector spaces over $\mathbb{Z}_{2}$) is either the zero element or is of rank
one. Specifically, $V^{\bullet}=\mathbb{Z}_{2}[a_{1},b_{1})$ means to
construct an $\mathbb{R}$-graded family of vector spaces,
$\\{V^{t}\\}_{t\in\mathbb{R}}$, so that

$V^{t}=\left\\{\begin{matrix}\mathbb{Z}_{2},&t\in[a_{1},b_{1})\\\
0,&\text{otherwise}\end{matrix}\right.,$

With this example in mind, we define a map $U^{\bullet}\to V^{\bullet}$ that
shifts the $\mathbb{R}$-grading by a distance $\delta$. A persistence module
homomorphism of (persistence) degree $\delta$ is a collection of maps
$\varphi^{t}\colon U^{t}\to V^{t+\delta}$ such that the following diagram is
commutative;

${U^{s}}$${U^{t}}$${V^{s+\delta}}$${V^{t+\delta}.}$$\scriptstyle{u_{s}^{t}}$$\scriptstyle{\varphi^{s}}$$\scriptstyle{\varphi^{t}}$$\scriptstyle{v_{s+\delta}^{t+\delta}}$

A special case is the $\delta$-shift map from a persistence module to itself,
i.e. for each real number $t$, $\mathbb{I}_{V}^{\delta}\colon V^{t}\to
V^{t+\delta}$ is a persistence module homomorphism of degree $\delta$. We say
that two persistence modules, $U^{\bullet}$ and $V^{\bullet}$ are
$2\delta$-interleaved if there exist homomorphisms $\varphi^{t}\colon U^{t}\to
V^{t+\delta}$ and $\psi^{t}\colon V^{t}\to U^{t+\delta}$, both of degree
$\delta$, such that $\psi^{t+\delta}\circ\varphi^{t}=\mathbb{I}_{U}^{2\delta}$
and $\varphi^{t+2\delta}\circ\psi^{t}=\mathbb{I}_{V}^{2\delta}$.

This allows us to equip the functor category of persistence modules with an
extended pseudometric.

######  Definition 2.

The _interleaving distance_ between two persistence modules is given by

$d(V^{\bullet},U^{\bullet})=\inf\\{\epsilon\,|\,V^{\bullet}\text{ and
}U^{\bullet}\text{ are }2\epsilon\text{-interleaved}\\},$

and we set $d(V^{\bullet},U^{\bullet})=\infty$ if $V^{\bullet}$ and
$U^{\bullet}$ are not $\epsilon$-interleaved for any $\epsilon\geq 0$.

See [BS14] for an explanation of why this function constitutes a pseudometric
on persistence modules.

####  2.1.4. Barcodes

The most important example of a persistence module is an interval module,
because the Legendrian contact homology of a knot can be uniquely expressed as
a direct sum of interval modules. This claim follows from Theorem 1.2, above,
and the following.

######  Theorem 2.2.

_(Uniqueness theorem of Krull, Remak, Schmidt, and Azumaya cf.[CdSGO16])_
Suppose that a persistence module $V^{\bullet}$ can be decomposed into
interval modules in two ways as

$V^{\bullet}=\bigoplus_{\ell\in L}\mathbb{Z}_{2}J_{\ell}=\bigoplus_{m\in
M}\mathbb{Z}_{2}K_{m}$

where $L$ and $M$ are index sets that will be finite in the examples at hand,
and where each $J_{\ell}$ and each $K_{m}$ is an interval of real numbers.
Then there is a bijection $\sigma\colon L\to M$ satisfying
$J_{\ell}=K_{\sigma(\ell)}$ for each $\ell\in L$.

We call the decomposition of the LCH persistence module into interval modules
a _barcode_. The uniqueness theorem will permit us to speak of “the” barcode
of a Legendrian knot. In practice, barcodes for Legendrian knots which admit
augmentations may be drawn as a collection of some finite-length intervals and
some infinite-length intervals. This is an information-dense way of drawing
the persistent Legendrian contact homology. Examples of barcodes as drawings
are given in Figure 2(b) and Figure 3(b), and the whole of Section 3 is
devoted to explaining how to compute barcodes, both in general and in specific
examples.

Theorem 2.2 does say that we can permute the intervals in the interval module
decomposition that we are calling the barcode. The uniqueness theorem _does
not_ say that the barcode is insensitive to the assignment of heights, as
described in the introduction.

In this work, when we apply the interleaving distance to two barcodes, we mean
that we are applying the interleaving distance to the persistence module
associated to each.

###  2.2. Legendrian contact homology

We now establish notation for the Legendrian contact homology of a Legendrian
knot $\Lambda\subset\mathbb{R}^{3}$, constructed from a Lagrangian diagram
planar isotopic to $\Pi(\Lambda)\subset\mathbb{R}^{2}$. This construction
takes the form of a differential graded algebra — known as the Chekanov-
Eliashberg DGA — the homology of which is an invariant of the Legendrian
isotopy class of $\Lambda$.

Legendrian contact homology fits into the symplectic field theory (SFT)
paradigm due to Eliashberg-Givental-Hofer [EGH00], with the Chekanov-
Eliashberg DGA carefully defined in [Che02]. Generalizations later appeared in
[ENS02], [EES05], [EENS13], [NR13], [EL17], but we will follow [Che02] in
presenting the DGA over $\mathbb{Z}_{2}$.

Viewed as a $2$-plane distribution in the tangent bundle of $\mathbb{R}^{3}$,
we denote the standard contact structure as
$\xi_{std}=\text{ker}(\alpha_{std})$. The manifold $\mathbb{R}^{3}$ together
with the standard contact structure is a contact manifold, written
$(\mathbb{R}^{3},\xi_{std})$.

####  2.2.1. Legendrian knots and their diagrams

As discussed in Section 1, the two most important projections associated to a
Legendrian knot $\Lambda\subset\mathbb{R}^{3}$ are the Lagrangian projection
$\Pi(\Lambda)$ and the front projection $F(\Lambda)$. We will be especially
interested in the Lagrangian projection.

While $\Pi(\Lambda)\subset\mathbb{R}^{2}$ is an immersed curve, we point out
that not every closed, immersed curve in $\mathbb{R}^{2}$ can be realized as a
Lagrangian projection. For instance, a closed, immersed curve which encloses a
nonzero signed area is not the Lagrangian projection of any Legendrian knot.
Nonetheless, it is standard practice to represent a Legendrian $\Lambda$ by a
knot diagram which is planar isotopic to $\Pi(\Lambda)$, but which may not
lift to a Legendrian in $\mathbb{R}^{3}$. The distinction between diagrams
which lift to Legendrians and those which do not will be important to us, so
we establish the following vocabulary.

######  Definition 3.

A _Lagrangian projection diagram_ (or _Lagrangian diagram_) is a knot diagram
in $\mathbb{R}^{2}$ which is planar isotopic to a Lagrangian projection
$\Pi(\Lambda)$ of some Legendrian knot $\Lambda\subset\mathbb{R}^{3}$. We
assume that $\Pi(\Lambda)$ retains the crossing information of $\Lambda$.

Where smooth knot diagrams have a strong Reidemeister theorem — namely, any
Reidemeister move on a smooth knot diagram may be realized by a smooth isotopy
of knots — Lagrangian projection diagrams have only a weak Reidemeister
theorem.

######  Theorem 2.3.

_(c.f.[Che02, Section 6])_ If $\Lambda,\Lambda^{\prime}$ are Legendrian
isotopic Legendrian knots in $\mathbb{R}^{3}$, then their Lagrangian
projection diagrams $\Pi(\Lambda),\Pi(\Lambda^{\prime})$ are related by a
sequence of Legendrian Reidemeister moves, depicted in Figure 1.

(a)

(b)

(c)

Figure 1. The three Legendrian Reidemeister moves in the Lagrangian
projection. Note that we call the move in which a bigon is removed the
“inverse” move II. This terminology will be important in Section 4.

We will see that Legendrian contact homology may be defined from a Lagrangian
projection diagram, while persistent Legendrian contact homology _a priori_
requires an honest Lagrangian projection.

####  2.2.2. The Chekanov-Eliashberg DGA

The definition of the Chekanov-Eliashberg DGA accepts as input an oriented
Legendrian knot $\Lambda\subset(\mathbb{R}^{3},\xi_{\mathrm{std}})$. Important
to our work in this paper is that a small Legendrian isotopy may be applied to
$\Lambda$ to ensure that its Lagrangian projection
$\Pi(\Lambda)\subset\mathbb{R}^{2}$ has singularities of only one type:
transverse double points, where the strands of $\Pi(\Lambda)$ meet
orthogonally. In fact, the starting point of our definition may be any
Lagrangian projection diagram for $\Lambda$ with this property.

The algebra $\mathcal{A}_{\Lambda}$ underlying the Chekanov-Eliashberg DGA
associated to $\Lambda$ is generated over $\mathbb{Z}_{2}$ by
$q_{1},\ldots,q_{n}$, where each generator $q_{i}$, $1\leq i\leq n$,
corresponds to a double point of $\Pi(\Lambda)$. The double points of
$\Pi(\Lambda)$ correspond to _Reeb chords_ in $\mathbb{R}^{3}$ with endpoints
on $\Lambda$ — these are arcs parallel to the $z$-axis, and serve as critical
points of the Reeb action functional described in Section 1.

The algebra $\mathcal{A}_{\Lambda}$ is associative, but not commutative. We
refer the reader to [EN20] for details about the grading and differential on
$\mathcal{A}_{\Lambda}$, but fix notation for these as follows: the grading of
a word $w\in\mathcal{A}_{\Lambda}$ is denoted $|w|\in\mathbb{Z}$, while its
differential is denoted $\partial_{\Lambda}(w)\in\mathcal{A}_{\Lambda}$. Up to
the small Legendrian isotopy which may be required at the beginning of the
construction, $(\mathcal{A}_{\Lambda},\partial_{\Lambda})$ is a well-defined
differential graded algebra.

As a DGA, $(\mathcal{A}_{\Lambda},\partial_{\Lambda})$ is not an invariant of
the Legendrian isotopy type of $\Lambda$. For instance, a Legendrian isotopy
which corresponds to a double point move in $\Pi(\Lambda)$ will either
introduce or delete two generators of the DGA. If, however,
$\Lambda,\Lambda^{\prime}$ are Legendrian isotopic, then the DGAs
$(\mathcal{A}_{\Lambda},\partial_{\Lambda})$ and
$(\mathcal{A}_{\Lambda^{\prime}},\partial_{\Lambda^{\prime}})$ are _stable
tame isomorphic_. Because our proof of Theorem 1.3 requires some explicit
manipulations of these DGAs, we recall now what it means to be stable tame
isomorphic. The basic story of Legendrian contact homology may be digested
without these technical definitions, however.

First, stabilization accounts for the possibility of introducing superfluous
generators.

######  Definition 4.

For any positive integer $k$, the _grading $k$ stabilization_ of a Chekanov-
Eliashberg DGA $(\mathcal{A}_{\Lambda}=\mathbb{Z}_{2}\langle
q_{1},\ldots,q_{n}\rangle,\partial_{\Lambda})$ is the DGA whose

  1. (1)

algebra $S_{k}\mathcal{A}_{\Lambda}$ is generated over $\mathbb{Z}_{2}$ by
$e_{k},e_{k-1},q_{1},\ldots,q_{n}$;

  2. (2)

grading is given by $|e_{k}|=k$, $|e_{k-1}|=k-1$, and agrees with the grading
of $\mathcal{A}_{\Lambda}$ for other generators;

  3. (3)

differential $\partial_{\Lambda}$ satisfies
$\partial_{\Lambda}(e_{k})=e_{k-1}$ and agrees with $\partial_{\Lambda}$ on
other generators.

After sufficiently many stabilizations, the Chekanov-Eliashberg DGAs of a pair
of Legendrian isotopic knots become tame isomorphic.

######  Definition 5.

An _elementary automorphism_ of a Chekanov-Eliashberg DGA
$(\mathcal{A}_{\Lambda},\partial_{\Lambda})$ with underlying algebra
$\mathbb{Z}_{2}\langle q_{1},\ldots,q_{n}\rangle$ is a chain map
$\phi\colon(\mathcal{A}_{\Lambda},\partial_{\Lambda})\to(\mathcal{A}_{\Lambda},\partial_{\Lambda})$
of the form

$\phi(q_{j})=\pm q_{j}+u,\qquad\phi(q_{i})=q_{i},i\neq j,$

for some $1\leq j\leq n$ and some $u\in\mathbb{Z}_{2}\langle
q_{1},\ldots,\widehat{q_{j}},\ldots,q_{n}\rangle$. A _tame isomorphism_
between two Chekanov-Eliashberg DGAs
$(\mathcal{A}_{\Lambda}=\mathbb{Z}_{2}\langle
q_{1},\ldots,q_{n}\rangle,\partial_{\Lambda})$ and
$(\mathcal{A}^{\prime}_{\Lambda}=\mathbb{Z}_{2}\langle
q^{\prime}_{1},\ldots,q^{\prime}_{n}\rangle,\partial^{\prime}_{\Lambda})$ is
the result of composing some number of elementary automorphisms of
$(\mathcal{A}_{\Lambda},\partial_{\Lambda})$ with the isomorphism
$q_{i}\mapsto q^{\prime}_{i}$.

We say that a pair of Chekanov-Eliashberg DGAs is stable tame isomorphic if
they become tame ismorphic after sufficiently many stabilizations have been
applied to each of them. The most important remark at present is that stable
tame isomorphisms are chain homotopy equivalences, and thus the _Legendrian
contact homology_

$\operatorname{LCH}_{*}(\Lambda):=H_{*}(\mathcal{A}_{\Lambda},\partial_{\Lambda})$

is a well-defined invariant of the Legendrian isotopy type of $\Lambda$.

####  2.2.3. Linearization

In general, $\operatorname{LCH}_{*}(\Lambda)$ can be difficult to compute and
need not have finite rank. This can be resolved by considering the _linearized
Legendrian contact homology_ , which we now define.

First, an _augmentation_ of $(\mathcal{A}_{\Lambda},\partial_{\Lambda})$ is an
algebra homomorphism $\epsilon\colon\mathcal{A}_{\Lambda}\to\mathbb{Z}_{2}$
with the following properties:

  * •

$\epsilon(a)=0$, for any $a\in\mathcal{A}_{\Lambda}$ of nonzero grading;

  * •

$\epsilon\circ\partial_{\Lambda}=0$.

There are Legendrian knots $\Lambda$ for which
$(\mathcal{A}_{\Lambda},\partial_{\Lambda})$ does not admit an augmentation —
for instance, those with $\operatorname{rot}(\Lambda)\neq 0$ — but the
property of admitting an augmentation is invariant under stable tame
isomorphism.

Given an augmentation $\epsilon$ of
$(\mathcal{A}_{\Lambda},\partial_{\Lambda})$, we now consider the automorphism
$\phi^{\epsilon}\colon\mathcal{A}_{\Lambda}\to\mathcal{A}_{\Lambda}$ defined
by

$\phi^{\epsilon}(q_{i}):=q_{i}+\epsilon(q_{i}),$

for each generator $q_{i}$, $1\leq i\leq n$. Conjugating the differential
$\partial_{\Lambda}$ produces a new differential

$\partial^{\epsilon}_{\Lambda}:=\phi^{\epsilon}\circ\partial_{\Lambda}\circ(\phi^{\epsilon})^{-1},$

making $(\mathcal{A}_{\Lambda},\partial_{\Lambda}^{\epsilon})$ a DGA. The
primary advantage held by $\partial^{\epsilon}_{\Lambda}$ over
$\partial_{\Lambda}$ is that $\partial^{\epsilon}_{\Lambda}(q_{i})$, which is
a formal sum of words in $q_{1},\ldots,q_{n}$, has constant term given by
$(\epsilon\circ\partial_{\Lambda})(q_{i})=0$, for each $q_{i}$, $1\leq i\leq
n$. This allows us to define a linear map $\partial_{1}^{\epsilon}\colon
A_{\Lambda}\to A_{\Lambda}$ by

$\partial_{1}^{\epsilon}(q_{i}):=\pi_{1}(\partial^{\epsilon}_{\Lambda}(q_{i})),$

where $A_{\Lambda}$ is the $\mathbb{Z}_{2}$-vector space with basis
$\\{q_{1},\ldots,q_{n}\\}$ and $\pi_{1}\colon\mathcal{A}_{\Lambda}\to
A_{\Lambda}$ is the linear map which sends to 0 any words of length not equal
to 1.

It is straightforward to verify that $\partial_{1}^{\epsilon}$ is a
differential on $A_{\Lambda}$, and we define the _linearized Legendrian
contact homology of $\Lambda$ with respect to $\epsilon$_ to be

$\operatorname{LCH}_{*}^{\epsilon}(\Lambda):=H_{*}(A_{\Lambda},\partial_{1}^{\epsilon}).$

It is important to observe that $\operatorname{LCH}_{*}^{\epsilon}(\Lambda)$
does, in general, depend on the choice of augmentation $\epsilon$. However, we
may still extract a Legendrian isotopy invariant.

######  Theorem 2.4 ([Che02, Theorem 3.3, Lemma 4.1]).

The collection of linearized Legendrian contact homologies

$\\{\operatorname{LCH}_{*}^{\epsilon}(\Lambda)\,|\,\epsilon\colon\mathcal{A}_{\Lambda}\to\mathbb{Z}_{2}\text{~{}is
an augmentation}\\}$

is a Legendrian isotopy invariant of $\Lambda$.

####  2.2.4. Filtering the Chekanov-Eliashberg DGA

Computing a persistent version of (linearized) Legendrian contact homology
will require a filtration at the chain level — that is, on the DGA
$(\mathcal{A}_{\Lambda},\partial_{\Lambda})$ or the differential vector space
$(A_{\Lambda},\partial_{1}^{\epsilon})$. A natural candidate for such a
filtration is the _action_ or _Reeb height_ discussed in Section 1. In
particular, the assignment

(2) $h(q):=\int_{\gamma}\alpha_{\mathrm{std}}=\int_{\gamma}dz-ydx,$

where $\gamma$ is the Reeb chord with ends on $\Lambda$ which projects to $q$,
fits into the Floer-theoretic perspective which first motivated Legendrian
contact homology. For a fixed (generic) Legendrian $\Lambda$, this is a
perfectly acceptable filtration, and one may proceed with the computation of
persistent homology.

In practice, however, this definition is of limited utility, since most
computations of Legendrian contact homology accept as input a Legendrian
isotopy class, rather than a fixed Legendrian knot. For instance, consider
that $\operatorname{LCH}_{*}^{\epsilon}(\Lambda)$ is typically computed from
the Lagrangian projection $\Pi(\Lambda)$. Given
$\Pi(\Lambda)\subset\mathbb{R}^{2}$, we may recover
$\Lambda\subset\mathbb{R}^{3}$ up to $z$-translation, and thus recover its
Reeb heights; but an arbitrary Lagrangian projection diagram of $\Lambda$ will
not generally lift to a Legendrian knot in $\mathbb{R}^{3}$, leaving us with
the question of how to define heights from a Lagrangian diagram.

Given a Lagrangian diagram $D\subset\mathbb{R}^{2}$, we will refer to the
bounded components of $\mathbb{R}^{2}\setminus D$ as _area patches_. Each area
patch $P$ has corners corresponding to Reeb chords, and these corners are
assigned _Reeb signs_ , defined as follows: as we traverse $\partial P$ with
the boundary orientation, a positive Reeb sign is assigned to each corner
where we move from an understrand to an overstrand; other corners are assigned
negative Reeb signs.

Given an explicit Lagrangian projection $\Pi(\Lambda)$, one may use Stokes’
theorem together with the Legendrian condition (1) to find that the (unsigned)
area of an area patch is equal to the sum of the Reeb heights of its corners,
weighted by the Reeb signs. This area may change under planar isotopy, but
must remain positive. Thus we may associate a linear inequality to each area
patch of a Lagrangian projection diagram. For instance, the area patches in
Figure 2(a) each lead to the inequality $h(q)>0$, while the topmost area patch
of Figure 3(a) has inequality

$h(q_{1})-h(q_{3})-h(q_{4})-h(q_{5})>0.$

When we speak of the _area inequalities_ of a Lagrangian diagram $D$ we are
referring to the collection of linear inequalities resulting from its area
patches. We recall that the goals of the present paper include:

  1. (1)

an algorithmic solution to the area inequalities, given a Lagrangian diagram
$D$;

  2. (2)

a description of how the solution set of the area inequalities changes under
Legendrian isotopy.

##  3\. Toy examples

This section is concerned with computing persistent Legendrian contact
homology in some simple cases. In Section 3.1 we describe the general
procedure for computing persistent LCH from a Lagrangian projection diagram,
and then we present two instructive examples.

###  3.1. The general case

We outline the steps needed to construct a barcode from a Lagrangian diagram,
provided the Chekanov-Eliashberg DGA admits an augmentation. Throughout, the
corresponding Legendrian isotopy type is denoted by $\Lambda$ and the
crossings of the diagram are denoted by $q_{1},\ldots,q_{n}$.

  1. (1)

First, calculate the non-linear Chekanov-Eliashberg DGA
$\mathcal{A}_{\Lambda}$. Pick a valid augmentation $\epsilon$ and use this to
produce a chain complex of $\mathbb{Z}_{2}$ vector spaces, denoted
$(A,\partial_{1}^{\epsilon})$, following the linearization procedure described
in Section 2.2.3.

  2. (2)

Assign a height $h(q_{i})>0$ to each generator $q_{i}$ of $A$. Because this
assignment is not included in the data of a Lagrangian diagram, one must make
a choice; we present one algorithm for making this choice in Section 5.

  3. (3)

Use the assignment made in Step 2 to upgrade $A$ to a persistence module of a
real parameter. Namely, for each $\ell\in\mathbb{R}$, let $A^{\ell}$ denote
the subspace of $A$ that is generated by those $q_{i}$ satisfying
$h(q_{i})\leq\ell$ and define

$A^{\bullet}=\bigoplus_{\ell\in\mathbb{R}}A^{\ell}.$

  4. (4)

Compute the homology of each chain complex
$(A^{\ell},\partial_{1}^{\epsilon})$ and define the persistence module

$F^{\bullet}\operatorname{LCH}^{\epsilon}_{\ast}(\Lambda)=\bigoplus_{\ell\in\mathbb{R}}H_{*}(A^{\ell},\partial_{1}^{\epsilon}).$

A transfer map on this persistence module will be induced from applying the
homology functor to the corresponding transfer map of $A^{\bullet}$. Note that
each vector space $F^{t}\operatorname{LCH}^{\epsilon}_{\ast}(\Lambda)$ also
has a $\mathbb{Z}$-grading, given by taking homology in different degrees. We
refer to $F^{\bullet}\operatorname{LCH}^{\epsilon}_{\ast}(\Lambda)$ as the
_persistent Legendrian contact homology_ of $\Lambda$ with the auxiliary data
of an augmentation $\epsilon$ and height assignment $h$.

It is our decomposition of
$F^{\bullet}\operatorname{LCH}^{\epsilon}_{\ast}(\Lambda)$ into interval
modules which allows us to depict the persistent LCH as a barcode. Moreover,
we will typically decorate our barcodes with (linear combinations of)
generators $q_{i}$, to indicate choices of homology bases.

###  3.2. The standard unknot

(a) A Lagrangian diagram of a Legendrian unknot. It has one Reeb chord labeled
$q$. We assign this chord a height of 1.

01$h$$q$$H_{1}$ (b) The barcode for the diagram of the unknot (left). This
barcode, has one ‘infinite’ bar in $H_{1}$, which is generated at height 1 by
$q$. The starting point of this bar depends on the height assigned to $q$.

Figure 2. The Legendrian unknot and its barcode.

For the standard Legendrian unknot whose Lagrangian diagram is pictured in
Figure 2(a), the algebra $\mathcal{A}_{\Lambda}$ has a single generator $q$,
with $|q|=1$, and a simple computation (c.f. [EN20, Example 3.5]) shows that
$\partial_{\Lambda}q=1$. There is only one augmentation
$\epsilon\colon\mathcal{A}_{\Lambda}\to\mathbb{Z}_{2}$, and it satisfies
$\epsilon(q)=0$. It follows that the augmented differential
$\partial_{1}^{\epsilon}$ is trivial. By declaring $h(q)=1$, we may upgrade
$A=\mathbb{Z}_{2}\langle q\rangle$ to a persistence module $A^{\bullet}$,
where

$A^{\ell}=\left\\{\begin{matrix}0&\ell<1\\\ \mathbb{Z}_{2}\langle
q\rangle&1\leq\ell\end{matrix}\right.$

Because $\partial_{1}^{\epsilon}$ is trivial,
$H_{*}(A^{\ell},\partial_{1}^{\epsilon})\cong A^{\ell}$, for every $\ell$, and
we see that $F^{\bullet}\operatorname{LCH}^{\epsilon}_{\ast}(\Lambda)$ is the
interval module $\mathbb{Z}_{2}[1,\infty)$. The persistent LCH of the unknot
is depicted in Figure 2(b), with the label $q$ on our infinite bar
representing our choice of homology basis $[q]$, as well as the fact that $q$
is “born” into existence when we begin considering generators of height 1 or
greater.

###  3.3. The standard trefoil

Our second (and more interesting) toy example is the standard Legendrian
trefoil whose Lagrangian diagram is pictured in Figure 3(a). With the
generators as labeled there, one may compute (c.f. [EN20, Example 3.6]) the
gradings

$|q_{1}|=|q_{2}|=1\quad\text{and}\quad|q_{3}|=|q_{4}|=|q_{5}|=0$

and the differential $\partial_{\Lambda}$

$\displaystyle\partial_{\Lambda}q_{1}$
$\displaystyle=1+q_{5}+q_{5}q_{4}q_{3}+q_{3}$
$\displaystyle\partial_{\Lambda}q_{2}$
$\displaystyle=1+q_{3}+q_{3}q_{4}q_{5}+q_{5}$
$\displaystyle\partial_{\Lambda}q_{3}$
$\displaystyle=\partial_{\Lambda}q_{4}=\partial_{\Lambda}q_{5}=0.$

(a) A Lagrangian diagram of a Legendrian trefoil derived from Ng’s resolution
applied to a front diagram of the trefoil. It has five Reeb chords labeled
$q_{1}$ through $q_{5}$. $q_{1}$ and $q_{2}$ are both assigned height 4. The
other three chords are assigned height 1.

014$h$$q_{5}$$q_{4}$$q_{1}+q_{2}$$H_{1}$$q_{3}+q_{5}$$q_{1},q_{2}$$H_{0}$ (b)
The barcode for the diagram of the trefoil (left). This barcode, has one
infinite bar in $H_{1}$, which is created at height 4, and two infinite bars
in $H_{0}$, both created at height 1. There is also a finite bar in $H_{0}$
created at height 1 and destroyed at height 4. The starting point and length
of these bars depend on the heights we assign to the Reeb chords.

Figure 3. The Legendrian trefoil and its barcode

The DGA $(\mathcal{A}_{\Lambda},\partial_{\Lambda})$ admits five augmentations
to $\mathbb{Z}_{2}$ (c.f. [EN20, Example 4.6]), but one may compute that all
of these give the linearized differential

$\displaystyle\partial_{1}^{\epsilon}q_{1}$ $\displaystyle=q_{5}+q_{3}$
$\displaystyle\partial_{1}^{\epsilon}q_{2}$ $\displaystyle=q_{5}+q_{3}$
$\displaystyle\partial_{1}^{\epsilon}q_{3}$
$\displaystyle=\partial_{1}^{\epsilon}q_{4}=\partial_{1}^{\epsilon}q_{5}=0.$

Through a process to be described in Section 5, we produce the heights
$h(q_{1})=h(q_{2})=4$ and $h(q_{3})=h(q_{4})=h(q_{5})=1$. Thus we have

$A^{\ell}=\left\\{\begin{matrix}0&\ell<1\\\ \mathbb{Z}_{2}\langle
q_{3},q_{4},q_{5}\rangle&1\leq\ell<4\\\ \mathbb{Z}_{2}\langle
q_{1},\ldots,q_{5}\rangle&4\leq\ell\end{matrix}\right.$

Notice that for $\ell<4$, $\partial_{1}^{\epsilon}$ restricts to $A^{\ell}$ as
the zero map, meaning that $H_{*}(A^{\ell},\partial_{1}^{\epsilon})\cong
A^{\ell}$ for $\ell<4$. For $\ell\geq 4$, the three degree-0 generators of
$A^{\ell}$ are joined by two generators of degree 1, and the sum of these
degree-1 generators is trivial in homology. That is,

$H_{k}(A^{\ell},\partial_{1}^{\epsilon})\cong\left\\{\begin{matrix}\mathbb{Z}_{2}^{2},&k=0\\\
\mathbb{Z}_{2},&k=1\end{matrix}\right.,\quad\text{for~{}}\ell\geq 4,$

and we may now express the persistent Legendrian contact homology as a direct
sum of interval modules:

$F^{\bullet}\operatorname{LCH}^{\epsilon}_{k}(\Lambda)\cong\left\\{\begin{matrix}\mathbb{Z}_{2}[0,\infty)\oplus\mathbb{Z}_{2}[0,\infty)\oplus\mathbb{Z}_{2}[0,4),&k=0\\\
\mathbb{Z}_{2}[4,\infty),&k=1\end{matrix}\right.$

Figure 3(b) presents the persistent LCH in barcode form. Once again we have
labeled the left endpoints of our bars with an arbitrary choice of homology
basis. The right endpoint of the finite bar is labeled with the generators
whose birth causes the change in homology.

##  4\. Proof of Theorem 1.3

Because of Theorem 2.3, we know that any Legendrian isotopy of Legendrian
knots can be realized in the Lagrangian projection as some sequence of
Legendrian Reidemeister moves and planar isotopies. We wish to show that the
persistent Legendrian contact homology barcode is an invariant of a Lagrangian
projection diagram of a Legendrian knot. It is for this reason that we
consider the Lagrangian projection of a knot modulo planar isotopies, and we
study how the barcode changes after a single Legendrian Reidemeister move is
applied to the diagram.

In this section we first consider those tame isomorphisms on the Chekanov-
Eliashberg DGA which are _semimonotonic_ — to be defined presently — and argue
that semimonotonic isomorphisms will induce a $2\delta$-interleaving on the
persistent LCH. See Proposition 4.3. We then argue that the tame isomorphism
induced by a single Reidemeister move will be semimonotonic. This work is done
in Sections 4.2 through 4.5.

Henceforth we will use the word _bigon_ to mean a bounded (interior) region of
a Lagrangian knot diagram that meets exactly two crossings. An example is
drawn in Figure 5b.i, below. The naming conventions for the Legendrian
Reidemeister moves are shown in Figure 1.

###  4.1. The Homological algebra of Reidemeister isotopies

Throughout this section we consider a filtered Chekanov-Eliashberg DGA
$\mathcal{A}_{\Lambda}$ generated over $\mathbb{Z}_{2}$ by
$q_{1},\ldots,q_{n}$. We will use the word _height_ to refer to the filtration
of $\mathcal{A}_{\Lambda}$, and remind the reader that a Legendrian isotopy of
the underlying knot $\Lambda$ will induce a _stable tame isomorphism_. See
Subsection 2.2.2.

######  Definition 6.

An elementary automorphism $\bar{\phi}:q_{i}\mapsto q_{i}+u$ is called
_semimonotonic_ if $u$ is a word in generators of height less than $q_{i}$. A
tame isomorphism is semimonotonic if the elementary automorphisms which
witness its tameness may be taken to be semimonotonic.

Recall from Subsection 2.2.3 that the computation of linearized LCH requires
conjugating the differential $\partial_{\Lambda}$ by a certain automorphism of
$\mathcal{A}_{\Lambda}$. Any tame isomorphism
$\bar{\phi}:\mathcal{A}_{\Lambda}\rightarrow\mathcal{A}_{\Lambda}$ analogously
induces a map

$\bar{\phi}_{1}^{\epsilon}:(A_{\Lambda},\partial_{1}^{\epsilon})\rightarrow(A_{\Lambda},\bar{\phi}_{1}^{\epsilon}\circ\partial_{1}^{\epsilon}\circ(\bar{\phi}_{1}^{\epsilon})^{-1}).$

on the linearized DGA $(A_{\Lambda},\partial_{1}^{\epsilon})$ with respect to
an augmentation $\epsilon\colon\mathcal{A}_{\Lambda}\to\mathbb{Z}_{2}$. The
following lemma verifies that if $\bar{\phi}$ is semimonotonic, then we in
fact have an induced map on persistence chain complexes.

######  Lemma 4.1.

Let $\bar{\phi}:\mathcal{A}_{\Lambda}\rightarrow\mathcal{A}_{\Lambda}$ be a
tame isomorphism. If $\bar{\phi}$ is semimonotonic, then
$(A_{\Lambda}^{\bullet},\bar{\phi}_{1}^{\epsilon}\circ\partial_{1}^{\epsilon}\circ(\bar{\phi}_{1}^{\epsilon})^{-1})$
is a well-defined persistence module of chain complexes and there is an
induced isomorphism of persistence modules

$\phi_{1}^{\epsilon,\bullet}:(A_{\Lambda}^{\bullet},\partial_{1}^{\epsilon})\rightarrow(A_{\Lambda}^{\bullet},\bar{\phi}_{1}^{\epsilon}\circ\partial_{1}^{\epsilon}\circ(\bar{\phi}_{1}^{\epsilon})^{-1}).$

###### Proof.

First, consider the case when $\phi$ is a semimonotonic elementary
automorphism with $\phi(q)=q+q_{i_{1}}\cdots q_{i_{k}}$. Then

$\bar{\phi}_{1}^{\epsilon}(q)=q+\sum_{\ell=1}^{k}\epsilon(q_{i_{1}})\cdots\widehat{\epsilon(q_{i_{\ell}})}\cdots\epsilon(q_{i_{k}})q_{i_{\ell}}$

and, by definition,

$h\left(q+\sum_{\ell=1}^{k}\epsilon(q_{i_{1}})\cdots\widehat{\epsilon(q_{i_{\ell}})}\cdots\epsilon(q_{i_{k}})q_{i_{\ell}}\right)=\max_{\ell}(h(q),h(\epsilon(q_{i_{1}})\cdots\widehat{\epsilon(q_{i_{\ell}})}\cdots\epsilon(q_{i_{k}})q_{i_{\ell}}))=h(q),$

so that $\bar{\phi}_{1}^{\epsilon}$ is height-preserving. Thus
$\bar{\phi}_{1}^{\epsilon}\circ\partial_{1}^{\epsilon}\circ(\bar{\phi}_{1}^{\epsilon})^{-1}$
is height-decreasing so
$(A_{\Lambda}^{\bullet},\bar{\phi}_{1}^{\epsilon}\circ\partial_{1}^{\epsilon}\circ(\bar{\phi}_{1}^{\epsilon})^{-1})$
is a well-defined chain complex. Clearly, $\bar{\phi}_{1}^{\epsilon}$ induces
a persistence isomorphism since
$(\bar{\phi}_{1}^{\epsilon})^{-1}=\bar{\phi}_{1}^{\epsilon}$ over
$\mathbb{Z}_{2}$.

If $\bar{\phi}$ is an arbitrary semimonotonic tame isomorphism, then
$\bar{\phi}_{1}^{\epsilon}$ is a composition of height-preserving elementary
automorphisms by the previous paragraph. For the exact same reasons as in the
elementary automorphism case, we see that
$(A_{\Lambda}^{\bullet},\bar{\phi}_{1}^{\epsilon}\circ\partial_{1}^{\epsilon}\circ(\bar{\phi}_{1}^{\epsilon})^{-1})$
is well-defined and $\bar{\phi}_{1}^{\epsilon}$ induces the desired
persistence isomorphism. ∎

We now see that a composition of semimonotonic elementary automorphisms
induces an isomorphism of persistent chain complexes. But an isotopy of
Legendrian knots will induce a tame isomorphism — obtained by composing some
number of elementary automorphisms with the isomorphism $q\mapsto q^{\prime}$
— and thus we need a criterion to ensure that the isomorphism $q\mapsto
q^{\prime}$ induces a $2\delta$-interleaving of persistent chain complexes.
The following lemma provides such a criterion.

######  Lemma 4.2.

Let $(\mathcal{A}_{\Lambda},\partial_{\Lambda})$ and
$(\mathcal{A}^{\prime}_{\Lambda},\partial^{\prime}_{\Lambda})$ be filtered
Chekanov-Eliashberg DGAs, with the former generated by $q_{1},\ldots,q_{n}$
and the latter generated by $q_{1}^{\prime},\ldots,q_{n}^{\prime}$. Let
$\sigma\colon(\mathcal{A}_{\Lambda},\partial_{\Lambda})\to(\mathcal{A}^{\prime}_{\Lambda},\partial^{\prime}_{\Lambda})$
be the DGA isomorphism defined by $\sigma(q_{i})=q_{i}^{\prime}$. If

$\displaystyle\delta\geq\min\\{|h^{\prime}(\sigma(q))-h(q)|\,\text{\emph{ such
that }}\,q\in A_{\Lambda}\\},$

where $h$ and $h^{\prime}$ denote the height filtrations in
$\mathcal{A}_{\Lambda}$ and $\mathcal{A}^{\prime}_{\Lambda}$, respectively,
then $\sigma$ induces a $2\delta$-interleaving of the persistent linearized
contact chain complexes $(A_{\Lambda}^{\bullet},\partial_{1}^{\epsilon})$ and
$((A^{\prime}_{\Lambda})^{\bullet},(\partial^{\prime})_{1}^{\epsilon})$.

###### Proof.

For any $t\in\mathbb{R}$ we define

$\varphi^{t}\colon(A_{\Lambda}^{t},\partial_{1}^{\epsilon})\to((A^{\prime}_{\Lambda})^{t+\delta},(\partial^{\prime})_{1}^{\epsilon})$

by $\varphi^{t}(q):=v_{h^{\prime}(\sigma(q))}^{t+\delta}(\sigma(q))$, for any
$q\in A_{\Lambda}^{t}$. Here
$v_{r}^{s}\colon((A^{\prime}_{\Lambda})^{r},(\partial^{\prime})_{1}^{\epsilon})\to((A^{\prime}_{\Lambda})^{s},(\partial^{\prime})_{1}^{\epsilon})$
is the linear transfer map, for any $r\leq s$. Our assumption on the size of
$\delta$ ensures that $h^{\prime}(\sigma(q))\leq h(q)+\delta\leq t+\delta$,
and thus $\varphi^{t}(q)$ is well-defined. We similarly define

$\psi^{t}\colon((A^{\prime}_{\Lambda})^{t},(\partial^{\prime})_{1}^{\epsilon})\to(A_{\Lambda}^{t+\delta},\partial_{1}^{\epsilon})$

by the formula
$\psi^{t}(q^{\prime}):=u_{h(\sigma^{-1}(q^{\prime}))}^{t+\delta}(\sigma^{-1}(q^{\prime}))$,
with $u_{r}^{s}$ denoting a transfer map on $A_{\Lambda}^{\bullet}$. Now for
any $q\in A_{\Lambda}^{t}$ we see that

$(\psi^{t+\delta}\circ\varphi^{t})(q)=\psi^{t+\delta}(v_{h^{\prime}(\sigma(q))}^{t+\delta}(\sigma(q)))=u_{h(q)}^{t+2\delta}(q)=\mathbb{I}_{A_{\Lambda}}^{2\delta}(q).$

We similarly see that
$\varphi^{t+\delta}\circ\psi^{t}=\mathbb{I}_{A^{\prime}_{\Lambda}}^{2\delta}$.
We have exhibited a $2\delta$-interleaving between the persistence modules
$A_{\Lambda}^{\bullet}$ and $(A^{\prime}_{\Lambda})^{\bullet}$, as desired. ∎

The previous two lemmas provide criteria to ensure that a tame isomorphism
induces a $2\delta$-interleaving of chain complexes. Because homology is
functorial, these criteria are sufficient to ensure a $2\delta$-interleaving
of the resulting homologies
$F^{\bullet}\operatorname{LCH}^{\epsilon}_{\ast}(\Lambda)$ and
$F^{\bullet}\operatorname{LCH}^{\epsilon}_{\ast}(\Lambda^{\prime})$. We
summarize this fact as follows.

######  Proposition 4.3.

Let
$\bar{\phi}\colon(\mathcal{A}_{\Lambda},\partial_{\Lambda})\to(\mathcal{A}_{\Lambda},\partial_{\Lambda})$
be a composition of semimonotonic elementary automorphisms, and let
$\sigma\colon(\mathcal{A}_{\Lambda},\partial_{\Lambda})\to(\mathcal{A}^{\prime}_{\Lambda},\partial^{\prime}_{\Lambda})$
be the DGA isomorphism identified in Lemma 4.2, satisfying the hypotheses
given there. Then $\phi=\sigma\circ\bar{\phi}$ induces a
$2\delta$-interleaving

$(\phi_{1}^{\epsilon,\bullet})_{*}\colon
F^{\bullet}\operatorname{LCH}_{\ast}^{\epsilon}(\Lambda)\to
F^{\bullet+\delta}\operatorname{LCH}_{\ast}^{\epsilon^{\prime}}(\Lambda^{\prime}).$

Finally, we observe that a filtered Chekanov-Eliashberg DGA may be stabilized
in a manner which preserves its homology, up to $2\delta$-interleaving.

######  Proposition 4.4.

Let $(\mathcal{A}_{\Lambda},\partial_{\Lambda})$ be a filtered Chekanov-
Eliashberg DGA, and let $(S_{k}\mathcal{A}_{\Lambda},\partial_{\Lambda})$ be
its grading-$k$ stabilization, with additional generators denoted $e_{k}$ and
$e_{k-1}$. If the filtration of $\mathcal{A}_{\Lambda}$ is extended to
$S_{k}\mathcal{A}_{\Lambda}$ in such a way that
$0<h(e_{k})-h(e_{k-1})<2\delta$, then the persistent linearized contact
homologies of $(\mathcal{A}_{\Lambda},\partial_{\Lambda})$ and
$(S_{k}\mathcal{A}_{\Lambda},\partial_{\Lambda})$ are $2\delta$-interleaved.

###### Proof.

For any $t\in\mathbb{R}$ we define homomorphisms

$\iota^{t}\colon(A_{\Lambda}^{t},\partial_{1}^{\epsilon})\to(S_{k}A_{\Lambda}^{t+\delta},\partial_{1}^{\epsilon})$

and

$r^{t}\colon(S_{k}A_{\Lambda}^{t},\partial_{1}^{\epsilon})\to(A_{\Lambda}^{t+\delta},\partial_{1}^{\epsilon}).$

The former is the natural inclusion operation followed by the transfer map
$u_{t}^{t+\delta}$, while the latter is defined by

$r^{t}(q):=\left\\{\begin{matrix}u_{t}^{t+\delta}(q),&q\neq e_{k},e_{k-1}\\\
u_{t}^{t+\delta}(0),&q=e_{k},e_{k-1}\end{matrix}\right..$

Notice that $r^{t+\delta}\circ\iota^{t}=\mathbb{I}_{A_{\Lambda}}^{2\delta}$,
so the induced map on homology is $\mathbb{I}_{H_{*}(A_{\Lambda})}^{2\delta}$.
On the other hand, $\iota^{t+\delta}\circ r^{t}$ does not agree with
$\mathbb{I}_{S_{k}A_{\Lambda}}^{2\delta}$, since, for instance,
$(\iota^{t+\delta}\circ r^{t})(e_{k-1})=0\neq u_{t}^{t+2\delta}(e_{k-1})$.
However, $e_{k-1}=\partial e_{k}$ vanishes at the level of homology, and thus
$\iota^{t+\delta}_{*}\circ
r^{t}_{*}=\mathbb{I}_{H_{*}(S_{k}A_{\Lambda})}^{2\delta}$. We conclude that
$\iota^{t}_{*}$ and $r^{t}_{*}$ give a $2\delta$-interleaving of
$H_{*}(A_{\Lambda})$ and $H_{*}(S_{k}A_{\Lambda})$. ∎

Propositions 4.3 and 4.4 give our strategy for proving Theorem 1.3. Namely,
any Legendrian isotopy will induce a stable tame isomorphism between Chekanov-
Eliashberg DGAs — that is, a tame isomorphism between stabilizations of the
Chekanov-Eliashberg DGAs. We will prove Theorem 1.3 by demonstrating that
there exist height assignments before and after the isotopy such that:

  1. (1)

any generator which survives the isotopy sees its height change by at most
$\delta$;

  2. (2)

any two cancelling generators which are created or destroyed by the isotopy
have heights that are at most $2\delta$ apart;

  3. (3)

the induced tame isomorphism of DGAs is semimonotonic.

The first two conditions ensure that the hypotheses of Lemma 4.2 and
Proposition 4.4 are satisfied, while the final condition allows us to apply
Lemma 4.1. In the ensuing four subsections, we apply this strategy to the
Legendrian isotopies depicted in Figure 1.

###  4.2. The case of a Reidemeister II move

Consider an arbitrary Lagrangian diagram of a Legendrian knot and consider any
possible Reidemeister II move on that diagram. Denote the Lagrangian diagrams
before and after the move as $\Pi(\Lambda)$ and $\Pi(\Lambda^{\prime})$, with
$\Pi(\Lambda^{\prime})$ having two crossings which do not appear in
$\Pi(\Lambda)$. Following Figure 5a.iii and Proposition 4.3, we refer to these
crossings as $a$ and $b$. From Remark 2.8 and Theorem 4.1 of [Kál05], we know
that the heights of $a$ and $b$ can be made arbitrarily close to one another
relative to the heights of the surrounding crossings. Here the ‘surrounding
crossings’ are the vertices of the area patches which meet $a$ or $b$ and
[Kál05, Theorem 4.1] allows us to ensure that $0<h(a)-h(b)<\delta$, for some
$\delta>0$ much smaller than the height of any surrounding crossing.

We first examine how the area inequalities associated to
$\Pi(\Lambda^{\prime})$ compare to those of $\Pi(\Lambda)$. Inequalities
associated to area patches not depicted in Figure 5(a) will persist as we move
from $\Pi(\Lambda)$ to $\Pi(\Lambda^{\prime})$. The inequalities associated to
the top- and bottommost area patches of Figure 5c.iii will differ from those
of the top- and bottommost area patches of Figure 5c.i by the addition or
subtraction of $h(a)-h(b)$. Similarly, the two middle area patches of
$\Pi(\Lambda^{\prime})$ will give area inequalities whose sum differs from
that of the middle area patch in $\Pi(\Lambda)$ by $h(a)-h(b)$.

Thus, examining the area patches in $\Pi(\Lambda)$ and
$\Pi(\Lambda^{\prime})$, we see that any inequality given by an area patch in
$\Pi(\Lambda)$ is a sum of inequalities from area patches in
$\Pi(\Lambda^{\prime})$, plus possibly the addition or the subtraction of
$h(a)-h(b)$, which we assumed to be arbitrarily small.

Thus any valid height assignment for $\Pi(\Lambda^{\prime})$ with
$h(a)<h(b)+\delta$ induces a height assignment for $\Pi(\Lambda)$, up to
shifting the heights of surrounding crossings by $\delta$ to compensate.
Because we have assumed $\delta>0$ to be significantly smaller than each of
these surrounding heights, this shift presents no problem.

We now describe how the barcode changes through the entire move. First there
is a stable tame isomorphism on the level of DGAs. Then there is a
reassignment of heights. From [Che02] we know that the stable tame isomorphism
consists of a stabilization followed by a composition of elementary
automorphisms, each of which has the form $q\mapsto q+\sum w_{i}$, where each
$w_{i}$ is a word in generators of height strictly smaller than $h(q)$. We are
identifying this map as a semimonotonic automorphism, so that Lemma 4.1
applies. Our above discussion of height assignments ensures that the height
reassignment will induce a shift of the heights of surrounding crossings by at
most $h(a)-h(b)<\delta$. Thus by Lemma 4.2 and Proposition 4.3, this induces a
$2\delta$-interleaving of the barcodes.

###  4.3. The case of an inverse Reidemeister II move

Recall that we wish to apply Proposition 4.3 to the case at hand. We consider
this move separately from the standard Reidemeister II move since it requires
a separate argument regarding the existence of height assignments before and
after the move which allow us to apply Lemma 4.2. However, the argument
regarding the induced stable tame isomorphism follows directly from Section
4.2. For any inverse Reidemeister II move, the standard Reidemeister move II
which undoes it gives a stable tame isomorphism $\phi$ that induces a
$2\delta$-interleaving. Thus the stable tame isomorphism for the inverse
Reidemeister move II $\phi^{-1}$ also induces a $2\delta$-interleaving. The
inverse of a Legendrian Reidemeister II move will remove a bigon from the knot
diagram, as depicted in Figure 5b.ii.

We cannot simply run the preceding argument in reverse; so consider an
arbitrary Lagrangian diagram of a Legendrian knot where the inverse of a
Legendrian Reidemeister II can be performed. Label the crossings of the
(aforementioned, pictured) bigon $a$ and $b$. The argument from Section 4.2
that the heights $h(a)$ and $h(b)$ can be made arbitrarily close to each other
still applies, so we can assume that $h(b)-h(a)<\delta$ for some small
$\delta$. Since the inverse Reidemeister move is possible, by [Kál05, Theorem
4.1] we know that the area patch above the bigon $R$ containing the top arc
$ab$ is positive, and its area can be assumed to be greater than $\delta$. See
Figure 5b.i for an illustration of the bigon $R$. This area, and the area
patch below $R$ containing the bottom arc $ab$, will only change by
$\pm|h(a)-h(b)|\leq\delta$. Consequently any area inequality involving these
generators will still be satisfied.

Additionally, that fact that a move II inverse can be done means there exists
a height assignment satisfying the positivity of the area patches $R_{a}$ and
$R_{b}$, while also keeping $a$ and $b$ within $\delta$ of each other.

Starting with such a height assignment for Figure 5b.i we note that deleting
$a$ and $b$ and keeping all other heights the same gives a valid height
assignment for Figure 5b.iii, since the area upper and lower area regions
change by at most $\pm|h(a)-h(b)|\leq\delta$, and although we subtract $a$
from the inequality on the left, we add the remaining generators from the
right area patch, which must add up to at least more than $h(a)-\delta$,
meaning the area inequality is still satisfied.

Thus there exists a height assignment before and after the move such that the
height of all generators apart from $a$ and $b$ remain unchanged. Hence, Lemma
4.2 and Proposition 4.3 imply a $2\delta$-interleaving of the barcodes that
one computes before and after the Reidemeister move.

a.i

a.ii

a.iii

(a) Reidemeister II
A very narrow ‘finger’ is created in the lower strand, which crosses
underneath the upper strand. The area of the bigon with endpoints $a$ and $b$,
as depicted in Figure 5a.iii, is arbitrarily small.

b.i

b.ii

b.iii

(b) Reidemeister II-1
The understrand is moved to position 5b.ii before performing the Reidemeister
move. Once $h(b)-h(a)<\delta$, the Reidemeister move is performed ending in
position 5b.iii.

c.i

c.ii

c.iii

(c) Reidemeister IIIb
The strand connecting the crossings $c$ and $a$ in Figure 5c.i is replaced
with the new arc in Figure 5c.iii, which is sufficiently close to the old
sides $cb$ and $ba$ so that the new heights $h(a^{\prime})$ and
$h(c^{\prime})$ are arbitrarily close to $h(c)+h(b)$ and $h(c)$, respectively.

Figure 5. Reidemeister moves II, II-1, and IIIb

###  4.4. The case of a Reidemeister IIIb move

When a Reidemeister move IIIb is possible, there is a Legendrian isotopy
realizing the planar isotopy from Figure 5c.i to Figure 5c.ii by [Kál05]. Such
an isotopy allows the area of the triangle $R$ to be made arbitrarily small;
as a result $h(a)$ can be made arbitrarily close to $h(b)+h(c)$ while leaving
$h(b)$ and $h(c)$ unchanged. For this reason, we assume that $h(a)>h(b)$ and
$h(a)>h(c)$. When performing the Reidemeister move to go from Figure 5c.ii to
Figure 5c.iii, we may satisfy the following three conditions.

  * •

The height $h(c^{\prime})$ can be made arbitrarily close to $h(c)$ by Green’s
theorem,

  * •

$h(b^{\prime})=h(b)$, and

  * •

$h(a^{\prime})$ can be made arbitrarily close to $h(b)+h(c)$.

By [Kál05] we can take the area of the triangle before and after the move,
respectively given by $h(b)+h(c)-h(a)$ and
$h(b^{\prime})+h(c^{\prime})-h(a^{\prime})$, to be less than $\delta/2<0$ and
similarly take $|h(c)-h(c^{\prime})|<\delta/2$. Thus we have

$-\delta/2<h(c^{\prime})-h(c)-(h(a)-h(a^{\prime}))<\delta/2.$

Hence $|h(a^{\prime})-h(a)|<\delta$. Thus we have a height assignment before
and after the Reidemeister move IIIb such that the height of each generator
changes by less than $\delta$ So by Lemma 4.2 it induces a
$2\delta$-interleaving.

Note that the induced isomorphism of the Reidemeister move IIIb isotopy from
$A_{\Lambda}$ to $A_{\Lambda^{\prime}}$ is given by $a\mapsto
a^{\prime}+\epsilon(c)b^{\prime}+\epsilon(b)c^{\prime}$ and $q\mapsto
q^{\prime}$ for $q\not=a$. But
$h(a^{\prime}+\epsilon(c)b^{\prime}+\epsilon(b)c^{\prime})=h(a^{\prime})$.
Thus the isomorphism is a single semimonotonic automorphism and by Lemma 4.1
this gives a $2\delta$-interleaving. Therefore separating the Reidemeister
move into two parts: a DGA isomorphism and a height reassignment, each give a
$2\delta$-interleaving of chain complexes and so Proposition 4.3 we have a
$2\delta$-interleaving of the barcodes.

###  4.5. The case of a Reidemeister IIIa move

When a Reidemeister move IIIa is possible, the change in height assignments is
similar to move IIIb. The movement is the same as Figure 5(c), except all of
the crossing data is flipped. The induced isomorphism of the move IIIb isotopy
is simply a relabeling $q\mapsto q^{\prime}$, which is semimonotonic and thus
by Lemma 4.1 gives a $2\delta$-interleaving. Since the heights can change by
at most $\delta$, it follows from Lemma 4.2 that the height reassignment is
also $2\delta$-interleaving. Once again, by Proposition 4.3, this induces a
$2\delta$-interleaving of the barcodes that one computes before and after the
Reidemeister move.

(a) A Reidemeister move II has been performed on the Lagrangian diagram from
Figure 3(a). The two additional crossings are denoted by $a$ and $b$.

012$2+\delta$4$h$$q_{5}$$q_{4}$$q_{4}+a$$q_{1}+q_{2}$$H_{1}$$q_{3}+q_{5}$$H_{0}$
(b) The barcode for the diagram in Fugure 6(a). It also has one infinite bar
in $H_{1}$, two infinite bars in $H_{0}$, and one finite bar in $H_{0}$ which
is created at height 1 and destroyed at height 4, compared to Figure 3(b); the
only difference is an additional finite bar of length $\delta$.

Figure 6. A diagram and its barcode for the Legendrian trefoil after a
Reidemeister move II has been performed

###  4.6. Returning to the trefoil example

Previously in Section 3.3 we computed the barcode of the standard Lagrangian
projection of the trefoil. Here we compute the barcode after a Redemeister
move II had been performed. Figure 3(b) is the barcode before this move, and
Figure 6(b) is the barcode after this move. Note that the only difference is
that a bar of length $\delta$ has been added to the post-move barcode. This
means that our choice of height assignment led to $2\delta$ interleaved
barcodes. The magnitude of the parameter $\delta$ is fixed by the area of the
bigon in Figure 6(a) that has corners at $a$ and $b$.

##  5\. The flooding algorithm

When compared to its unfiltered analogue, persistent Legendrian contact
homology has two primary shortcomings:

  1. (1)

Persistent LCH is not a Legendrian isotopy invariant.

  2. (2)

The computation of persistent LCH requires more data than is contained in a
Lagrangian diagram.

In particular, while a Lagrangian projection diagram allows us to compute the
grading of each generator of the Chekanov-Eliashberg DGA, such a diagram does
not encode the DGA’s filtration111The distinction between Lagrangian
projection diagrams and Lagrangian projections is important here, as one _can_
recover the filtration from $\Pi(\Lambda)$.. Of course these shortcomings are
related — while a Lagrangian projection diagram does not determine a unique
Legendrian knot, it determines a unique Legendrian isotopy type.

The effect of Legendrian isotopy on persistent LCH is the focus of Theorem 1.3
above. In order to compute persistent LCH directly from a Lagrangian diagram,
we might hope to find a canonical assignment of heights to the crossings of
the diagram. For a Lagrangian diagram with $n$ crossings, the set of
permissible height assignments $(h_{1},\ldots,h_{n})$ forms an open, unbounded
cone in the first orthant of $\mathbb{R}^{n}$, with vertex at the origin. By
declaring a minimum allowable height, as well as a minimum allowable area for
the area patches of the diagram, we obtain a closed, unbounded cone disjoint
from the coordinate hyperplanes of $\mathbb{R}^{n}$. A natural candidate for a
Lagrangian diagram’s canonical height assignment is given by the (unique)
vertex of this cone which is nearest the origin, and we anticipate future work
investigating the nature of persistent LCH using this height assignment.

In this section, however, we take up the more modest goal of producing _some_
— not necessarily canonical — height assignment via a short computation. While
a variety of efficient algorithms exist to solve the linear programming
problem described in the above paragraph, these can be labor-intensive to
carry out by hand. We describe in this section our _flooding algorithm_ ,
which can be quickly applied by a human. The algorithm does not succeed in
assigning heights to _every_ Lagrangian diagram, but it has the advantage of
halting for all Lagrangian diagrams. Moreover, the collection of diagrams for
which the algorithm assigns heights is quite large. For instance, Ng showed in
[Ng03] how to start with a front projection $F(\Lambda)$ and produce a
Lagrangian diagram for a knot Legendrian isotopic to $\Lambda$. The algorithm
described in this section successfully assigns heights to any Lagrangian
diagram obtained by applying Ng’s resolution. Thus, when one wishes to produce
a barcode from a Lagrangian diagram, the flooding algorithm often provides a
quick means of doing so.

We present our algorithm in subsection 5.1 and give a recursive implementation
in subsection 5.2. In subsection 5.3 we explain how to carry out the algorithm
on a Lagrangian diagram without explicitly writing down the related area
inequalities. We then use subsection 5.4 to demonstrate a choice of height
assignment once the algorithm terminates, and 5.5 to show that the algorithm
succeeds in assigning heights for any Lagrangian diagram obtained via Ng’s
resolution. Finally, in subsection 5.6 we give an example of a Lagrangian
diagram for which the flooding algorithm fails to assign heights.

###  5.1. The algorithm

Recall from subsection 2.2.4 that a Lagrangian diagram determines a collection
of area inequalities; each inequality ensures that a patch in the diagram has
positive area. Furthermore, we use the Legendrian condition, (1), to convert
the condition of positivity of area into an inequality on the sums of heights
of each crossing involved in an area patch. Suppose that the generators of
$\mathcal{A}_{\Lambda}$ are labeled $q_{1},q_{2},\cdots,q_{n}$, and that

$\mathcal{R}=\\{f_{1},f_{2},\ldots,f_{m}\\}$

is the resulting set of height inequalities for the given Lagrangian diagram,
with each $f_{i}$ taking the form

$\alpha_{i,1}h(q_{1})+\alpha_{i,2}h(q_{2})+\cdots+\alpha_{i,n}h(q_{n})>0,$

for some integers $\alpha_{i,j}\in\\{-2,-1,0,1,2\\}$.

Note that if some particular $h(q_{j})$ has non-negative coefficient
$\alpha_{i,j}$ in _every_ inequality $f_{i}$, then $h(q_{i})$ may be taken to
be arbitrarily large, and thus each inequality in which $h(q_{i})$ has a
positive coefficient may be solved — regardless of the heights assigned to
other generators. This observation allows us to ignore inequalities in which
$h(q_{i})$ has a positive coefficient, leading us to consider a new, smaller
set of inequalities in which $h(q_{i})$ does not appear. After solving this
smaller set of inequalities, we may assign $h(q_{i})$ a value as large as is
necessary to solve the previously ignored inequalities.

In fact, we will apply the above reduction simultaneously to all heights which
appear with only non-negative coefficients, considering the corresponding
generators to be our first tier of generators. We then repeat the process,
defining a second tier of generators which appear with only non-negative
coefficients in our smaller collection of inequalities. The algorithm
terminates if we can successfully assign every generator $q_{j}$ to a tier, at
which time particular height values may be chosen.

###  5.2. Recursive implementation

The flooding algorithm is implemented in a recursive manner as follows. We
start with the collection $\mathcal{R^{\prime}}=\mathcal{R}$ of inequalities,
defined above.

  1. (1)

Let $k=1$.

  2. (2)

Set $T_{k}=\left\\{q_{j}\mid\alpha_{i,j}\geq 0\text{ in every inequality
}f_{i}\in\mathcal{R^{\prime}}\right\\}$.

  3. (3)

If $T_{k}=\emptyset$, stop; and in this case the algorithm is said to fail.
Otherwise, for each $q_{j}\in T_{k}$, remove $f_{i}$ from
$\mathcal{R^{\prime}}$, for every $i$ where $\alpha_{i,j}>0$.

  4. (4)

If $\mathcal{R}^{\prime}\neq\emptyset$, increase $k$ by $1$ and return to Step
2. If $\mathcal{R}^{\prime}=\emptyset$, define

$T_{k+1}=\\{q_{i}\mid q_{i}\notin T_{\ell},\,\text{for
all}\,\ell=1,\ldots,k\\}$

and let $M=k+1$ be the total number of tiers.

Because the set of generators $\\{q_{1},\ldots,q_{n}\\}$ and the set of
inequalities $\mathcal{R}$ are both finite, this recursive process will
terminate. Provided the algorithm does not fail, the result will be a
partition of the set of generators into $M$ disjoint ‘tiers’ that we called
$T_{k}$:

$T_{1}\sqcup\cdots\sqcup T_{M}=\\{q_{1},\ldots,q_{n}\\}.$

###  5.3. Diagrammatic implementation

The algorithm above could be applied to any collection of linear inequalities,
so let us describe the algorithm in terms of Lagrangian diagrams, motivating
its name. Each inequality in our set $\mathcal{R}$ corresponds to an area
patch of our Lagrangian diagram, and our algorithm begins with none of these
area patches having been ‘flooded’ — though we declare the unbounded exterior
region of our Lagrangian diagram to be flooded.

Now recall from Section 2 that each crossing $q_{i}$ has two quadrants with
positive Reeb sign and two quadrants with negative Reeb sign. At step $k$ of
our algorithm, we add to tier $T_{k}$ all those crossings whose negative
quadrants both lie in flooded regions, and we then flood all regions for which
these crossings serve as vertices. The flooding step corresponds to removing
the inequalities from our list, and the algorithm will succeed in producing a
height assignment if all area patches are eventually flooded. See Figure 7 for
an example of this diagrammatic implementation where the algorithm halts
without assigning all crossings to tiers.

###  5.4. Obtaining a valid height assignment from the flooding algorithm

As written, the algorithm above does not in fact assign heights to the
crossings of our Lagrangian diagram, but only sorts them into tiers. The point
of these tiers is that, once heights have been assigned to all crossings in
tiers $T_{k+1},\ldots,T_{M}$, we may satisfy the inequalities of $\mathcal{R}$
by assigning sufficiently large heights to the crossings in $T_{k}$. For the
sake of completeness, we now give an explicit assignment of heights in this
manner.

Let $T_{1},\ldots,T_{M}$ be the tiers produced by the flooding algorithm; we
denote by $|T_{i}|$ the cardinality of $T_{i}$. We define integers
$h_{1},\ldots,h_{M}$ recursively by declaring $h_{M}=1$ and then

(3)
$h_{k}=1+\sum_{i=k+1}^{M}2h_{i}\,|T_{i}|,\quad\text{for~{}}k=M-1,M-2,\ldots,1.$

Finally, we assign height $h_{k}$ to each crossing in tier $T_{k}$, for $1\leq
k\leq M$.

######  Proposition 5.1.

Declaring each crossing in tier $T_{k}$ to have height $h_{k}$, for $1\leq
k\leq M$, gives a valid height assignment for the Lagrangian diagram.

###### Proof.

Consider a crossing $q_{j}$ in the tier $T_{k}$. Because the crossings in
tiers $T_{1},\ldots,T_{k-1}$ will see their heights assigned after that of
$q_{j}$, we need only concern ourselves with inequalities involving crossings
from tiers $T_{k+1},\ldots,T_{M}$ when assigning the height $h(q_{j})$. Our
construction of the tiers ensures that $h(q_{j})$ appears with non-negative
coefficient in any such inequality. Indeed, consider an inequality

$\alpha_{i,1}h(q_{1})+\alpha_{i,2}h(q_{2})+\cdots+\alpha_{i,n}h(q_{n})>0$

with the property that $\alpha_{i,j}>0$ and $\alpha_{i,\ell}=0$ whenever
$q_{\ell}$ is a crossing in $T_{1}\sqcup\cdots\sqcup T_{k-1}$. The number of
negative terms in the left hand side of this inequality is bounded above by
$|T_{k+1}|+\cdots+|T_{M}|$, and each coefficient $\alpha_{i,\ell}$ satisfies
$|\alpha_{i,\ell}|\leq 2$. The assignment made in Equation 3 then ensures that
this inequality is satisfied. In this manner we see that all inequalities
involving the crossings of $T_{k}\sqcup\cdots\sqcup T_{M}$ are satisfied and,
continuing inductively, that all inequalities of $\mathcal{R}$ are satisfied
by our height assignment. ∎

######  Remark 5.2.

If any of our tiers contain more than one crossing, this height assignment
will be resonant, in that distinct generators of our DGA will have the same
filtration value. This can be easily overcome with a small perturbation.

###  5.5. Proof of termination

As discussed, the flooding algorithm does not always succeed in assigning
heights to the crossings of a Lagrangian diagram. In particular, if
$\mathcal{R}$ is a collection of inequalities in the variables
$h(q_{1}),\ldots,h(q_{k})$ in which each variable appears at least once with a
negative coefficient, then we are unable to extract a first tier of ‘free
variables’ from $\mathcal{R}$. However, we now show that any Lagrangian
diagram obtained by applying Ng’s resolution to a front diagram will yield a
set of area inequalities for which our algorithm terminates.

Let us fix a front diagram on which Ng’s resolution will be applied and denote
by $\mathcal{R}$ the collection of area inequalities corresponding to the
resulting Lagrangian diagram. In the front diagram, we enumerate the right
cusps and crossings right-to-left as $c_{1},\ldots,c_{n}$, so that

$x(c_{1})>x(c_{2})>\cdots>x(c_{n}).$

Obtaining distinct $x$-coordinates for these generators may require a small
planar isotopy, but this can be done without modifying $\mathcal{R}$.

We now show by strong induction on $k$ that each generator $c_{k}$ will be
assigned by our algorithm to some tier of generators. Our base case is
$c_{1}$, which must be a right cusp outside of this diagram. Applying Ng’s
resolution turns this generator into a crossing $q_{1}$ whose negative
quadrants both lie in the region outside the Lagrangian diagram, and thus are
flooded when our algorithm commences. That is, the $h(q_{1})$ will not have
negative coefficient in any inequality of $\mathcal{R}$, and it follows that
our algorithm will assign $q_{1}$ to the first tier of crossings.

For our inductive step, we claim that if all generators $c_{i}$ are added to a
tier by our algorithm, for $1\leq i<k$, then $c_{k}$ is added to a tier by our
algorithm. Indeed, regardless of whether $c_{k}$ is a right cusp or a
crossing, Ng’s resolution will produce a crossing in the Lagrangian diagram
whose topmost and bottommost quadrants have negative Reeb signs. Moreover,
each of these quadrants will either lie outside of the Lagrangian diagram or
lie in an area patch which meets a crossing $c_{i}$, $1\leq i<k$. In either
case, the quadrant lies in an area patch which will be flooded by our
algorithm, and thus our algorithm will assign $c_{k}$ to a tier.

Inductively, we see that each generator $c_{1},\ldots,c_{n}$ is assigned by
our algorithm to a tier, and thus the algorithm terminates.

###  5.6. An example where the flooding algorithm fails

Figure 7. Example of a Lagrangian diagram for which the flooding algorithm
fails to terminate.

For Lagrangian diagrams which are not obtained via Ng’s resolution, the
flooding algorithm is not guaranteed to succeed in assigning heights to
crossings. (The algorithm still halts, however.) Indeed, the Lagrangian
diagram depicted in Figure 7 has area inequalities given by

$\displaystyle h_{1}$ $\displaystyle>0$ $\displaystyle h_{3}+h_{4}-h_{5}$
$\displaystyle>0$ $\displaystyle h_{9}+h_{7}-h_{8}$ $\displaystyle>0$
$\displaystyle h_{2}$ $\displaystyle>0$ $\displaystyle h_{5}+h_{6}-h_{4}$
$\displaystyle>0$ $\displaystyle h_{8}-h_{9}$ $\displaystyle>0$ $\displaystyle
h_{1}-h_{4}-h_{3}$ $\displaystyle>0$ $\displaystyle h_{7}-h_{6}-h_{5}$
$\displaystyle>0$ $\displaystyle h_{8}+h_{6}-h_{7}$ $\displaystyle>0$
$\displaystyle h_{2}-h_{3}+h_{5}-h_{7}-h_{9}$ $\displaystyle>0$

Here we have written $h_{i}$ for $h(q_{i})$. The flooding algorithm succeeds
in creating two tiers of crossings — $T_{1}=\\{q_{1},q_{2}\\}$ and
$T_{2}=\\{q_{3}\\}$ — after which our collection of inequalities is

$\displaystyle h_{5}+h_{6}-h_{4}$ $\displaystyle>0$ $\displaystyle
h_{8}+h_{6}-h_{7}$ $\displaystyle>0$ $\displaystyle h_{8}-h_{9}$
$\displaystyle>0$ $\displaystyle h_{7}-h_{6}-h_{5}$ $\displaystyle>0$
$\displaystyle h_{9}+h_{7}-h_{8}$ $\displaystyle>0$

Because each height appears with negative coefficient at least once, the
flooding algorithm halts without successfully assigning every crossing to a
tier.

This third stage of the flooding algorithm is depicted in Figure 7, where
shading indicates those area patches which have been flooded. Because every
crossing with unflooded quadrants has at least one _negative_ unflooded
quadrant, we cannot flood any more area patches, and the algorithm halts. We
think of the regions which share the vertices $q_{5}$ and $q_{6}$ as ‘islands’
which we are unable to flood, and any Lagrangian diagram with such a region
will cause the flooding algorithm to fail.

##  6\. Strong Morse inequalities

As the reader can well see, the computation of Legendrian contact homology
involves associating a Morse chain complex to the critical points of the
Legendrian action functional, Equation 2. In this section, we push the Morse
theory further, and develop a relationship between two polynomials in Theorem
6.1. This relationship will be called the strong Morse inequalities in the
context of persistent LCH.

The Morse-Chekanov polynomial of a Legendrian knot $\Lambda$ is defined to be
the Laurent polynomial

(4) $\displaystyle MC(z)=\sum_{k=-\infty}^{\infty}\\#\\{\text{Reeb chords of
$\Lambda$ in grading }k\\}z^{k}.$

Similarly, given an augmentation
$\epsilon:{\mathcal{A}_{\Lambda}}\rightarrow\mathbb{Z}_{2}$, the Poincaré-
Chekanov polynomial is defined as the Laurent polynomial

(5) $\displaystyle
PC^{\epsilon}(z)=\sum_{k=-\infty}^{\infty}\dim_{\mathbb{Z}_{2}}\left(\operatorname{LCH}_{k}^{\epsilon}(\Lambda)\right)z^{k}.$

######  Theorem 6.1 (Strong Morse Inequality).

Given an augmentation
$\epsilon:({\mathcal{A}_{\Lambda}},\partial_{\Lambda})\rightarrow(\mathbb{Z}_{2},0)$,
we have that

$MC(z)-PC^{\epsilon}(z)=(z+1)R(z)$

where $R(z)=\sum_{k}\\#\\{\text{Finite bars in
}F^{\bullet}\operatorname{LCH}^{\epsilon}_{k}(\Lambda)\\}z^{k}.$

###### Proof.

By the rank-nullity theorem, we have that

$\displaystyle MC(z)-PC^{\epsilon}(z)$
$\displaystyle=\sum_{k=-\infty}^{\infty}(\operatorname{null}(\partial^{\epsilon}_{1,k})+\operatorname{rank}(\partial^{\epsilon}_{1,k})-\operatorname{null}(\partial^{\epsilon}_{1,k})+\operatorname{rank}(\partial^{\epsilon}_{1,k+1}))z^{k}$
$\displaystyle=\sum_{k=-\infty}^{\infty}(\operatorname{rank}(\partial^{\epsilon}_{1,k})+\operatorname{rank}(\partial^{\epsilon}_{1,k+1}))z^{k}$
$\displaystyle=\sum_{k=-\infty}^{\infty}\operatorname{rank}(\partial^{\epsilon}_{1,k})z^{k}+\operatorname{rank}(\partial^{\epsilon}_{1,k})z^{k-1}$
$\displaystyle=\sum_{k=-\infty}^{\infty}\operatorname{rank}(\partial^{\epsilon}_{1,k})(z+1)z^{k-1}$
$\displaystyle=(z+1)\left(\sum_{k=-\infty}^{\infty}\operatorname{rank}(\partial^{\epsilon}_{1,k})z^{k-1}\right).$

Since $\partial^{\epsilon}$ is strictly height decreasing it follows that no
homology class can be born and die simultaneously. This shows that
$\operatorname{rank}(\partial^{\epsilon}_{1,k})$ is the number of finite bars
in $F^{\bullet}\operatorname{LCH}^{\epsilon}_{k-1}(\Lambda)$ proving the
claim. ∎

From this theorem, one can see that some aspects of persistence are accessible
from the standard Legendrian contact homology. However, the data coming from
the actual Reeb chord heights and the generators of the finite bars in
$F^{\bullet}\operatorname{LCH}^{\epsilon}_{\ast}(\Lambda)$ is absent in the
above result. Similar to how the normal Poincaré polynomial doesn’t recover
all the information given by (singular) homology, $R(z)$ is not enough to
recover the data given by persistent Legendrian contact homology.

######  Remark 6.2.

As $t\in\mathbb{R}$ varies, generators of $A_{\Lambda}$ are born at
$A_{\Lambda}^{t}$ generating an infinite bar of $A_{\Lambda}^{\bullet}$ in
degree $k$ and correspondingly contributing a $z^{k}$ to $MC(z)$. Some of
these infinite bars will correspond to infinite bars in
$F^{\bullet}\operatorname{LCH}^{\epsilon}_{k}(\Lambda)$ and contributing a
$z^{k}$ to $PC^{\epsilon}(z)$. Left over are infinite bars in
$A_{\Lambda}^{\bullet}$ that correspond to killing a sum of generators of
smaller height corresponding to a single finite bar in
$F^{\bullet}\operatorname{LCH}^{\epsilon}_{\ast}(\Lambda)$ and contributing a
factor of $t^{k}+t^{k-1}$ to $MC(z)-PC^{\epsilon}(z)$. Overall, the difference
of the number of generators in $A_{\Lambda}$ and the number of generators in
$\operatorname{LCH}^{\epsilon}(\Lambda)$ contributes twice the number of
finite bars in $F^{\bullet}\operatorname{LCH}^{\epsilon}(\Lambda)$, and this
agrees with Theorem 6.1 by setting $z=1$ and noticing that
$R(1)=\frac{1}{2}(MC(1)-PC^{\epsilon}(1))$.

######  Example 6.3.

We can now compute the Morse-Chekanov and the Poincaré-Chekanov polynomials
for the examples given in section 3. We find that

$MC(z)-PC^{\epsilon}(z)=0=(z+1)\cdot 0,$

in the case of the unknot, and

$MC(z)-PC^{\epsilon}(z)=z+1=(z+1)\cdot 1,$

in the case of the trefoil. The reader should compare these differences to the
finite bars found in the barcodes of Figure 2(b) and Figure 3(b),
respectively.

### Acknowledgments

This project was initiated as part of a Research Experience for Undergraduates
in the School of Mathematics at Georgia Institute of Technology. The REU was
supported by Georgia Tech’s College of Sciences and NSF grants #1745583,
#1851843, #2244427, and arranged by Igor Belegradek. The authors are grateful
to all who made this REU possible. DI was additionally supported by the NSF-
Simons Southeast Center for Mathematics and Biology (SCMB) through the grants
National Science Foundation DMS1764406 and Simons Foundation/SFARI 594594, and
would like to thank Orsola Capovilla-Searle and Ziva Myer for many helpful
conversations. WS was partially supported by NSF grant DMS-2104144 and would
like to thank Jennifer Hom for that.

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