\section{Pairing calculations}\label{sec:pairings}

Throughout this section, let us fix some \(\alpha\in\Z^{>0}\). We will study the Lagrangian Floer homology of the special curve \(\textcolor{red}{\Special_\alpha(0;\TEIV,\TEI)}\) and other special and rational curves, restricting ourselves to the relative univariate Alexander and \(\delta\)-grading. 
Before doing so, we introduce some notation.

\begin{definition}
Given some \(k\in\Z^{>0}\), let \(\Contiguous{k}\) denote the contiguous, absolutely bigraded vector space of dimension \(k\), supported in Alexander gradings \(1,\dots,k\) and \(\delta\)-grading 0.
\end{definition}

\begin{lemma}\label{lem:computation:specials}
	Let \(\beta\in\Z^{>0}\) with \(\beta\leq\alpha\).  Suppose \(\HFr(\textcolor{red}{\Special_\alpha(0;\TEIV,\TEI)},\textcolor{blue}{\Special_\beta(0;\TEIV,\TEI)})\) is skeletal. Then 
	\[
	\HFr(\textcolor{red}{\Special_\alpha(0;\TEIV,\TEI)},\textcolor{blue}{\Special_\beta(0;\TEIV,\TEI)})
	=
	\delta^0 t^{-\alpha} \Contiguous{2\beta}
	\oplus
	\delta^{\pm1} t^\alpha \Contiguous{2\beta}
	\]
	as relatively bigraded vector spaces. 
	Moreover, the punctures \(\TEI\) and \(\TEIV\) of \(\FourPuncturedSphere\) are oriented in the same direction. 
\end{lemma}

\begin{lemma}\label{lem:computation:mixed}
	Let \(n\in\Z\). Suppose \(\HFr(\textcolor{red}{\Special_\alpha(0;\TEIV,\TEI)},\textcolor{blue}{\Rational(\tfrac{1}{n})})\) is skeletal. Then 
	\[
	\HFr(\textcolor{red}{\Special_\alpha(0;\TEIV,\TEI)},\textcolor{blue}{\Rational(\tfrac{1}{n})})
	=
	\Contiguous{2\alpha}
	\]
	as relatively bigraded vector spaces. 
\end{lemma}

\begin{remark}
	In the graphical notation from Remark~\ref{rem:gaps}, the two vector spaces from Lemmas~\ref{lem:computation:specials} and~\ref{lem:computation:mixed} look as follows:
	\[\setlength{\tabcolsep}{10pt}
		\begin{tabular}{cc}
			\(\HFr(\textcolor{red}{\Special_\alpha(0;\TEIV,\TEI)},\textcolor{blue}{\Special_\beta(0;\TEIV,\TEI)})\)
			&
			\(\HFr(\textcolor{red}{\Special_\alpha(0;\TEIV,\TEI)},\textcolor{blue}{\Rational(\tfrac{1}{n})})\)
			\\
			\(
			\vc{
				\begin{tikzpicture}[scale=0.5]
				\draw[->] (0,0) node [left] {\phantom{\(A\)}} -- (16,0) node[right]{\(A\)};
				\draw (1,-0.25) -- (1,0.25) node[above] {\(\bullet\)};
				\draw (2,-0.25) -- (2,0.25) node[above] {\(\bullet\)};
				\draw (3,0.25) node [above] {\(\cdots\)};
				\draw (4,-0.25) -- (4,0.25) node[above] {\(\bullet\)};
				\draw (5,-0.25) -- (5,0.25) node[above] {\(\bullet\)};
				\draw [decorate,decoration={brace}] (5.25,-0.4) -- (0.75,-0.4) node[midway, below] {\(\scriptstyle 2\beta\)};
				\draw (6,-0.25) -- (6,0.25);
				\draw (7,-0.25) -- (7,0.25);
				\draw (8,0.25) node [above] {\(\cdots\)};
				\draw (9,-0.25) -- (9,0.25);
				\draw (10,-0.25) -- (10,0.25);
				\draw [decorate,decoration={brace}] (10.25,-0.4) -- (5.75,-0.4) node[midway, below] {\(\scriptstyle 2(\alpha-\beta)\)};
				\draw (11,-0.25) -- (11,0.25) node[above] {\(\circ\)};
				\draw (12,-0.25) -- (12,0.25) node[above] {\(\circ\)};
				\draw (13,0.25) node [above] {\(\cdots\)};
				\draw (14,-0.25) -- (14,0.25) node[above] {\(\circ\)};
				\draw (15,-0.25) -- (15,0.25) node[above] {\(\circ\)};
				\draw [decorate,decoration={brace}] (15.25,-0.4) -- (10.75,-0.4) node[midway, below] {\(\scriptstyle 2\beta\)};
		\end{tikzpicture}}
			\)
			&
			\(
			\vc{
				\begin{tikzpicture}[scale=0.5]
				\draw[->] (0,0) node [left] {\phantom{\(A\)}} -- (6,0) node[right]{\(A\)};
				\draw (1,-0.25) -- (1,0.25) node[above] {\(\bullet\)};
				\draw (2,-0.25) -- (2,0.25) node[above] {\(\bullet\)};
				
				\draw (3,0.25) node [above] {\(\cdots\)};
				
				\draw (4,-0.25) -- (4,0.25) node[above] {\(\bullet\)};
				\draw (5,-0.25) -- (5,0.25) node[above] {\(\bullet\)};
				\draw [decorate,decoration={brace}] (5.25,-0.4) -- (0.75,-0.4) node[midway, below] {\(\scriptstyle 2\alpha\)};
		\end{tikzpicture}}
			\)
		\end{tabular}
	\]
	In the first vector space, the \(\delta\)-grading of the generators \(\bullet\) is constant, the same is true for the \(\delta\)-grading of the generators \(\circ\), and the difference between these two \(\delta\)-gradings is \(\pm1\).  
	The sign is immaterial for the proof of the main theorem. As will be apparent from the proof below, this sign only depends on the orientation of the punctures \(\TEI\) and \(\TEIV\).
\end{remark}

\begin{figure}[t]
	\(\PairingIrrIrrParallel\)
	\caption{Calculation of \(\HFr(\textcolor{red}{\Special_\alpha(0;\TEIV,\TEI)},\textcolor{blue}{\Special_\beta(0;\TEIV,\TEI)})\) for the proof of Lemma~\ref{lem:computation:specials}}\label{fig:computation:specials}
\end{figure}

\begin{proof}[Proof of Lemma~\ref{lem:computation:specials}]
	Figure~\ref{fig:computation:specials} shows the diagram from which we will compute the vector space \(\HFr(\textcolor{red}{\Special_\alpha(0;\TEIV,\TEI)},\textcolor{blue}{\Special_\beta(0;\TEIV,\TEI)})\) in the case that \(\beta\) is even. 
	If \(\beta\) is odd, there is a very similar diagram that can be obtained by rotating the outer part by \(\pi\) in the drawing plane such that the outermost blue curve segments lie on the right. 
	We claim that \(\HF(\textcolor{red}{\Special_\alpha(0;\TEIV,\TEI)},\textcolor{blue}{\Special_\beta(0;\TEIV,\TEI)})\) is freely generated by the intersection points in this diagram. 
	
	
	For each \(\varepsilon=1,\dots,\beta\), \(z\in\{x,y\}\) and \(k\in\{1,2\}\), there is a bigon from \(z_{k1}^\varepsilon\) to \(z_{k2}^\varepsilon\) covering the puncture \(\TEI\) once. 
	So by Lemma~\ref{lem:Alex:bigon} and Remark~\ref{rem:delta:HF}, each of these pairs of generators sits in consecutive Alexander gradings and a single \(\delta\)-grading.
	To compute \(\HFr(\textcolor{red}{\Special_\alpha(0;\TEIV,\TEI)},\textcolor{blue}{\Special_\beta(0;\TEIV,\TEI)})\), it therefore suffices to restrict our attention to the intersection points \(z_{k1}^\varepsilon\), where \(\varepsilon=1,\dots,\beta\), \(z\in\{x,y\}\) and \(k\in\{1,2\}\). The following diagram shows these generators arranged in the 2-dimensional plane according to their Alexander gradings:
	\[
	\begin{tikzpicture}[xscale=0.85,yscale=0.425,decoration=brace]
	\draw [->] (-9.25,-1.5) -- (-7.5,2) node [above] {\(\Z e_{\TEIV}\subset\GeneralAlex\)};
	\draw [->] (-9.25,-0.5) -- (-7.5,-4) node [below] {\(\Z e_{\TEI}\subset\GeneralAlex\)};
	
	\node (X1_1) at (-8,-1) {$x_{11}^1$};
	\node (X2_1) at (-7,+1) {$x_{21}^1$};
	\node (X1_2) at (-6,-1) {$x_{11}^2$};
	\node (X2_2) at (-5,+1) {$x_{21}^2$};
	\node (X1_3) at (-4,-1) {};
	\node (X2_3) at (-3,+1) {};
	\node (X1_b) at (-2,-1) {$x_{11}^\beta$};
	\node (X2_b) at (-1,+1) {$x_{21}^\beta$};
	
	\node (Y2_1) at (8,+1) {$y_{21}^1$};
	\node (Y1_1) at (7,-1) {$y_{11}^1$};
	\node (Y2_2) at (6,+1) {$y_{21}^2$};
	\node (Y1_2) at (5,-1) {$y_{11}^2$};
	\node (Y2_3) at (4,+1) {};
	\node (Y1_3) at (3,-1) {};
	\node (Y2_b) at (2,+1) {$y_{21}^\beta$};
	\node (Y1_b) at (1,-1) {$y_{11}^\beta$};
	
	\draw (X1_1) -- (X2_1);
	\draw (X1_2) -- (X2_2);

	\draw (X1_b) -- (X2_b);
	\draw (X1_1) -- (X1_2);
	\draw (X1_2) -- (X1_3);
	\draw (X1_3) -- (X1_b);
	
	\draw (Y1_1) -- (Y2_1);
	\draw (Y1_2) -- (Y2_2);

	\draw (Y1_b) -- (Y2_b);
	\draw (Y1_1) -- (Y1_2);
	\draw (Y1_2) -- (Y1_3);
	\draw (Y1_3) -- (Y1_b);
	
	\draw [dashed] (X1_b) -- (Y1_b);
	
	\node (x) at (-0.5,-4) {};
	\draw [<->] (x) -- (Y1_b) node [pos=0.5, below right] {\footnotesize\((\alpha-\beta+1)\)};
	\draw [<->] (x) -- (X1_b) node [pos=0.5, below left] {\footnotesize\((\alpha-\beta+1)\)};

	\end{tikzpicture}
	\]
	The lines connecting these generators indicate certain domains from which we compute the relative positions of all generators: For each \(\varepsilon=1,\dots,\beta\) and \(z\in\{x,y\}\), there is a bigon from \(z_{11}^\varepsilon\) to \(z_{21}^\varepsilon\) covering the puncture \(\TEIV\) once; these bigons correspond to the diagonal lines and preserve the \(\delta\)-grading. For every \(\varepsilon=1,\dots,\beta-1\), there are domains covering both punctures \(\TEI\) and \(\TEIV\) connecting \(x_{11}^\varepsilon\) to \(x_{11}^{\varepsilon+1}\) and \(y_{11}^{\varepsilon+1}\) to \(x_{11}^{\varepsilon}\); these correspond to the solid horizontal lines and also preserve the \(\delta\)-grading.  Finally, there is a domain from \(x_{11}^\beta\) to \(y_{11}^{\beta}\) covering the punctures \(\TEI\) and \(\TEIV\) \((\alpha-\beta+1)\) times; it corresponds to the dashed horizontal line in the centre of the diagram above and changes the \(\delta\)-grading by one.
	Note that the other half of intersection points, i.e.\ the generators \(\{z_{k2}^\varepsilon\}\), is obtained by shifting the generators \(\{z_{k1}^\varepsilon\}\) by one unit in the \(e_{\TEI}\)-direction.  In particular, the only pair of generators \((x_{ij}^\varepsilon,y_{k\ell}^{\nu})\) that share the same Alexander grading is \((x_{22}^\beta,y_{11}^\beta)\), and this only if \(\alpha=\beta\).
	 

	There are no immersed annuli with non-negative multiplicities in this diagram, which can be seen, for example, by considering the induced boundary orientation on \(\textcolor{blue}{\Special_\beta(0;\TEIV,\TEI)}\) near the puncture \(\TEI\). Therefore, the intersection points generate the complex whose homology is \(\HF(\textcolor{red}{\Special_\alpha(0;\TEIV,\TEI)},\textcolor{blue}{\Special_\beta(0;\TEIV,\TEI)})\).  To compute the differentials, observe that the \(\delta\)-grading is constant on the intersection points \(x_{k\ell}^\varepsilon\), and so is the \(\delta\)-grading on \(y_{k\ell}^\varepsilon\).
	
	So for grading reasons, only the generators \(x_{22}^\beta\) and \(y_{11}^\beta\) can be connected by a potentially non-vanishing differential, and this only if \(\alpha=\beta\). In this case, there are in fact two bigons between these two intersection points, so the differential always vanishes.  
	(In fact, for \(\alpha=\beta\), the generator \(x_{22}^\beta\) corresponds to the identity morphism of \(\textcolor{red}{\Special_\alpha(0;\TEIV,\TEI)}=\textcolor{blue}{\Special_\beta(0;\TEIV,\TEI)}\) in the Fukaya category.)
	
	Suppose the punctures \(\TEI\) and \(\TEIV\) are oriented differently, i.e.\ we quotient \(\GeneralAlex\) by \(e_{\TEI}+e_{\TEIV}\) when passing to the univariate Alexander grading. 
	Then these generators are concentrated in only two Alexander gradings, so \(\HFr(\textcolor{red}{\Special_\alpha(0;\TEIV,\TEI)},\textcolor{blue}{\Special_\beta(0;\TEIV,\TEI)})\) is not skeletal. 
	Hence the punctures are oriented in the same direction, i.e.\ we quotient \(\GeneralAlex\) by \(e_{\TEI}-e_{\TEIV}\). 
	In this case,  \(\HFr(\textcolor{red}{\Special_\alpha(0;\TEIV,\TEI)},\textcolor{blue}{\Special_\beta(0;\TEIV,\TEI)})\) has the desired form.  
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:computation:mixed}]
	Figure~\ref{fig:computation:mixed} shows all intersection points \(x_\varepsilon\), \(\varepsilon=1,\dots,4\alpha\), between these curves in minimal position.
	Observe that there is a 
	bigon from \(x_{2\alpha+1-\varepsilon}\) to \(x_{2\alpha+\varepsilon}\) 
	covering the puncture \(\TEI\) for all \(\varepsilon=1,\dots,2\alpha\).
	Therefore \(\HFr(\textcolor{red}{\Special_\alpha(0;\TEIV,\TEI)},\textcolor{blue}{\Rational(\tfrac{1}{n})})\) is isomorphic to the relatively bigraded vector space generated by \(\{x_{2\alpha+\varepsilon}\}_{\varepsilon=1,\dots,2\alpha}\).  
	We claim that all generators have the same \(\delta\)-grading and their Alexander grading is as follows:	
	\[
	\begin{tikzpicture}[xscale=0.85,yscale=0.425,decoration=brace]
	\draw [->] (-9.25,-1.5) -- (-7.5,2) node [above] {\(\Z e_{\TEIV}\subset\GeneralAlex\)};
	\draw [->] (-9.25,-0.5) -- (-7.5,-4) node [below] {\(\Z e_{\TEI}\subset\GeneralAlex\)};
	
	\node (X8) at (-8,-1) {$x_{4\alpha}$};
	\node (X7) at (-7,+1) {$x_{4\alpha-1}$};
	\node (X6) at (-6,-1) {$x_{4\alpha-2}$};
	\node (X5) at (-5,+1) {$x_{4\alpha-3}$};
	\node (X4) at (-4,-1) {};
	\node (X3) at (-3,+1) {};
	\node (X2) at (-2,-1) {$x_{2\alpha+2}$};
	\node (X1) at (-1,+1) {$x_{2\alpha+1}$};
	
	\draw (X1) -- (X2);
	\draw [dashed] (X2) -- (X3);
	\draw (X3) -- (X4);
	\draw [dashed] (X4) -- (X5);
	\draw (X5) -- (X6);
	\draw [dashed] (X6) -- (X7);
	\draw (X7) -- (X8);
	\end{tikzpicture}
	\]
	As in the proof of Lemma~\ref{lem:computation:specials}, the solid and dashed lines between the generators in this picture correspond to certain domains: The solid lines represent bigons from \(x_{2\alpha+2\varepsilon}\) to \(x_{2\alpha+2\varepsilon-1}\) 
	covering the puncture \(\TEIV\), where \(\varepsilon=1,\dots,\alpha\). The dashed lines correspond to the bigons from \(x_{2\alpha+2\varepsilon+1}\) to \(x_{2\alpha+2\varepsilon}\) 
	covering the puncture \(\TEI\), where \(\varepsilon=1,\dots,\alpha-1\). 
	Note that all generators sit in the same \(\delta\)-grading by Remark~\ref{rem:delta:HF}.
	Now, if the two punctures \(\TEIV\) and \(\TEI\) are oppositely oriented and \(\alpha>1\), \(\HFr(\textcolor{red}{\Special_\alpha(0;\TEIV,\TEI)},\textcolor{blue}{\Rational(\tfrac{1}{n})})\) is not skeletal. Otherwise, the vector space is contiguous. 
\end{proof}

\begin{lemma}\label{lem:reduction:specials}
	For any \(\beta\in\Z^{>0}\), \(s\in\QPI\smallsetminus\{0\}\), and \(\TEi,\TEj\in\{\TEI,\TEII,\TEIII,\TEIV\}\),  \(\HFr(\textcolor{red}{\Special_\alpha(0;\TEIV,\TEI)},\textcolor{blue}{\Special_\beta(s;\TEi,\TEj)})\) is not skeletal.
\end{lemma}

\begin{lemma}\label{lem:reduction:mixed}
	Let \(s\in\QPI\smallsetminus\left(\{0\}\cup\{\tfrac{1}{n}\mid n\in\Z\}\right)\). Then  \(\HFr(\textcolor{red}{\Special_\alpha(0;\TEIV,\TEI)},\textcolor{blue}{\Rational(s)})\) is not skeletal.
\end{lemma}


\begin{figure}[tb]
	\centering
	\begin{subfigure}{0.54\textwidth}
		\centering
		\(\PairingIrrRat\)
		\caption{\(\HFr(\textcolor{red}{\Special_\alpha(0;\TEIV,\TEI)},\textcolor{blue}{\Rational(\tfrac{1}{n})})\)}\label{fig:computation:mixed}
	\end{subfigure}
	\begin{subfigure}{0.4\textwidth}
		\centering
		\(\PairingIrrIrrNonparallel\)
		\caption{\(\HFr(\textcolor{red}{\Special_\alpha(0;\TEIV,\TEI)},\textcolor{blue}{\gamma})\)}\label{fig:reduction}
	\end{subfigure}
	\caption{Computations for the proof of (a) Lemma~\ref{lem:computation:mixed} and (b)  Lemmas~\ref{lem:reduction:specials} and~\ref{lem:reduction:mixed} }\label{fig:specials}
\end{figure}

\begin{proof}[Proof of Lemmas~\ref{lem:reduction:specials} and~\ref{lem:reduction:mixed}]
	Let \(\textcolor{blue}{\gamma}\) be either equal to \(\textcolor{blue}{\Special_\beta(s;\TEi,\TEj)}\) from Lemma~\ref{lem:reduction:specials} or  \(\textcolor{blue}{\Rational(s)}\) from Lemma~\ref{lem:reduction:mixed}. Let us put \(\textcolor{red}{\Special_\alpha(0;\TEIV,\TEI)}\) and \(\textcolor{blue}{\gamma}\) into minimal position. We claim that there are portions of \(\textcolor{red}{\Special_\alpha(0;\TEIV,\TEI)}\) and \(\textcolor{blue}{\gamma}\) that look like Figure~\ref{fig:reduction}, where the two curve segments of \(\textcolor{blue}{\gamma}\) stay parallel until they are joined up near some puncture. 
	To see this, let \(a\) be the horizontal arc connecting the punctures \(\TEI\) and \(\TEIV\). 
	If \(\textcolor{blue}{\gamma}\) is special, it is a figure-eight curve which sits in the neighborhood of some immersed arc connecting a puncture to itself. It suffices to see that this arc intersects \(a\) non-trivially. Suppose it does not. Then its slope is either 0 or \(\tfrac{1}{n}\) for some \(n\in\Z\). The first case contradicts our assumption on \(s\). In the second case, either \(\TEi\not\in\{\TEI,\TEIV\}\) or \(\TEj\not\in\{\TEI,\TEIV\}\). Without loss of generality, let us assume the former. Then we may reinterpret \(\textcolor{blue}{\gamma}\) as a figure-eight curve which sits in the neighborhood of a different immersed arc of the same slope as the first, but one which starts and ends at the puncture \(\TEi\neq\TEI,\TEIV\). This new arc intersects \(a\) non-trivially.
	A similar argument applies if \(\textcolor{blue}{\gamma}\) is rational. In this case, we can think of \(\textcolor{blue}{\gamma}\) as the boundary of a tubular neighborhood of an embedded arc, which connects a pair of distinct punctures. We may choose this arc such that at least one of its ends is different from \(\TEI\) and \(\TEIV\). Since in this case, \(s\neq\tfrac{1}{n}\) for all \(n\in\Z\) and \(s\neq0\) by assumption, this arc also intersects \(a\) non-trivially.
	
	
	
	
	
	We now compute the Lagrangian Floer homology from this picture. 
	For any \(k,\ell\in\{1,2\}\), there are bigons from \(x_{k\ell}\) to \(y_{k\ell}\) covering a single puncture. So \(\HFr(\textcolor{red}{\Special_\alpha(0;\TEIV,\TEI)},\textcolor{blue}{\gamma})\) contains a summand which is isomorphic to a vector space generated by the four intersection points \(x_{k\ell}\), \(k,\ell\in\{1,2\}\). One can easily see that this summand is (relatively) bigraded isomorphic to \(V^{\otimes2}\), which is not skeletal. 
\end{proof}
{"config": "arxiv", "file": "2006.03521/sections/PairingCalculations.tex"}