\section{Examples}  \label{sec:example}


In this section, we apply the methods developed in Section \ref{sec:MCF} to compute induced augmentations and augmentation sets for some particular Legendrian fillings.  In Section \ref{sec:6-1}, we give examples of oriented Legendrian fillings inducing augmentations that cannot be induced by any oriented 
embedded Lagrangian fillings.  In Section \ref{sec:6-2}, we prove Theorem \ref{thm:1-4} by constructing $2n$ Legendrian fillings of the Legendrian torus knot, $T(2,2n+1)$, each with a single degree $0$ Reeb chord, and distinguished by their induced augmentation sets.


\subsection{Initial examples}  \label{sec:6-1}

Before launching into examples, we summarize the method that we use for computing induced augmentations and induced augmentation sets arising from a (compact) Legendrian filling, $\Sigma$.  For a $1$-dimensional Legendrian, $\Lambda$, with admissible transverse decomposition $\mathcal{E}_\pitchfork$, Proposition \ref{prop:bijectMCFAug} and Corollary \ref{cor:augmentationset} provide a commutative diagram of bijections and inclusions
\begin{equation}  \label{eq:}
\begin{array}{ccccc}
\mathit{MCF}_\rho(\Lambda)/{\sim} & \cong & \mathit{Aug}_\rho(\Lambda, \mathcal{E}_{||})/{\sim} & \cong & \mathit{Aug}_\rho(\widetilde{\Lambda})/{\sim} \\  
\cup & & \cup & & \cup \\
I^\mathit{MCF}_\Sigma  & \cong & I^\comp_\Sigma & \cong & I_{\widetilde{\Sigma}}
\end{array}
\end{equation} 
where, in the top row, from left to right the sets are the equivalence classes of MCFs of $\Lambda$; homotopy classes of augmentations of the cellular DGA of $(\Lambda, \mathcal{E}_{||})$; and homotopy classes of augmentations of the LCH DGA $\mathcal{A}(\widetilde{\Lambda})$ where $\widetilde{\Lambda}$ is Legendrian isotopic to $\Lambda$.  (It is the standard geometric model from Section \ref{sec:4-5}.)  The bottom row consists of the 
(equivalent) versions of the induced augmentation set in these three different settings (where $\widetilde{\Sigma}= G_{\mathit{std}}(\Sigma)$ is $\Sigma$ with its positive end extended to become a conical cobordism).  
The MCFs in $I^\mathit{MCF}_\Sigma$ are those that arise from restricting a $2$-dimensional MCF, $\mathcal{C}_\Sigma$, for $\Sigma$ to $\Lambda$.   
Thus, elements of $I^\mathit{MCF}_\Sigma$ (and so also induced augmentations in $I_{\widetilde{\Sigma}}$) may be constructed by specifying a $2$-dimensional MCF for $\Sigma$, and then observing its restriction to $\Lambda$.  
To keep track of the different MCFs of $\Lambda$ we use the further bijection from Proposition \ref{prop:Aform} with augmentations of the Ng resolution,
\[
\mathit{MCF}_\rho(\Lambda)/{\sim} \, \cong \, \mathit{Aug}(\Lambda_{res})
\]
following Section \ref{sec:Aform}.
That is, once a $2$-dimensional MCF has been restricted to the Legendrian boundary $\Lambda$, if necessary, we transform it by an equivalence into $A$-form to arrive at 
an augmentation of $\Lambda_\mathit{res}$.
 
For the rest of the section, all of the augmentations we talk about are of $\Lambda_\mathit{res}$, and we will refer to Reeb chords of $\Lambda_\mathit{res}$ as crossings and right cusps of the front projection.


\begin{example} An augmentation that can be induced by a Legendrian filling
for a knot that does not have any oriented embedded Lagrangian fillings.
\end{example}


The work \cite[Section 8.3]{PanRu1} gives an infinite family of such knots without explicitly computing the induced augmentations. One of them is  the knot $4_1$ shown in Figure \ref{fig:4_1}
where a conical Legendrian filling is constructed by a clasp move  (see Figure \ref{fig:PinchMoves} below for a discussion of local moves for constructing Legendrian cobordisms) followed by Reidemeister I moves and an unknot move; the pictures are slices of the front projection of the corresponding compact filling, $\Sigma \subset J^1([0,1]\times M)$, with $t \in [0,1]$ constant.  As shown in \cite[Section 8.3]{PanRu1}, the clasp move can be realized by a conical Legendrian cobordism with a single Reeb chord, so that a conical version of $\Sigma$ (after an appropriate Legendrian isotopy) is a disk with a single double point in  the Lagrangian projection.
Figure \ref{fig:4_1} indicates the MCF for $\Sigma$ as a  movie of handleslides.  Note that the handleslides that appear after the Reidemeister I moves are as required in Axiom \ref{ax:endpoints} (3).    
Using the correspondence between A-form  MCFs and augmentations of $\Lambda_\mathit{res}$, the induced augmentation $\e_1$ sends
the crossings $b_1$ and $b_2$ to $1$ and all other Reeb chords to $0$.
Note that a $\Z$-valued Maslov potential for $4_1$ extends (uniquely) over $\Sigma$, and the MCF for $\Sigma$ is $0$-graded.   Correspondingly, $\epsilon_1$ is the unique
$0$-graded augmentation of $\Lambda_\mathit{res}$.
As discussed in \cite[Section 8.3]{PanRu1}, the $4_1$ knot does not 
have  any oriented embedded Lagrangian fillings due to a restriction on the Thurston-Bennequin number.


\begin{figure}[!ht]
\labellist
\small
\pinlabel $b_1$ at 95 45
\pinlabel $b_2$ at 95 63 
\pinlabel $\downarrow$ at 175  65
\pinlabel $\downarrow$ at 175  -5
\pinlabel $\emptyset$ at 175 -20
\endlabellist
\includegraphics[width=3in]{Images/4-1}
\vspace{0.2in}

\caption{A conical Legendrian filling of Legendrian $4_1$ constructed by a movie. The arrow indicates the negative $t$ direction. Red vertical line segments indicates handle slides in the MCF.}
\label{fig:4_1}
\end{figure}

We note that $4_1$ does admit {\it non-orientable} embedded Lagrangian fillings.  Such a filling, constructed via applying two pinch moves (see Figure  \ref{fig:PinchMoves}) and then an unknot move, is pictured in Figure \ref{fig:4_1NonOrient}.  It induces the $1$-graded augmentation $\e_2$ that sends  $b_1, b_2, a$, and $c$ to $1$ and other Reeb chords to $0$.


\begin{question}  Can the augmentation $\e_1$ induced by the Legendrian filling from Figure \ref{fig:4_1} also be induced by a non-orientable embedded Lagrangian filling?
\end{question}
We suspect that the answer is no, but we do not know of any obstructions to such a non-orientable filling that are currently available in the literature.


\begin{figure}[!ht]
\labellist
\small
\pinlabel $a$ [t] at 18 32
\pinlabel $c$ [l] at 90 32 
\pinlabel $\emptyset$ at 268 36
\endlabellist
\includegraphics[width=3in]{Images/4-1NonOrient}
\vspace{0.2in}

\caption{A non-orientable conical Legendrian filling of Legendrian $4_1$ with embedded Lagrangian projection.  A $1$-graded MCF inducing the augmentation $\e_2$ is pictured.}
\label{fig:4_1NonOrient}
\end{figure}


\begin{example} Two augmentations for the same knot, one can be induced by an embedded Lagrangian filling and one cannot.
\end{example}


Figure \ref{fig:8_21} shows a Legendrian of knot type
 $m(8_{21})$.
  The knot has a genus one embedded Lagrangian filling constructed by doing three pinch moves (see Figure \ref{fig:PinchMoves}) and closing up the two disjoint unknots
(see the first row in Figure \ref{fig:8_21}).
The associated MCF induces the augmentation $\epsilon_1$ that sends $b_1, b_4, b_5$ to $1$ and others to $0$.
 
 \begin{figure}[!ht]
\labellist
\small
\pinlabel $b_1$ at 35 85
\pinlabel $b_2$ at 70 90
\pinlabel $b_3$ at  115 88
\pinlabel $b_4$ at 35 35
\pinlabel $b_5$ at  73 37
\pinlabel $b_6$ at 113 37

\pinlabel $b_1$ at 35 248
\pinlabel $b_2$ at 70 253
\pinlabel $b_3$ at  115 253
\pinlabel $b_4$ at 35 200
\pinlabel $b_5$ at  73 198
\pinlabel $b_6$ at 113 198

\pinlabel $\emptyset$ at 600 60
\pinlabel $\emptyset$ at 600 220
\endlabellist
\hspace{-1in}\includegraphics[width=5in]{Images/8_21}
\caption{The first row gives an embedded Lagrangian filling of $m(8_{21})$ constructed by a movie. The second row gives a conical Legendrian filling of $m(8_{21})$ through a movie. The arrow indicates negative $t$ direction.}
\label{fig:8_21}
\end{figure}

 
 One the other hand, one can construct a conical Legendrian filling through doing three clasp moves and then close up the unknot as shown in the movie in the second row of Figure \ref{fig:8_21}.
 This is a Legendrian disk with three Reeb chords. 
 The associated MCF as  shown in Figure \ref{fig:8_21} induces the augmentation
 $\e_2$ that sends $b_1,b_2,b_4$ and $b_5$ to $1$ and other Reeb chords to $0$.
 This augmentation cannot be induced by an embedded orientable Lagrangian filling since its linearized homology has Poincare polynomial
$t^{-1}+4+2t$.
 By Seidel's Isomorphism \cite{Ekh, DR}, if an augmentation was induced by an embedded orientable genus $g$ filling, using the $\Z/2$-grading on linearized homology arising from a $\Z/2$-valued Maslov potential on the filling (such a Maslov potential exists by orientability, see Remark \ref{rem:orient}), its Poincare polynomial with $\Z/2$-grading should be $t+2g$. 


\subsection{Proof of Theorem \ref{thm:1-4}} \label{sec:6-2}


The theorem provides examples of knots with an arbitrary  (finite) number of immersed fillings, all having isomorphic DGAs,  
but distinguished by induced augmentation sets.

\medskip

\noindent {\bf Theorem \ref{thm:1-4}.}
{\it For each $n \geq 1$, there exists $2n$ distinct conical Legendrian fillings, $\Sigma_1, \ldots, \Sigma_{2n}$ of the max-$\mathit{tb}$ Legendrian torus knot $T(2, 2n+1)$
such that 
\begin{itemize}
\item[(i)] the $\Sigma_i$ are all orientable with genus $n-1$ and have $\Z$-valued Maslov potentials,
\item[(ii)] each $\Sigma_i$ has a single Reeb chord of degree $0$,
\item[(iii)] and the induced augmentation sets satisfy $I_{\Sigma_i} \neq I_{\Sigma_j}$ when $i \neq j$.
\end{itemize}
}

\medskip


\begin{figure}[!ht]
\labellist
\small
\pinlabel $b_1$ at  20 98
\pinlabel $b_2$ at  38 98
\pinlabel $b_3$ at 55 98
\pinlabel $b_{2n+1}$ at 150 98
\pinlabel $(a)$ at 80 60
\pinlabel $(b)$ at  280 100
\pinlabel $(c)$ at 280 -5
\pinlabel $c_1$ at 225 145
\pinlabel $c_{2n-1}$ at 353 143
\endlabellist
\includegraphics[width=6in]{Images/torus}
\vspace{0.1in}

\caption{Part $(a)$ is a Legendrian $(2,2n+1)$ torus knot and $(b)$ is a Legendrian $(2, 2n-1)$ torus knot.  Part $(c)$ is the unknot got after doing $2n-2$ pinch moves to $T(2, 2n-1)$.}
\label{fig:torus}
\end{figure}

Note that each of the DGAs $\mathcal{A}(\Sigma_i)$ is $\Z$-graded with a single generator in degree $0$ and the differential is necessarily zero for grading reasons.
Thus, they are all isomorphic.

\begin{proof}

Consider the Legendrian torus knot $T(2, 2n+1)$ as shown in Figure \ref{fig:torus} part $(a)$. There are $2n$ ``eye shapes'' that one can do a clasp move on to get a $(2,2n-1)$ Legendrian torus knot as shown in part $(b)$.
One can then do $2n-2$ pinch moves to the $T(2,2n-1)$ knot and close the unknot in part $(c)$.
Let $\Sigma_i$ be the conical Legendrian filling of $T(2,2n+1)$ that is constructed by performing the first clasp move at the ``eye shape" between $b_i$ and $b_{i+1}$.
According to the MCF shown in Figure \ref{fig:torus} part $(b)$ and $(c)$, the pinching procedure on $T(2,2n-1)$ induces an augmentation of $T(2,2n-1)$ that only sends $c_1$ to $1$ and all others to $0$.

The clasp move produces a degree $0$ Reeb chord, $a$, (see Proposition 8.3 of \cite{PanRu1} for the degree computation) of the resulting conical Legendrian surface $\Sigma_i$, and the rest of the cobordism can be constructed without Reeb chords. Thus, $\alg(\Sigma_i)$ has  two augmentations, $a \mapsto 0$ and $a \mapsto 1$, and it follows that $I_{\Sigma_{i}}$ can have at most two elements.  In fact, $I_{\Sigma_{i}}$ does have two elements induced by the two  MCFs of the conical Legendrian surface shown in Figure \ref{fig:claspMCF}.  
For a subset $I$ of the set $\{1, \cdots 2n+1\}$, denote by $\e_{I}$ the algebra map from $\alg(T(2,2n+1))$ to $\Z/2$ that only sends $\{b_i| i\in I\}$ to $1$ and all other Reeb chords to $0$.
Then, Figure \ref{fig:claspMCF} shows $$I_{\Sigma_i}=\begin{cases}\{\e_{\{1\}}, \e_{\{3\}}\}, & \mbox{ if }i=1\\
\{\e_{\{1\}}, \e_{\{1,i, i+2\}}\}, & \mbox{ if }1<i<2n\\
\{\e_{\{1\}}, \e_{\{1, 2n\}}\}, & \mbox{ if }i=2n.
\end{cases}$$ 

\begin{figure}[!ht]
\labellist
\small
\pinlabel $(a)$ at 25 -5
\pinlabel $(b)$ at 105 -5
\pinlabel $(c)$ at 180 -5

\pinlabel $b_1$ at 13 111
\pinlabel $b_2$ at 33 111
\pinlabel $b_3$ at 43 111
\pinlabel $c_1$ at  43 65

\pinlabel $b_1$ at 13 47
\pinlabel $b_2$ at 33 47
\pinlabel $b_3$ at 43 47
\pinlabel $c_1$ at  43 3


\pinlabel $b_1$ at 88 111
\pinlabel $b_i$ at 95 111
\pinlabel $b_{i+1}$ at 115 111
\pinlabel $b_{i+2}$ at 125 111
\pinlabel $c_1$ at  88 65
\pinlabel $c_{i}$ at  125 65

\pinlabel $b_1$ at 88 47
\pinlabel $b_i$ at 95 47
\pinlabel $b_{i+1}$ at 115 47
\pinlabel $b_{i+2}$ at 125 47
\pinlabel $c_1$ at  88 3
\pinlabel $c_{i}$ at  125 3

\pinlabel $b_1$ at 160 111
\pinlabel $b_{2n-1}$ at 168 111
\pinlabel $b_{2n}$ at 177 111
\pinlabel $b_{2n+1}$ at 200 111
\pinlabel $c_1$ at  162 65
\pinlabel $c_{2n-1}$ at  172 65

\pinlabel $b_1$ at 160 47
\pinlabel $b_{2n-1}$ at 168 47
\pinlabel $b_{2n}$ at 177 47
\pinlabel $b_{2n+1}$ at 200 47
\pinlabel $c_1$ at  162 3
\pinlabel $c_{2n-1}$ at  172 3


\endlabellist
\includegraphics[width=6in]{Images/claspMCF}
\vspace{0.1in}

\caption{Two MCFs for the clasp move constructing $\Sigma_i$ pictured in the base projection.  These MCFs extend the MCF from Figure \ref{fig:torus} (b) and (c) to all of $\Sigma_i$.  Columns $(a), (b),(c)$ are for $i=1$, $1<i<2n$ and $i=2n$, respectively. The black dotted lines represent double points (in the front projection) and red lines represent handleslides.  }
\label{fig:claspMCF}
\end{figure}


\end{proof}
{"config": "arxiv", "file": "2006.16436/Sec6.tex"}