\CF_{J_{c_-}}(L_5)
\ar[r,"\id"]
\ar[dr,dashed]
\ar[d]
& \CF_{J_{c_-}}(L_5)
\ar[d]
\\
\CF_{J_{c_-}}(L_6)
\ar[r, "\id"]
\CF_{J_{c_-}}(L_6)
\end{tikzcd}
\label{eq:Psi_1->3-cx-structures}
\end{equation}

The hyperbox in Equation~\eqref{eq:Psi_1->3-cx-structures} realizes a chain homotopy between the model of $\Psi_{\frD_1\to \frD_3}$, computed with respect to almost complex structures which are maximally pinched along $c_+,$ $c_-$ and $c_0$, and a model of $\Psi_{\frD_1\to \frD_3}$ which is computed using almost complex structures which are maximally pinched along $c_+,$ $c_-$ and $c_\delta$.

In the model of $\Psi_{\frD_1\to \frD_3}$ appearing along the right hand side of Equation~\eqref{eq:Psi_1->3-cx-structures}, the holomorphic polygon counts all occur on a multi-stabilization of the diagram on $\Sigma$ or $\Sigma\# \bar \Sigma$. The stabilization results for triangles imply that the two length 1 maps along the top of ~\eqref{eq:Psi_1->3} compose to give a map which is intertwined with the identity under the canonical isomorphisms $\CF(\cH)\iso \CF(\cH_i)$ (similar to~\eqref{eq:canonical-iso}). (This is the content of \cite{JTNaturality}*{Propositions~9.31, 9.32}).

It remains to show that none of the diagonal maps make non-trivial contribution to the compression of~\eqref{eq:Psi_1->3}. To this end, we make the following claims, which together imply that the compression has trivial diagonal map:
\begin{enumerate}
\item\label{compression-claim-1} The compression of the top two levels of~\eqref{eq:Psi_1->3} has trivial diagonal map.
\item\label{compression-claim-2} The compression of the third and fourth levels of~\eqref{eq:Psi_1->3} has trivial diagonal map.
\item\label{compression-claim-3} The bottom-most level of ~\eqref{eq:Psi_1->3} has trivial diagonal map.
\end{enumerate}
Consider first the claim~\eqref{compression-claim-3}. The diagonal map is a quadrilateral counting map on the connected sum $(\Sigma,\as_u,\as_u,\bs_u,\bs_u)\#(\Sigma_0,\as_0^H,\as_0,\bs_0,\bs_0^H)$. Applying the stabilization result from Proposition~\ref{prop:multi-stabilization-counts} to the setting of holomorphic quadrilaterals, we may destabilize the map to the diagram $(\Sigma,\as_u,\as_u,\bs_u,\bs_u)$. By the small translate theorem for quadrilaterals \cite[Proposition 11.5]{HHSZExact}, this map vanishes.

Next, consider~\eqref{compression-claim-1}. For this claim, we observe that the stabilization results for holomorphic triangles imply that the length 1 holomorphic triangle maps preserve the subspaces spanned by the top-degree generator (and furthermore, act tensorially on these subspaces). Furthermore, the same argument as for claim~\eqref{compression-claim-3} implies that the holomorphic quadrilateral maps vanish on elements which have a factor which is the top degree generator. Since the top degree generator is the output of the 1-handle maps on the top row, this implies that the compression has trivial diagonal map. Claim~\eqref{compression-claim-2} follows from an analogous argument. This completes the proof of the claim about $\Psi_{\frD_1\to \frD_3}$.

We now consider the claim about $\Psi_{g(\frD_3)\to \frD_1}$. This map may be computed by compressing the following diagram
\begin{equation}
\begin{tikzcd}[labels=description, row sep=1cm]
\CF(\as,\bs)
\ar[r]
\ar[d, "F_{1}^{\bar{\b},\bar{\b}}"]
& \CF(\as,\bs)
\ar[r]
\ar[d, "F_{1}^{\bar{\b},\bar{\b}}"]
& \CF(\as,\bs)
\ar[d,"F_{1}^{\bar{\b},\bar{\b}}"]
\\
\CF(\as \bar{\bs}, \bs \bar{\bs})
\ar[r]
\ar[d]
\ar[dr,dashed]
& \CF(\as\bar{\bs}, \bs \bar{\bs})
\ar[r]
\ar[d]
\ar[dr,dashed]
& \CF(\as \bar{\bs}, \bs \bar{\bs})
\ar[d]
\\
\CF(\as \bar \bs,g(\Ds))
\ar[r]
\ar[d]
\ar[dr,dashed]
& \CF(\as\bar \bs,g(\Ds))
\ar[r]
\ar[d]
\ar[dr,dashed]
& \CF(\as\bar \bs,\Ds)
\ar[d]
\\
\CF(\as\bar \bs, \as \bar \as)
\ar[r]
\ar[d, "F_3^{\a,\a }"]
\CF(\as \bar \bs, \as \bar \as)
\ar[r]
\ar[d, "F_3^{\a,\a }"]
\CF(\as \bar{\bs}, \as \bar{\as})
\ar[d, "F_3^{\a,\a}"]
\\
\CF(\bar{\bs}, \bar{\as})
\ar[r]
\ar[drr,dashed]
\ar[d]
& \CF(\bar{\bs},\bar{\as})
\ar[r]
& \CF(\bar{\bs},\bar{\as})
\ar[d]
\\
\CF(\bar \bs,\bar \as)
\ar[r]
\CF(\bar \bs, \bar \as)
\ar[r]
\CF(\bar \bs,\bar \as)
\end{tikzcd}
\label{eq:g(D)->D}
\end{equation}
The description in~\eqref{eq:g(D)->D} is not the most convenient for our purposes. Instead, we describe a sequence of doubling curves $g(\Ds_0)=\Ds_0^0,\dots, \Ds_0^6=\Ds_0$ on $\Sigma_0\# \bar \Sigma_0$, such that each $\Ds_0^i$ is obtained from $\Ds_0^{i-1}$ by a handleslide. Furthermore, we have the following properties for each $i\in \{0,\dots 6\}$:
\begin{enumerate}
\item The diagram
\[
(\Sigma_0\# \bar \Sigma_0,\as_0 \bar \as_0, \bs_0 \bar \bs_0, \Ds_0^i,\Ds_0^{i+1},w)
\]
is weakly admissible.
\item Each of the complexes $\CF(\as_0 \bar \bs_0, \Ds_0^i)$, $\CF(\as \bar \as, \Ds_0^i)$, $\CF(\bs \bar \bs, \Ds_0^i)$, and $\CF(\Ds_0^i,\Ds_0^{i+1})$ has vanishing differential.
\end{enumerate}
The curves $\Ds_0^0,\dots \Ds_0^6$ are shown in Figure~\ref{fig:handleswaps-Delta}. We leave it to the reader to verify the above claims. (Compare Lemma~\ref{lem:admissibility-handleswap-1}).

We compute the map $\Psi_{g(\frD_3)\to \frD_1}$ as a composition of 6 maps, corresponding to the transition maps for changing $\Ds_0^i$ to $\Ds_0^{i+1}$, each of which is computed by compressing a hyperbox similar to~\eqref{eq:g(D)->D}. The same argument as for $\Psi_{\frD_1\to \frD_3}$ implies that each of these maps is intertwined with the identity map by the canonical isomorphisms in ~\eqref{eq:canonical-iso}. The proof is complete.
\end{proof}

\begin{figure}[p]
\centering
\input{handleswaps_delta-2.pdf_tex}
\caption{The curves $g(\Ds_0)=\Ds_0^0,\dots, \Ds_0^6=\Ds_0$, which relate $g(\Ds_0)$ and $\Ds_0$ by a sequence of handleslides. Each arrow indicates the subsequent handleslide.}
\label{fig:handleswaps-Delta}
\end{figure}

\section{Completing the proofs of naturality and functoriality} \label{sec:final-overview}

In this section, we complete our proofs of naturality and functoriality.

\subsection{Naturality}

If $$ and $'$ are two arbitrary doubling-enhanced Heegaard diagrams (equipped with a framing at the basepoint), we define our transition map $Ψ_→'$ by composing the maps for an arbitrary sequence stabilizations, elementary equivalences of attaching curves, and pointed isotopies of the Heegaard diagram in $(Y,w)$.

We now prove Theorem \ref{thm:final-naturality}, showing that this transition map $Ψ_→'$ defined above is independent up to chain homotopy from the sequence of Heegaard moves.

\begin{proof}[Proof of Theorem \ref{thm:final-naturality}] By \cite{JTNaturality}*{Theorem~2.38} and as in Section \ref{sec:naturality}, it suffices to show that the transition maps we have defined satisfy the axioms of a strong Heegaard invariant \cite{JTNaturality}*{Definition~2.32}.  We verify these axioms presently.

The \emph{functoriality} axiom asserts two claims. Firstly, it asserts that we have well defined morphisms associated to handleslide equivalences of alpha curves and morphisms associated to handleslide equivalences of beta curves (for a fixed Heegaard surface). This part of the axiom follows from Theorem~\ref{thm:naturality-fixed-Sigma}. (We remind the reader that unlike in the setting of ordinary Heegaard Floer homology, it is not particularly natural to define an involutive transition map which changes just the alpha curves, or just the beta curves, since the transition map is defined by compressing hyperboxes as in~\eqref{def:transition-map-elementary-handleslide}. Such hyperboxes necessarily require  changing both the alpha and beta curves simultaneously).

The second part of the functoriality axiom asserts that the morphisms for stabilizations are inverse to the morphisms for destabilizations. This is proven in Proposition~\ref{prop:stabilization-naturality}.

The next axiom from \cite{JTNaturality}*{Definition~3.32} is the \emph{commutativity} axiom, concerning commutativity of the distinguished rectangles from \cite{JTNaturality}*{Definition~2.29}. In our present case, we verify that there is no monodromy around the distinguished rectangles in Proposition~\ref{prop:simply-connected}.
In our present setting, we may rephrase the distinguished rectangles from \cite{JTNaturality} as diagrams of embedded, doubling enhanced Heegaard diagrams with the following shape, satisfying one of five configurations:
\[
\begin{tikzcd}
\frD_1
\ar[r, "e"]
\ar[d, "f"]
\frD_2
\ar[d, "g"]
\\
\frD_3
\ar[r, "h"] & \frD_4
\end{tikzcd}
\]
We presently enumerate the five configurations, and also prove that there is no monodromy:
\begin{enumerate}
\item $e$, $f$, $g$ and $h$ are handleslide equivalences. In our setting, we must consider changes of both of the attaching curves, as well as changes of the doubling datum. Note these are also referred to as \emph{strong equivalences} in the literature, e.g.~in \cite{MOIntegerSurgery}. The diagram commutes up to chain homotopy by Theorem~\ref{thm:naturality-fixed-Sigma}. We note that in \cite{JTNaturality}, the stated rectangle has the property that $e$ and $h$ are $\alpha$-equivalences, while $f$ and $g$ are $\beta$-equivalences. However the rectangle therein is also a rectangle of \emph{isotopy diagrams}, so we must consider changes of both the alpha and beta curves along each edge.
\item $e$ and $h$ are handleslide equivalences of attaching curves and doubling curves, while $f$ and $g$ are both stabilizations. Commutativity follows from Proposition~\ref{prop:stabilization-naturality}.
\item $e$ and $h$ are handleslide equivalences, while $f$ and $g$ are diffeomorphisms preserving the framing $\xi$. Commutativity around such rectangles is tautological.
\item $e$, $f$, $g$ and $h$ are all stabilizations. Furthermore, there are disjoint 3-balls $B_1$ and $B_2$, such that $e$ and $h$ correspond to a stabilization in $B_1$, while $f$ and $g$ correspond to a stabilization in $B_2$. Commutativity around such rectangles follows from Proposition~\ref{prop:stabilization-naturality}.
\item The maps $e$ and $h$ are stabilizations, while $f$ and $g$ are diffeomorphisms preserving the framing. Furthermore, the stabilization ball for $h$ is the image of the stabilization ball for $e$ under the diffeomorphism $g$. Commutativity around such rectangles is tautological.
\end{enumerate}

The remaining axioms of \cite{JTNaturality}*{Definition~3.32}, continuity and handleswap invariance, were verified in  Proposition~\ref{prop:continuity} and Theorem~\ref{thm:handleswap}.
\end{proof}

\subsection{Functoriality}

We now finish defining our cobordism maps and sketch the proof of Theorem \ref{thm:well-defined-cobordism-map}, showing that these maps are well-defined. Our proof is modeled on \cite{OSTriangles}. Since our approach is standard, we only provide an overview and sketch the points at which our construction deviates from \cite{OSTriangles}.

We begin with the construction, and will shortly sketch a proof of invariance. Suppose $W$ is a cobordism from $Y_1$ to $Y_2$, and assume basepoints in $Y_1$ and $Y_2$, as well as a framed path $$ connecting them, have been chosen. We assume that $W$, $Y_1$, $Y_2$ are connected. We pick a Morse function $fW→[0,1]$. We may assume that the indices of the critical points of $f$ are non-decreasing, and we assume all critical values are distinct and that $f$ has critical points only of index 1, 2 and 3. Next, we pick a gradient like vector field $v$ such that our path $$ is a flow-line. A choice of gradient like vector field gives a diffeomorphism
\[
W\iso W(\bS_2)\cup W(\bS_1)\cup W(\bS_0),
\]
where $_i$ are framed $i$-dimensional links and $_0⊂Y_1$, $_1⊂Y_1(_0)$, and $_2⊂Y_1(_0)(_1)$. If $∈^c(W)$ is self-conjugate, we define our cobordism map as the composition
\[
\CFI(W,\xi, \frs)=\CFI(W(\bS_2))\circ \CFI(W(\bS_1), \frs|_{W(\bS_1)})\circ \CFI(W(\bS_0)).
\]

We now prove Theorem \ref{thm:well-defined-cobordism-map}. As a first step, we show that the 2-handle map is well-defined:

\begin{lem}\label{lem:well-defined-2handle}
Our 2-handle map $\CFI(W(\bS_1),\frs|_{W(\bS_1)})$ is independent of the choice of diagram subordinate to a particular bouquet. Furthermore, it is also invariant from the choice of bouquet, as well as handleslides amongst the components of $\bS_1$.
\end{lem}
\begin{proof}
Ozsv\'{a}th and Szab\'{o} \cite{OSTriangles}*{Lemma~4.5} described five moves which relate any two bouquets of a fixed link. Invariance under the first four of these moves follows immediately from invariance of our map from handleslides amongst the curves in our Heegaard triple, Proposition~\ref{prop:2-handles-handle-slides}. The final move described by Ozsv\'{a}th and Szab\'{o} is stabilization of the Heegaard triple. Invariance from this move is a formal consequence of our definition of the stabilization map. Indeed we defined the naturality map for stabilizations as the composition of the 1-handle map, followed by a canceling 2-handle map. Both the 1-handle map and map for a canceling 2-handle map commute with the 2-handle map $\CFI(W(\bS_1), \frs|_{W(\bS_1)})$ by Propositions~\ref{prop:composition-law} and ~\ref{prop:commute-1handles/2-handles}. Hence our 2-handle map is independent of the choice of diagram subordinate to a fixed bouquet.

Next, we note that independence from the choice of bouquet, as well as independence from handleslides amongst the components of $\bS_1$ both follow from independence of the map from handleslides of the attaching curves, which we proved in Proposition~\ref{prop:2-handles-handle-slides}. Compare \cite{OSTriangles}*{Lemma~4.8}.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:well-defined-cobordism-map}]
Firstly, we verify that the construction is independent of the choice of gradient-like vector field $v$ for $f$. Subsequently we sketch that the cobordism map is invariant from $f$.

The space of gradient like vector fields for a fixed Morse function is connected. Since we have already decomposed $W$ along level sets which separate critical points of different indices, the codimension 1 singularities of a path of gradient like vector fields are handleslides amongst link components of the same indices. For critical points of index 1 or 3, invariance under these moves can instead be proven by reordering the attachment of the handles using the composition law for 1-handles and 3-handles, Proposition~\ref{lem:commute-1/3-handles}, since a handleslide is the same as an isotopy after attaching one of the handles. For 2-handles, we cannot add additional level sets because of the $\Spin^c$ decomposition, so instead handleslide invariance is proven directly in Lemma~\ref{lem:well-defined-2handle}.

We now consider invariance under $f$. By standard Cerf theory, \cite{CerfStratification} \cite{KirbyCalculus} any two Morse functions $f_0$ and $f_1$ with critical points ordered monotonically may be connected with a 1-parameter family $(f_{t})_{t\in [0,1]}$ whose critical points are also ordered monotonically, except at finitely many points where a birth-death singularity occurs. Furthermore, if $f_0$ and $f_1$ have no index 0 or 4 critical points, then each $f_t$ may also be chosen to have no index 0 or 4 critical points. We now show that the maps are invariant under index 1/2 handle cancellations, as well as index 2/3 handle cancellations. Invariance under such handle cancellation is essentially automatic from the definition of our naturality map for stabilizations. Indeed, consider the case that $W$ has a Morse function with only index 2 critical points, whose descending manifolds intersect $Y_0$ in a framed link $\bS_1$. Let $\bS_0\subset Y_0$ be a 0-sphere, and let $\bK$ be a framed knot in $Y_0(\bS_0)$ which intersects the co-core of $\bS_0$ in exactly 1-point. Assume $\bS_1$ is disjoint from $\bS_0$ and $\bK$. Using the composition law of Propositions~\ref{prop:composition-law}, we obtain
\[
\CFI(W(\bS_1\cup \bK))\circ \CFI(W(\bS_0))\simeq \CFI(W(\bS_1))\circ \CFI(W(\bK))\circ \CFI(W(\bS_0)):=\CFI(W(\bS_1))\circ \sigma,
\]
where $\sigma$ is the naturality map for stabilization (and we are omitting $\Spin^c$ structures from the notation). Invariance under index 2/3 handle cancelations follows entirely analogously. The proof is complete.
\end{proof}

\section{The cobordism map for $S^2\times S^2$}
\label{sec:example}

In this section we compute the cobordism map for $S^2×S^2$.
We prove the following:
\begin{prop}
Let $W$ denote $S^2\times S^2$, with two 4-balls removed. Let $\frs$ denote the unique self-conjugate $\Spin^c$ structure on $W$. Then the cobordism map
\[
\CFI(W,\frs)\colon \bF[U,Q]/Q^2\to \bF[U,Q]/Q^2
\]
is multiplication by $Q$.
\end{prop}
Using the composition law, we may decompose $W$ as two 2-handle cobordisms, $W=W_2∘W_1$. The first 2-handle is attached along a 0-framed unknot in $S^3$ to form $W_1$. The second 2-handle is attached along an $S^1$ fiber of $S^1×S^2$ to form $W_2$. Note that, in both cases, one end of the cobordism is a copy of $S^3$, on which the map $Φ$ is nullhomotopic. This implies that the choice of framings of the basepoints of the three-manifolds and along choices of paths connecting them do not affect the final computation, and we therefore omit them.

We start with the standard Heegaard triple $(Σ,',,)$ for $W_1$. In Figure~\ref{fig:s2s2-1}, we show the diagram $(Σ# Σ̅,',,,)$. Therein, the top half of the figure is $Σ$, and the bottom half is $Σ̅$. This figure also encodes $(Σ,',,)$.

The first quadrilateral counting map in Equation~\eqref{eq:hyperbox-2-handles} is $h^̱̅→Δ_→'$.  The Heegaard quadruple $(Σ# Σ̅, ' , , ,)$ is depicted in Figure \ref{fig:s2s2-1}. This diagram is weakly admissible.  The input for $h^̱̅→Δ_→'$ from $(,)$ is $Θ^+_̱̅,Δ$, and the input from $(',)$ is $Θ^+_',$.  We need only evaluate $h^̱̅→Δ_→'(Θ_,̱̅^+)$, since $F_1^,(Θ_,)=Θ_,̱̅^+$, and for grading reasons $h^̱̅→Δ_→'(Θ_,̱̅^+)$ is a multiple of $Θ_',Δ^+$. Hence, we need only consider holomorphic quadrilaterals in
\[
\pi_2(\Theta^+_{\a'\bar{\b},\a\bar{\b}},\Theta_{\a\bar{\b},\b\bar{\b}}^+,\Theta^+_{\b\bar{\b},\Delta},\Theta_{\a'\bar{\b},\Delta}^+)
\]
There is an index $-1$ domain (shaded). This class has a holomorphic representative for a unique conformal class of rectangle.  Indeed, for each slit length (along the alpha and beta arcs in the interior of $P_2$)  there is a map of a varying conformal class of rectangle to the domain of $P_2$.  Meanwhile, there is a holomorphic rectangle mapping to $P_1$ for a unique conformal structure.

\begin{figure}[H]
\centering
\input{s2s2-1.pdf_tex}
\caption{The Heegaard quadruple $(\Sigma\# \bar{\Sigma},\as'\bar{\bs},\as\bar{\bs},\bs\bar{\bs},\Ds)$.  The generators of interest are $\Theta^+_{\a'\bar{\b},\a\bar{\b}}=\{\phi,\phi^+\}$, $\Theta_{\a\bar{\b},\b\bar{\b}}^+=\{\theta,\theta^+\}$, $\Theta^+_{\b\bar{\b},\Delta}=\{\tau,\tau^+\}$ and $\Theta_{\a'\bar{\b},\Delta}^+=\{z,z^+\}$. }\label{fig:s2s2-1}
\end{figure}

By inspection, there are no other positive domains and so
\[
h^{\b\bar{\b}\to \Delta}_{\a\bar{\b}\to \a'\bar{\b}}(\Theta_{\a\bar{\b},\b\bar{\b}}^+)=\Theta_{\a'\bar{\b},\Delta}^+.
\]

For $h^Δ→''_→'$ we need only consider summands in the image containing $={x_0^-,x_1^+}$, since grading considerations imply that the only other generator that can appear in the image of $h^Δ→''̅_→'∘f^̱̅→Δ_∘F_1^,$ is ${x_0^+,x_1^-}$, which is annihilated by $F_3^','$.

\begin{figure}[H]
\centering
\input{s2s2-2.pdf_tex}
\caption{The Heegaard quadruple $(\Sigma,\as'\bar{\bs},\as\bar{\bs},\Ds,\as'\bar{\as}')$ in the definition of $h^{\Delta\to \a'\bar{\a'}}_{\a\bar{\b}\to \a'\bar{\b}}$, with some intersection points labeled.  The generators of interest are $\Theta^+_{\Delta,\a'\bar{\a}'}=\{\tau^+,\tau\}$,  $\Theta^+_{\a\bar{\b},\Delta}=\{\theta,\theta'\}$,  $\Theta^+_{\a'\bar{\b},\a\bar{\b}}=\{ \phi^+,\phi\}$ and $\xs=\{x_0^-,x_1^+\}$. 	}\label{fig:s2s2-2}
\end{figure}

\begin{figure}[H]
\centering
\input{s2s2-3.pdf_tex}
\caption{The Heegaard quadruple $(\Sigma,\as\bar{\bs},\Ds,\as'\bar{\as}',\as\bar{\as}')$ in the definition of $H^{\Dt\to \a\bar{\a}'}_{\a\bar{\b}}$, with some intersection points labeled.  The generators of interest are $\Theta^+_{\Delta,\a'\bar{\a}'}=\{\tau^+,\tau\}$,  $\Theta^+_{\a'\bar{\a}',\a\bar{\a}'}=\xs_2:=\{x,x_2\}$,  $\ys=\{ \phi_2,\phi_3\}$ and $\Theta_{\a\bar{\b},\Delta}=\{\theta,\theta'\}$. 	}\label{fig:s2s2-3}
\end{figure}

One checks directly that there are no positive domains among those generators for $h^Δ→''̅_→'$ as well as $H^Δ→'_$.

It follows that the map $F_W_1,|_W_1$ takes the form
\[\left(
\begin{tikzcd}[column sep=1cm]
\CFI(S^3)
\ar[r, "F_{W_1,\frs|_{W_1}}"]
\CFI(S^1\times S^2)
\end{tikzcd}\right)
\simeq
\begin{tikzcd}[column sep=2cm, row sep=1cm, labels=description]
\CF(S^3)
\ar[r, "1\mapsto \Theta^-"]
\ar[d, "Q(1+\iota)"]
\ar[dr,dashed, "1\mapsto Q\Theta^+"]
\CF(S^1\times S^2)
\ar[d, "Q(1+\iota)"]
\\
\ar[r, "1\mapsto \Theta^-"]
Q\cdot\CF(S^1\times S^2)
\end{tikzcd}
\]

We now consider the $2$-handle cobordism $W$ from $S^1×S^2→S^3$.  This is determined by the Heegaard triple $(Σ,”,',)$, where $',$ are as before, and $”$ is a copy of $$.  Let $θ^-_',$ be the lower-degree generator of $^-(Σ,',)$ and $θ^+_̱̅,̱̅$ be the top-degree generator of $^-(Σ̅,,)$.  We need to calculate $h^̱̅→̱Δ_'̱̅→”̱̅(θ^-_',×θ^+_̱̅,̱̅)$, $h^Δ,””_'→”(Θ^-_',Δ)$, and $H^Δ→'”_'(Θ^-_',Δ)$.

We consider diagrams for each of those quadruples.  Indeed, the first quadruple is represented in Figure \ref{fig:s2s2-1} with the role of the $'$ and $$ curves exchanged.  A direct calculation shows there are no positive domains for $π_2(Θ^+_”,',θ^-_',×θ^+_̱̅,̱̅,Θ^+_̱̅,Δ,Θ_”,Δ)$ so $h^̱̅→̱Δ_'̱̅→”̱̅(θ^-_',×θ^+_̱̅,̱̅)=0$.  Entirely similar remarks apply for $h^Δ,””_'→”(Θ^-_',Δ)$, and $H^Δ→'”_'(Θ^-_',Δ)$.
Computing as above, we obtain that
\[\left(
\begin{tikzcd}[column sep=1cm]
\CFI(S^1\times S^2)
\ar[r, "F_{W_2,\frs|_{W_2}}"]
\CFI(S^3)
\end{tikzcd}\right)
\simeq
\begin{tikzcd}[column sep=2cm, row sep=1cm, labels=description]
\CF(S^1\times S^2)
\ar[r, "\Theta^+\mapsto 1{,} \Theta^-\mapsto 0"]
\ar[d, "Q(1+\iota)"]
\CF(S^3)
\ar[d, "Q(1+\iota)"]
\\
Q\cdot\CF(S^1\times S^2)
\ar[r, "\Theta^+\mapsto 1{,} \Theta^-\mapsto 0"]
\end{tikzcd}
\]

Composing the $(W_2,_2)∘(W_1,_1)$, we obtain multiplication by $Q$.

\begin{rem}
Although $W_1$ is obtained topologically by turning the cobordism $W_2$ around, the maps $\CFI(W_1,\frs_1)$ and $\CFI(W_2,\frs_2)$ are not dual over $\bF[U,Q]/Q^2$. Compare \cite{OSTriangles}*{Theorem~3.5}. Note that our definition of the 2-handle map in Equation~\eqref{eq:hyperbox-2-handles} does not naturally dualize to another 2-handle map.
\end{rem}

\section{Knots and links} \label{sec:knots}

In this section, we describe how to adapt the naturality and functoriality results of the previous sections to the case of knots and links, and in particular prove Theorems \ref{thm:link-naturality} and \ref{thm:link-functoriality}.

If $L$ is a link, in this section we write $(L)$ for the pair $((L), ι_L)$. Analogously to the case of knots, our construction requires a choice of orientation on $L$, so there are potentially $2^ℓ$ different potential models for $ι_L$. To simplify the notation, we assume that an orientation is fixed, and we consider only the two orientations which are either coherent, or opposite to our preferred orientation. We write $ι_L,+$ and $ι_L,-$ for these models of the link involution.

In analogy to our construction in the case of closed 3-manifolds, we will describe several expanded models of the link involution $ι̃_L,±$ in Section~\ref{sec:expanded-involution-knots}. To streamline the presentation, we only define transition maps with respect to the expanded models (unlike in the case of closed 3-manifolds, where we defined both non-expanded and expanded models). Since we only consider the expanded models of the involution in this section, we will write $$ for the expanded, doubling enhanced link diagram we consider in Section~\ref{sec:expanded-involution-knots}. This notation departs from previous sections, where we wrote $$ for expanded models of the involution. Similarly, we write $Ψ_ _1→_2$ for the expanded models of the transition maps in this section, instead of $Ψ̃__1→_2$.

In this section, we show how to adapt the techniques of the earlier sections to prove the following naturality theorem, which implies Theorem \ref{thm:link-naturality}.

\begin{thm}
\label{thm:naturality-knots-links}
Suppose that $L$ is a link in a 3-manifold.
\begin{enumerate}
\item The transition maps $ \Psi_{ \frD\to \frD}$ are well-defined up to homotopies of morphisms of $\iota_L$-complexes.
\item $\Psi_{ \frD\to  \frD}\simeq \id_{\cCFLI( \frD)}$.
\item If $\frD_1$, $ \frD_2$ and $\frD_3$ are three expanded, doubling enhanced Heegaard link diagrams for $(Y,L)$, then
\[
\Psi_{\frD_2\to\frD_3}\circ  \Psi_{ \frD_1\to \frD_2}\simeq \Psi_{ \frD_1\to \frD_3}.
\]
\end{enumerate}
\end{thm}

Additionally, we prove the following functoriality theorem, which is an expanded version of Theorem \ref{thm:link-functoriality}.

\begin{thm}
\label{thm:functoriality}
Suppose that $(W,\Sigma)$ is a link cobordism from $(Y_1,L_1)$ to $(Y_2,L_2)$. Suppose further that $\Sigma$ consists of a collection of annuli, each with one boundary component in $Y_1$, and one boundary component in $Y_2$. Suppose further that each annulus is decorated with two parallel longitudinal  arcs, $\frs\in W$ is a $\Spin^c$ structure such that $\bar \frs=\frs+\PD[\Sigma]$. Then the cobordism map $\cCFLI(W,\Sigma,\frs)$ defined via the doubling model is an enhanced $\iota_L$-homomorphism, and is well-defined up to enhanced $\iota_L$-homotopy.
\end{thm}

\subsection{Expanded models of the knot and link involutions}
\label{sec:expanded-involution-knots}

In this section, we describe our expanded model of the knot and link involution, similar to the expanded 3-manifold involution which appeared in Section~\ref{sec:expanded-def}. This is the model we use to prove Theorem~\ref{thm:naturality-knots-links}.

The construction is as follows. Suppose that $L$ is a link in $Y$, and  $(Σ,,,,)$ is a Heegaard link diagram for $(Y,L)$ such that each component of $L$ contains exactly two basepoints. We pick an auxiliary point $p∈Σ∖(∪∪{w,z})$, at which we will form the connected sum of $Σ$ and $Σ̅$.

We write $c$ for the curve which is a meridian of the connected sum tube, as in the 3-manifold case. There is a natural 1-handle map
\[
F_1^{c\bar \b,c \bar \b}\colon \cCFL(\Sigma,\as,\bs,\ws,\zs)\to \cCFL(\Sigma\# \bar \Sigma, \as c \bar \bs, \bs c\bar \bs,\ws\cup \bar \zs, \zs\cup \bar \ws).
\]

We pick a collection of properly embedded and pairwise disjoint arcs $δ_1,…, δ_2g+2|L|-1$ on $Σ∖N(p)$, such that after cutting $Σ∖N(p)$ along $δ_1,…, δ_2g+2|L|-1$, we are left with $2|L|$ punctured disks. We assume that each disk contains exactly one basepoint from $∪$.  Equivalently, we may assume that $δ_1,…, δ_2g+2|L|-1$ forms a basis of $H_1(Σ∖(∪), p)$. We double the arcs $δ_1,…, δ_2g+2|L|-1$ to obtain the attaching curves $$. See Figure~\ref{fig:11} for an example.

The complex $(Σ# Σ̅, c , , ∪, ∪)$ represents an $|L|$-component unlink in $#^g S^1×S^2$, where each component contains 4 basepoints. There are two canonical homology classes, represented by cycles $Θ^_c̱ ̱̅, $ and $Θ^_c̱ ̱̅, $. These are the generators of the top degree of homology with respect to the gradings $_$ and $_$, respectively. See \cite{ZemCFLTQFT}*{Lemma~3.7}. We define
\[
f_{\a c \bar \b}^{\b c \bar \b\to\Dt;\zs}(-)=f_{\a c \bar \b, \b c \bar \b, \Dt}(-,\Theta^{\ws}_{\b c \bar \b ,\Dt})
\]
and we similarly define $f_c ̱̅^c̱ ̱̅→;$ to be the holomorphic triangle map with special input $Θ_c̱ ̱̅, ^$. (The reason that the $$-map has the $$-generator as its input is due to the interpretation of the map in terms of decorated link cobordisms from \cite{ZemCFLTQFT}).

Similarly, there are two distinguished classes $Θ_,c ^,Θ_,c ^$ in $(Σ# Σ̅, , c , ∪, ∪)$.

We define the following two expanded models of the knot involution:
\[
\begin{split}
\tilde\iota_{L,+}&=\eta_K\circ F_{3}^{\a c, \a c} \circ f_{\a c \bar \a}^{\Dt\to \a c \bar \a;\ws} \circ f_{\a c \bar \a}^{\b c \bar \b \to \Dt;\zs}\circ F_1^{c \bar \b, c \bar \b}\\
\tilde\iota_{L,-}&=\eta_K\circ F_{3}^{\a c, \a c} \circ f_{\a c \bar \a}^{\Dt\to \a c \bar \a;\zs} \circ f_{\a c \bar \a}^{\b c \bar \b \to \Dt;\ws}\circ F_1^{c \bar \b, c \bar \b}.
\end{split}
\]

\begin{figure}[h]
\centering
\input{fig11.pdf_tex}
\caption{A doubly pointed Heegaard knot diagram with a special point $p$ (left), and the double (right).}
\label{fig:11}
\end{figure}

\begin{lem}
\label{lem:expanded-transition-links}
The maps $\tilde \iota_{L,-}$ and $\tilde \iota_{L,+}$ are homotopic to the maps $\iota_{L,-}$ and $\iota_{L,+}$, respectively, which were defined in Section~\ref{sec:doubling-knots}.
\end{lem}
\begin{proof}
We focus on the case when $L$ is a doubly pointed knot $K$, since the proof for links is not substantially different.

The proof is to identify each of the above compositions with a cobordism map from \cite{ZemCFLTQFT}. The map $F_1^{c \bar \b, c \bar \b}$ is equal to a composition of $g$ 4-dimensional 1-handle maps, as well as the birth cobordism map for adding an unknot component. The map $F_3^{\a c, \a c}$ is similarly equal to the composition of $g$ 3-handle maps, as well as a death cobordism map which deletes an unknot.

We now consider the triangle maps.   The map $f_{\a c \bar \a}^{\b c \bar \b\to \Dt;\zs}$ is the composition of $g$ 4-dimensional 2-handle maps (as in the ordinary doubling model of the involution), as well as the saddle cobordism map which attaches a band which has both of its ends in a component of $K\setminus \{w,z\}$ which is oriented from $z$ to $w$ (i.e. which lies in the beta handlebody). Furthermore, the band is contained in the type-$\zs$ subsurface of the link cobordism. The map $f_{\a c \bar \a}^{\Dt\to \a c \bar \a;\ws}$ has a similar description, except that the band is instead in the type-$\ws$ subsurface.

To make the identifications in the main statement, we argue as follows. By canceling the 1-handles and 2-handles, we may identify the composition $f_{\a c \bar \a}^{\b c \bar \b\to \Dt;\zs}\circ F_1^{c \bar \b, c \bar \b}$ with the composition of a birth cobordism, followed by a $\ws$-saddle map. By \cite{ZemCFLTQFT}*{Proposition~8.5}, we may identify this composition with the quasi-stabilization map $S^+_{\bar z, \bar w}$, from \cite{ZemCFLTQFT}*{Section~4} and \cite{ZemQuasi}. Analogously, we may identify the composition $F_3^{\a c, \a c}\circ f_{\a c \bar \a}^{\Dt\to \a c \bar \a;\ws}$ with $T^-_{z,w}$. The composition $T^-_{z,w}\circ S^+_{\bar z, \bar w}$ is homotopic to the basepoint moving map which sends $w$ to $\bar z$ and $z$ to $\bar w$, both moving in the positive direction by \cite{ZemCFLTQFT}*{Section~4.4}. See Figure~\ref{fig:9}.
\end{proof}

\begin{figure}[h]
\centering
\input{fig9.pdf_tex}
\caption{Decorations on a cylindrical link cobordism $[0,1]\times K$ corresponding to a composition of a quasi-stabilization followed by a quasi-destabilization.}
\label{fig:9}
\end{figure}

\begin{rem} The compositions $\eta_K\circ F_{3}^{\a c, \a c} \circ f_{\a c \bar \a}^{\Dt\to \a c \bar \a;\zs} \circ f_{\a c \bar \a}^{\b c \bar \b \to \Dt;\zs}\circ F_1^{c \bar \b, c \bar \b}$ and $\eta_K\circ F_{3}^{\a c, \a c} \circ f_{\a c \bar \a}^{\Dt\to \a c \bar \a;\ws} \circ f_{\a c \bar \a}^{\b c \bar \b \to \Dt;\ws}\circ F_1^{c \bar \b, c \bar \b}$ are null-homotopic. Indeed the same argument as above identifies the two maps with the link cobordism maps for a cylinder which contains a null-homotopic loop in the dividing set. Such a cobordism map induces the trivial map, since it may be identified with one of the compositions $S_{w,z}^- S_{w,z}^+$ or $T_{w,z}^- T_{w,z}^+$, which both vanish by \cite{ZemCFLTQFT}*{Lemma~4.13}.
\end{rem}

\begin{rem}
As previously mentioned, if $L$ has $\ell$-components, there are $2^\ell$ natural models of the link involution, depending on how we twist each link component. The expanded model described above can also be used to realize any of these models.  We consider the diagram $(\Sigma\# \bar \Sigma, \bs c \bar \bs, \Ds, \ws\cup \bar \zs, \zs\cup \bar \ws)$. Suppose that $\mathbb{O}$ is a map from the set of link components of $L$ to the set $\{+,-\}$ (corresponding to a choice of whether to twist in the positive or negative direction). We let $\bZ(\bO)$  consist of the basepoints from $\ws\cup \bar \zs$, for components where $\bO=-$, as well as the basepoints from $\zs\cup \bar \ws$ for components where $\bO=+$. Let $\bW(\bO)$ denote the complement. Then each collection $\bZ(\bO)$ and $\bW(\bO)$ determine Maslov gradings on $\cCFL(\Sigma\# \bar \Sigma, \bs c \bar \bs, \Ds, \ws\cup \bar \zs, \zs\cup \bar \ws)$, and we let $\Theta^{\bZ(\bO)}$ and $\Theta^{\bW(\bO)}$ be the corresponding top-degree generators. A straightforward extension of Lemma~\ref{lem:expanded-transition-links} shows that the involution $\tilde \iota_{L,\bO}$ is computed by using $\Theta^{\bZ(\bO)}$ for the first triangle map, and $\Theta^{\bW(\bO)}$ for the second.
\end{rem}

\subsection{Naturality for knots and links}
\label{sec:naturality-knots-and-links}

Most of the ideas from the previous sections carry over to define transition maps and cobordism maps for the model of involutive link Floer homology described in Section~\ref{sec:expanded-involution-knots}.

The main new technical subtlety is that the model of involutive link Floer homology in Section~\ref{sec:expanded-involution-knots} has an additional piece of data: the choice of connected sum point $p$. If $p$ and $p'$ are two choices of special points on $Σ$, a choice of path on $Σ$ from $p$ to $p'$ can be used to relate the two models where the connected sum is taken at $p$ or $p'$, by using a point-pushing diffeomorphism along this path. In principle, a transition map defined in this manner might depend on the choice of path. To handle this issue, we introduce further expansions of the doubling model, as follows.

Suppose that $p={p_1,…, p_n}$ is a collection of marked points on $Σ∖(∪∪∪)$. We may form an expanded model of the involution, as follows. Firstly, we consider the surface $Σ#_p Σ̅$, where we attach a connected sum tube at each point in $p$. Let $c={c_1,…, c_n}$ denote a collection of meridians of the tubes. We pick properly embedded arcs $δ_1,…, δ_2g+2|L|+|p|-2$, which form a basis of $H_1(Σ∖(w∪z), p)$. It is straightforward to see that if $$ is constructed by doubling $δ_1,…, δ_2g+2|L|+|p|-2$, then $(Σ#_p Σ̅, c , , ∪, ∪)$ is a diagram for $(Y,L)$, where each component of $L$ is given four basepoints.

Given a collection of points $p$ as well as doubling arcs, as above, the construction from Section~\ref{sec:expanded-involution-knots} adapts verbatim to produce a model of the link involution. It remains to relate the models for different choices of $p$.

Suppose $=(Σ,,,,)$ is a Heegaard link diagram, and $p$ is a subset of $Σ∖(∪∪∪)$, and $q$ is a point on $Σ∖(∪∪∪∪p)$. Suppose that  $_p$ and $_p∪{q}$ are doubling enhancements of $$, which use the connected sum points $p$ and $p∪{q}$, respectively. We now define \emph{expansion} and \emph{contraction} morphisms
\[
\scE_{q}\colon \cCFKI(\frD_{\ve{p}})\to \cCFKI(\frD_{\ve{p}\cup \{q\}})\quad \text{and} \quad \scC_{q}\colon \cCFKI(\frD_{\ve{p}\cup \{q\}})\to \cCFKI(\frD_{\ve{p}}).
\]

We define the map $_q$ in Equation~\eqref{eq:def-E_p}. Our definition of $_q$ is similar to our definition of the 1-handle map, and $_q$ is defined similarly to the 3-handle map.
\begin{equation}
\scE_q:= \begin{tikzcd}[column sep=2cm, row sep=2cm, labels=description]
\cCFK(\as,\bs)
\ar[rr, "\id"]
\ar[d,"F_1^{c_p\bar \b,c_p\bar \b}"]
\cCFK(\as,\bs)
\ar[d, "F_1^{c_p c_q \bar \b,c_p c_q\bar \b}"]
\\
\cCFK(\as\ve{c}_{\ve{p}}\bar \bs,\bs\ve{c}_{\ve{p}}\bar \bs)
\ar[r, "F_1^{c_q,c_q}"]
\ar[d, "f_{\a c_p \bar \b }^{\b c_p \bar \b\to \Dt;\zs}"]
\cCFK(\as\ve{c}_{\ve{p}}c_{q}\bar \bs,\bs\ve{c}_{\ve{p}}c_q\bar \bs)
\ar[r, "\id"]
\ar[d, "f_{\a c_p c_q \bar \b}^{\b c_p c_q \bar \b\to \Dt c_q;\zs}"]
\ar[dr,dashed]
\cCFK(\as\ve{c}_{\ve{p}}c_{q}\bar \bs,\bs\ve{c}_{\ve{p}}c_q\bar \bs)
\ar[d, "f_{\a c_p c_q \bar \b}^{\b c_p c_q \bar \b\to \Dt';\zs} "]
\\
\cCFK(\as\ve{c}_{\ve{p}}\bar \bs,\Ds)
\ar[r, "F_1^{c_q,c_q}"]
\ar[d,"f_{\a c_p \bar \b}^{\Dt\to \a c_p \bar \a;\ws}"]
\cCFK(\as\ve{c}_{\ve{p}}c_q\bar \bs,\Ds c_q)
\ar[r, "f_{\a c_p c_q \bar \b}^{\Dt c_q\to \Dt'}"]
\ar[d, "f_{\a c_p c_q \bar \b}^{\Dt c_q\to \a c_p c_q \bar \a;\ws}"]
\ar[dr,dashed]
\cCFK(\as\ve{c}_{\ve{p}}c_q\bar \bs,\Ds')
\ar[d,"f_{\a c_p c_q \bar \b}^{\Dt'\to \a c_p c_q \bar \a;\ws}"]
\\
\cCFK(\as\ve{c}_{\ve{p}}\bar \bs,\as \ve{c}_{\ve{p}} \bar \as)
\ar[r, "F_1^{c_q,c_q}"]
\ar[d,"F_3^{a c_p,a c_p}"]
\cCFK(\as\ve{c}_{\ve{p}}c_q\bar \bs,\as \ve{c}_{\ve{p}} c_q \bar \as)
\ar[r, "B_\tau"]
\cCFK(\as\ve{c}_{\ve{p}}c_q\bar \bs,\as \ve{c}_{\ve{p}} c_q \bar \as)
\ar[d, "F_3^{\a c_p c_q, \a c_p c_q}"]
\\
\cCFK(\bar \bs, \bar \as)
\ar[rr, "\id"]
\cCFK(\bar \bs, \bar \as)
\end{tikzcd}
\label{eq:def-E_p}
\end{equation}
In the definition of $_p$, we write $τ$ for a closed curve on $Σ#_p∪{q} Σ̅$ which is symmetric with respect to the $_2$-symmetry, and has a single transverse intersection with $c_q$.

\begin{lem}\,
\begin{enumerate}
\item\label{eq:E-C-1} The maps $\scE_{q}$ and $\scC_{q}$ commute with the transition maps for handleslides and isotopies of the attaching curves. Furthermore, $\scE_q$ and $\scC_q$ are independent of the choice of $\tau$, up to chain homotopy.
\item\label{eq:E-C-2} If $p\neq q$, then
\[
[\scE_p,\scC_q]\simeq 0,\quad [\scE_p,\scE_q]\simeq 0\quad \text{and} \quad [\scC_p,\scC_q]\simeq 0.
\]
\item\label{eq:E-C-3} $\scE_p \circ \scC_p\simeq\id$ and $\scC_p\circ \scE_p\simeq \id$.
\item\label{eq:E-C-4} If $\phi$ is a surface isotopy of $\Sigma$ which moves $p$ to $q$ and fixes all of the other connected sum points and basepoints, then
\[
\phi_*\simeq \scC_p\circ \scE_q.
\]
\end{enumerate}
\end{lem}
\begin{proof}
The proof of part~\eqref{eq:E-C-1} is essentially the same as for the 1-handle and 3-handle maps. See Lemmas~\ref{lem:1-handle-map-ind-tau} and~\ref{lem:1-handles-and-handleslides}.

The proofs of claims~\eqref{eq:E-C-2} and \eqref{eq:E-C-3} amount to constructing five hyperboxes (one for each relation). All of these hyperboxes have a similar flavor, so we describe the hyperbox for the relation $\scC_q\circ \scE_q\simeq \id$, and leave the remaining hyperboxes to the reader.

The hyperbox we build will correspond to the hypercube
\[
\begin{tikzcd}[labels=description]\cCFKI(\frD_{\ve{p}})\ar[r, "\scE_q"] \ar[d, "\id"] \ar[dr,dashed]& \cCFKI(\frD_{\ve{p}\cup \{q\}}) \ar[d, "\scC_q"]\\
\cCFKI(\frD_{\ve{p}})\ar[r, "\id"]& \cCFKI(\frD_{\ve{p}})
\end{tikzcd}
\]
We build four constituent hyperboxes, each corresponding to one of the four levels of Equation~\eqref{eq:def-E_p}. The highest level is the following diagram:
\[
\begin{tikzcd}[
column sep={2cm,between origins},
row sep=.8cm,
fill opacity=.7,
text opacity=1,
\cCFK(\as,\bs)
\ar[dr, "\id"]
\ar[ddd, "F_1^{c_p\bar \b,c_p \bar \b}"]
\ar[rr, "\id"]
&[.7 cm]
\ar[rd, "\id"]
\ar[ddd,"F_1^{c_pc_q\bar \b,c_pc_q \bar \b}"]
&[.7 cm]
\\
&[.7 cm]
\cCFK(\as,\bs)
\ar[rr,line width=2mm,dash,color=white,opacity=.7]
\ar[rr, "\id"]
&[.7 cm]
\cCFK(\as, \bs)
\ar[ddd,"F_1^{c_p\bar \b,c_p \bar \b}"]
\\
\\
\cCFK(\as\ve{c}_{\ve{p}}\bar \bs, \bs \ve{c}_{\ve{p}} \bar \bs)
\ar[dr, "\id"]
\ar[rr, "F_1^{c_q,c_q}"]
&[.7 cm]\,&
\cCFK(\as \ve{c}_{\ve{p}} c_q \bar \bs, \bs \ve{c}_{\ve{p}} c_q \bar \bs)
\ar[dr, "F_3^{c_q,c_q}\circ B_\tau"]
\\
& [.7 cm]
\cCFK(\as \ve{c}_{\ve{p}}\bar \bs, \bs\ve{c}_{\ve{p}} \bar \bs)
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu,"F_1^{\bar \b, \bar \b}"]
\ar[rr, "\id"]
&[.7 cm]
\cCFK(\as \ve{c}_{\ve{p}}\bar \bs, \bs \ve{c}_{\ve{p}}\bar \bs)
\end{tikzcd}
\]
The bottom-most level is very similar to the above, so we omit it.

We now consider the hyperbox corresponding to the second level from the top in Equation~\eqref{eq:def-E_p}:
\begin{equation}
\begin{tikzcd}[
column sep={2.2cm,between origins},
row sep={1.5cm,between origins},
fill opacity=.8,
text opacity=1,
execute at end picture={
\foreach \Number in  {A,B,C,D,E}
{\coordinate (\Number) at (\Number.center);}
\filldraw[black] (A) node[rectangle,fill=white, text opacity=1]{$\cCFK(\as \ve{c}_{\ve{p}} \bar \bs, \bs \ve{c}_{\ve{p}} \bar \bs)$};
\filldraw[black] (B) node[rectangle,fill=white, text opacity=1]{$\cCFK(\as \ve{c}_{\ve{p}} c_q \bar \bs, \bs \ve{c}_{\ve{p}} c_q \bar \bs)$};
\filldraw[black] (C) node[rectangle,fill=white, text opacity=1]{$\cCFK(\as \ve{c}_{\ve{p}} c_q \bar \bs, \bs \ve{c}_{\ve{p}} c_q \bar \bs)$};
\filldraw[black] (D) node[rectangle,fill=white, text opacity=1]{$\cCFK(\as \ve{c}_{\ve{p}} \bar \bs, \bs \ve{c}_{\ve{p}} \bar \bs)$};
\filldraw[black] (E) node[rectangle,fill=white, text opacity=1]{$\cCFK(\as \ve{c}_{\ve{p}} \bar \bs, \bs \ve{c}_{\ve{p}} \bar \bs) $};
|[alias=A]|\phantom{\cCFK(\as \ve{c}_{\ve{p}} \bar \bs, \bs \ve{c}_{\ve{p}} \bar \bs)}
\ar[ddd]
\ar[rr, "F_1^{c_q,c_q}"]
\ar[ddrr, "\id"]
\cCFK(\as \ve{c}_{\ve{p}}c_{q}\bar \bs, \bs \ve{c}_{\ve{p}}c_{q} \bar \bs)
\ar[rr, "\id"]
\ar[ddd, shift right]
\ar[dr, "B_\tau"]
\ar[ddddr,dashed]
\ar[dddrr,dashed]
\cCFK(\as \ve{c}_{\ve{p}}c_{q}\bar \bs, \bs \ve{c}_{\ve{p}}c_{q} \bar \bs)
\ar[dr, "B_\tau"]
\ar[ddd]
\ar[ddddr,dashed]
\\
|[alias=B]|\phantom{\cCFK(\as \ve{c}_{\ve{p}} c_q \bar \bs, \bs \ve{c}_{\ve{p}} c_q \bar \bs)}
\ar[rr, "\id", pos=.2]
\ar[ddd]
\ar[dr, "F_3^{c_q,c_q}"]
|[alias=C]|\phantom{\cCFK(\as \ve{c}_{\ve{p}} c_q \bar \bs, \bs \ve{c}_{\ve{p}} c_q \bar \bs)}
\ar[dr, "F_3^{c_q,c_q}"]
\ar[ddd]
\\
&&|[alias=D]|\phantom{\cCFK(\as \ve{c}_{\ve{p}} \bar \bs, \bs \ve{c}_{\ve{p}} \bar \bs)}
\ar[rr, "\id",pos=.7]
|[alias=E]|\phantom{\cCFK(\as \ve{c}_{\ve{p}} \bar \bs, \bs \ve{c}_{\ve{p}} \bar \bs)}
\ar[rr, "\id",pos=.7]
\cCFK(\as \ve{c}_{\ve{p}} \bar \bs, \bs \ve{c}_{\ve{p}} \bar \bs)
\\
\cCFK(\as \ve{c}_{\ve{p}} \bar \bs, \Ds)
\ar[ddrr, "\id"]
\ar[rr, "F_1^{c_q,c_q}"]
\cCFK(\as \ve{c}_{\ve{p}} c_q \bar \bs, \Ds c_q)
\ar[rr]
\ar[dr, "B_\tau"]
\ar[drrr,dashed]
\cCFK(\as \ve{c}_{\ve{p}} c_q \bar \bs, \Ds')
\ar[dr]
\\
\cCFK(\as \ve{c}_{\ve{p}} c_q \bar\bs, \Ds c_q )
\ar[rr, "\id"]
\ar[dr, "F_3^{c_q,c_q}"]
\cCFK(\as \ve{c}_{\ve{p}} c_q \bar \bs, \Ds c_q)
\ar[dr, "F_3^{c_q,c_q}"]
\\
\,&&
\cCFK(\as \ve{c}_{\ve{p}} \bar \bs, \Ds)
\ar[rr,"\id"]
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu, pos=.6]
\cCFK(\as \ve{c}_{\ve{p}} \bar \bs, \Ds)
\ar[rr,"\id"]
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7, shift left]
\ar[from=uuu, shift left]
\cCFK(\as \ve{c}_{\ve{p}} \bar \bs, \Ds)
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu]
\end{tikzcd}
\label{eq:main-hypercube-C_q-E_q=id}
\end{equation}
In Equation~\eqref{eq:main-hypercube-C_q-E_q=id}, each unlabeled arrow is a triangle or quadrilateral map. The back right hypercube also has a length 3 map, which is not labeled. We now explain the back-right hypercube. This hypercube is obtained by pairing $\as \ve{c}_{\ve{p}} c_q \bar \bs$ with the following hypercube of beta attaching curves:
\begin{equation}
\begin{tikzcd}[
column sep={2cm,between origins},
row sep=.8cm,
fill opacity=.7,
text opacity=1,
\bs \ve{c}_{\ve{p}} c_q \bar \bs
\ar[dr, "\tau"]
\ar[ddd, "\Theta_{\b c_p c_q \bar \b, \Dt c_q}^{\zs}"]
\ar[rr, "1"]
\ar[dddrr,dashed]
\ar[ddddrrr,dotted]
&[.7 cm]
&\bs \ve{c}_{\ve{p}} c_q \bar \bs
\ar[rd, "\tau"]
\ar[ddd,"\Theta_{\b c_p c_q \bar \b, \Dt'}^{\zs}"]
\ar[ddddr,dashed]
&[.7 cm]
\\
&[.7 cm]
\bs \ve{c}_{\ve{p}} c_q \bar \bs
\ar[rr,line width=2mm,dash,color=white,opacity=.7]
\ar[rr, "1"]
&[.7 cm]
\bs \ve{c}_{\ve{p}} c_q \bar \bs
\ar[ddd,"\Theta_{\b c_p c_q \bar \b, \Dt c_q}^{\zs}"]
\\
\\
\Ds c_q
\ar[dr, "\tau"]
\ar[rr, "\Theta_{\Dt c_q,\Dt'}", pos=.4]
\ar[drrr,dashed, "\eta"]
&[.7 cm]\,&
\Ds'
\ar[dr, "\Theta_{\Dt',\Dt c_q}"]
\\
& [.7 cm]
\Ds c_q
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu,"\Theta_{\b c_p c_q \bar \b, \Dt c_q}^{\zs}"]
\ar[rr, "1"]
&[.7 cm]
\Ds c_q
\end{tikzcd}
\label{eq:3-d-hyperucbe-[C,E]}
\end{equation}
The above hypercube of beta attaching curves is constructed as follows. Length 2 chains are constructed by adapting Lemma~\ref{lem:1-handle-bottom-hypercube}. Once these are chosen, a length 3 chain may be chosen by the standard cube-filling procedure. Namely, if $C$ is the sum of all other terms of the length 3 relation, we know $\d C=0$ because of the hypercube relations on the other proper faces, and $C$ is in $\cCFK(\bs \ve{c}_{\ve{p}} c_q \bar \bs, \Ds c_q)$, a complex for an unlink in a connected sum of several copies of $S^1\times S^2$, of $\gr_{\zs}$-grading equal to that of the top degree generator of homology. By potentially adding a copy of the top degree generator to the 2-chain labeled $\eta$, in Equation~\eqref{eq:3-d-hyperucbe-[C,E]}, the hypercube relations may be satisfied.

The hyperbox for the third level of Equation~\eqref{eq:def-E_p} is constructed very similarly. Indeed, if we switch the roles of the alpha and beta curves, then the hypercube for the third level of Equation~\eqref{eq:def-E_p} is obtained by dualizing Equation~\eqref{eq:main-hypercube-C_q-E_q=id}. Stacking these hyperboxes and compressing gives the relation $\scC_q\circ \scE_q\simeq \id$.

One important point to check is that the diagonal maps on the second and third hypercubes can be chosen to match. For example, to build the hypercube in Equation~\eqref{eq:main-hypercube-C_q-E_q=id}, we may potentially have to add the top degree generator to $\eta$ so that the hypercube relations are satisfiable. A similar issue occurs in constructing the length 2 map on the top face of the third hypercube as well. Nonetheless, as in the proof of Lemma~\ref{lem:1-handles-and-handleslides}, addition of the top degree generator to $\eta$ in Equation~\eqref{eq:3-d-hyperucbe-[C,E]} has no effect on the diagonal map in the compression of Equation~\eqref{eq:main-hypercube-C_q-E_q=id}. Hence, the compression of Equation~\eqref{eq:main-hypercube-C_q-E_q=id} may be stacked on top of the corresponding hypercube below it.

We now consider part~\eqref{eq:E-C-4}. This follows from the first claims, as we now explain. Let $\lambda$ be a path from $p$ to $q$. Let $\phi_{q\to p}$ denote the surface isotopy which moves $q$ to $p$ along $\lambda$, and is supported in a small neighborhood of $\lambda$. It is sufficient to show that
\[
\phi_{q\to p}\circ \scC_p\circ \scE_q\simeq \id.
\]
Let $x_0$ be a point in the interior of $\lambda$. We compute as follows, using the previous relations:
\[
\begin{split}
\phi_{q\to p}\circ \scC_p\circ \scE_q&\simeq  \phi_{q\to p}\circ \scC_p\circ \scC_{x_0} \circ \scE_{x_0}\circ  \scE_q\\
&\simeq \phi_{q\to p} \circ \scC_{x_0} \circ\scE_{q} \circ \scC_{p}\circ \scE_{x_0}\\
&\simeq  \scC_{x_0} \circ \phi_{q\to p} \circ\scE_{q} \circ \scC_{p}\circ \scE_{x_0}\\
&\simeq \scC_{x_0} \circ \scE_{p} \circ \scC_{p}\circ \scE_{x_0}\\
&\simeq \id.
\end{split}
\]
Note that commuting $\phi_{q\to p}$ past $\scC_{x_0}$ in the second line is possible since the diffeomorphism $\phi_{q\to p}$ is isotopic to one which fixes $x_0$. Next, $\phi_{q\to p}\circ \scE_q\simeq \scE_p$ by naturality of the expansion and contraction maps with respect to changes of attaching curves, which is part~\eqref{eq:E-C-1} of the statement.  This establishes part~\eqref{eq:E-C-4}.
\end{proof}

\subsection{Functoriality for knots and links}

In this section, we describe several aspects of the proof of Theorem~\ref{thm:functoriality}. Firstly, to construct cobordism maps for $(W,Σ)(Y_1,L_1)→(Y_2,L_2)$ when $Σ$ is a collection of annuli such that each component of $Σ$ intersects $Y_1$ and $Y_2$ non-trivially, it is sufficient to describe cobordism maps on link Floer homology for 4-dimensional handles attached to the complement of a link.

Cobordism maps for 4-dimensional handles attached in the complement of a link are defined similarly to the maps for cobordisms between closed 3-manifolds (cf. \cite{ZemCFLTQFT}*{Section~5}). The 1-handle and 3-handle maps are defined as in Section~\ref{sec:1-and-3-handles}, and the 2-handle maps are defined similarly to Section~\ref{sec:2-handles}. One caveat is that there is an asymmetry in the definition of the 2-handle maps with respect to the basepoints, in that we choose the 2-handle maps to count holomorphic triangles which represent classes satisfying $_(ψ)=$. Note that by \cite{ZemAbsoluteGradings}*{Lemma~3.8}, we have
\[
\]
In particular, the final hypercube we use in constructing the cobordism maps for our 2-handle map will have the following form:
\[
\begin{tikzcd}[column sep=3cm, labels=description]
\cCFL(\bar \Sigma, \bar \bs, \bar \as)
\ar[r, "f_{\bar \b;\frs}^{\bar \a\to \bar \a'}"]
\ar[d, "\eta_L"]
\cCFL(\bar \Sigma, \bar \bs, \bar \as')
\ar[d,"\eta_L"]
\\
\cCFL(\Sigma,\as, \bs)
\ar[r, "f_{\a\to \a', \bar \frs+\PD[\Sigma]}^{\b}"]
\cCFL(\Sigma,\as',\bs)
\end{tikzcd}
\]

\section{2-handle maps and the knot surgery formula}
\label{sec:knot-surgery}

In this section, we compute another family of involutive cobordism maps. If $K⊂S^3$, and $n∈$, we will compute the map for the 2-handle cobordism from $S_2n^3(K)$ to $S^3$. Our description will be in terms of the knot surgery formula of \cite{HHSZExact}, and is in a similar spirit to the non-involutive description of the 2-handle map of Ozsv\'{a}th and Szab\'{o} \cite{OSIntegerSurgeries}*{Theorem~1.1}.

We suppose that $K⊂S^3$ is a knot, and we consider the 2-handle cobordism $W'_m(K)$ from $S^3_m(K)$ to $S^3$. This cobordism is $$ if and only if $m$ is even. Furthermore, if $m=2n$, then the unique self-conjugate $^c$ structure on $W'_2n(K)$ is the one which satisfies $⟨c_1(),Σ⟩=0$, where $Σ$ is obtained by capping $K$ with the core of the 2-handle. This $^c$ structure restricts to $[n]∈^c(S^3_2n(K))$ under the isomorphism $^c(S^3_2n(K))/2n$.

Our formula is phrased in terms of the mapping cone formula of Ozsv\'{a}th and Szab\'{o} \cite{OSIntegerSurgeries} and its involutive analog from \cite{HHSZExact}. We review these constructions briefly. We write $A_s(K)$ for the subcomplex of $^∞(K)$ generated by elements $^i ^j x$ where $A(x)+j-i=0$ and $j≥-s$ and $i≥0$. We write $B_s(K)$ for the subspace generated by similar monomials, with no restriction on $j$. We write $B̃_s(K)$ for the subspace generated by monomials which satisfy $j≥-s$, but with no restriction on $i$. We write $v_sA_s→B_s$ and $ṽ_sA_s→B̃_s$ for the canonical inclusions. We recall from \cite{OSIntegerSurgeries} that the mapping cone formula requires a choice of homotopy equivalence from $B̃_s(K)$ to $B_s+m(K)$. We write $_m$ for this map. We write $(K)$ for the direct product over $s$ of $A_s(K)$, tensored with the power series ring $[[U]]$. Similarly $(K)$ denotes the direct product of the complexes $B_s(K)$, tensored with $[[U]]$.

We recall that Ozsv\'{a}th and Szab\'{o} proved that if $K⊂S^3$ and $m∈$, then
\[
\ve{\CF}^-(S_m^3(K))\iso \Cone(v+h_m\colon \bA(K)\to \bB(K)),
\]
where $h_m=_mṽ$. In \cite{HHSZExact}, we extended this to the involutive setting, and proved that
\[
\ve{\CFI}^-(S^3_m(K))\simeq \begin{tikzcd}[column sep=2cm]\bA(K)\ar[dr,dashed, "H_m\tilde v"]\ar[r, "v+h_m"]\ar[d, "Q(\id+\iota_{\bA})"]& \bB(K)\ar[d, "Q(\id+\iota_{\bB})"]\\
Q\bA(K)\ar[r, "v+h_m"]& Q \bB(K)
\end{tikzcd}
\]
We refer the reader to \cite{HHSZExact}*{Section~3.5} for the definitions of $ι_$ and $ι_$. The map $H_m$ is defined by picking a map $H_mB̃_s(K)→B_-s(K)$ satisfying
\[
U^s\iota_K+\frF_m U^s \iota_K \frF_m=[\d, H_m].
\]
See \cite{HHSZExact}*{Section~3.5}.

With the above notation in place, we now state our formula for the 2-handle map. Once again, because one end of the cobordism is the three-sphere, the formula is not affected by choices of framings, and we omit them from the notation.
\begin{thm}\label{thm:knot-surgery-cobordism-map} Let $\frs$ be the unique self-conjugate $\Spin^c$ structure on the 2-handle cobordism $W'_{2n}(K)$ from $S^3_{2n}(K)$ to $S^3$. Write $\mathit{BI}_n(K):=\Cone(Q(\id+\iota_{\bB})\colon B_n(K)\to Q B_n(K))$, and let
\[
J\colon \bXI_n(K)\to \BI_n(K)
\]
be the map $J:=v_n \Pi^A_n+Q\Pi_n^B\iota_{\bX}$, where $\Pi_s^A$ is projection onto $A_s\oplus Q A_s$ and $\Pi_s^B$ is similar. Then there is a hypercube of $\bF[U,Q]/Q^2$-equivariant maps
\[
\begin{tikzcd}[row sep=2cm, column sep=3cm, labels=description]
\CFI^-(S_{2n}^3(K)) \ar[dr,dashed] \ar[d,"\simeq"]\ar[r, "{\CFI(W'_{2n}(K),\frs)}"]& \CFI^-(S^3) \ar[d, "\simeq"]\\
\bXI_{2n}(K)\ar[r, "J"]& \BI_n(K)
\end{tikzcd}
\]
\end{thm}
\begin{rem} In Theorem~\ref{thm:knot-surgery-cobordism-map}, $\iota_{\bX}$ denotes the involution on the mapping cone, i.e. $\iota_{\bX}=\iota_{\bA}+\iota_{\bB}+H_{2n} \tilde v$. Note $\iota_{\bA}$ vanishes when composed with $\Pi_n^B$, and hence makes no contribution to the map $J$.
\end{rem}

We will focus on the case that $n≠0$. The case that $n=0$ is similar.

We begin by considering the non-involutive analog of the above theorem, which is similar to \cite{OSIntegerSurgeries}*{Theorem~1.1}.  Ozsv\'{a}th and Szab\'{o} constructed a map from $^-(S^3_2n(K))$ to $(^-(S^3_2n+m(K))→^-(S^3))$, which we think of as a hypercube
\begin{equation}
\begin{tikzcd}
\ve{\CF}^-(S^3_{2n}(K))\ar[d] \ar[dr,dashed, "k"]\\
\ve{\CF}^-(S^3_{2n+m}(K))\ar[r]& \underline{\ve{\CF}}^-(S^3).
\label{eq:surgery-quasi-iso-main}
\end{tikzcd}
\end{equation}
In the above, $^-(S^3)$ denotes a version of twisted Floer homology which is isomorphic to  $^-(S^3)⊗[T]/(T^m-1)$.

We may consider instead a version of the above hypercube with twisted coefficients lying in $[T]$.   We will write $^-(S^3)$ for this version. We can also construct an analog of Equation~\eqref{eq:surgery-quasi-iso-main} over $[T]$, though the resulting hypercube has a different shape:
\begin{equation}
\begin{tikzcd}
\ve{\CF}^-(S^3_{2n}(K))\ar[d] \ar[dr,dashed, "k"] \ar[r, "F"]& \tilde{\ve{\CF}}{}^-(S^3)\ar[d, "M"]\\
\ve{\CF}^-(S^3_{2n+m}(K))\ar[r, "G"]& \tilde{\ve{\CF}}{}^-(S^3).
\label{eq:surgery-quasi-iso-pre-quotient}
\end{tikzcd}
\end{equation}
In Equation~\eqref{eq:surgery-quasi-iso-pre-quotient}, $F$ denotes
\[
F=\sum_{\frs\in \Spin^c(W_{2n}'(K))} T^{A_{W_{2n}',\Sigma}(\frs)}\CF(W_{2n}'(K),\frs)
\]
\[
A_{W_{2n}',\Sigma}(\frs):=\frac{\langle c_1(\frs),\widehat \Sigma\rangle +2n}{2}.
\]
\[
M=\sum_{s\in 2\Z+1} U^{m(s^2-1)/8}T^{m(s+1)/2}.
\]

Note that $M=0$ if we set $T^m=1$. Hence, the hypercube in Equation~\eqref{eq:surgery-quasi-iso-main} is obtained from Equation~\eqref{eq:surgery-quasi-iso-pre-quotient}   if we set $T^m=1$ and remove the top right corner of the hypercube.

The formula for $F$ is derived as follows. Ozsv\'{a}th and Szab\'{o} proved Equation~\eqref{eq:surgery-quasi-iso-main} by considering a Heegaard quadruple $(Σ,_3,_2,_1,,w,z)$, with a special genus 1 surgery region, as in \cite{OSIntegerSurgeries}*{Section~3}. In the diagrams $(Σ,_1,)$ and $(Σ,_2,)$, the point $w$ and $z$ are immediately adjacent. The diagram $(Σ,_1,,w)$ represents $S^3_2n(K)$. The diagram $(Σ,_2,,w)$ represents $S^3_2n+m(K)$. The diagram $(Σ,_3,,w,z)$ represents $(S^3,K)$. One obtains the hypercube in Equation~\eqref{eq:surgery-quasi-iso-pre-quotient} by considering the degenerations of holomorphic quadrilaterals. The quantity $U^m(s^2-1)/8T^m(s+1)/2$ arises from a model count of holomorphic triangles on the diagram $(Σ,_3,_2,_1,w,z)$ (see \cite{OSIntegerSurgeries}*{Section~3}).

We now fix $δ≫0$, and we set $U^δ=0$. Write $^δ$ for $^-/U^δ$, and make similar notation for maps. We assume that $m≫0$, so the only terms of $F$ with $U$-power less than $δ$ have $s∈{1,-1}$. Hence
\[
\\ \frs\in \Spin^c(W_{2n}'(K))} T^{A_{W_{2n}',\Sigma}(\frs)} \CF^\delta(W_{2n}'(K),\frs).
\]

On $D(-m,1)$, $^c$ structures may be identified with $2+1$, where $s∈2+1$ corresponds to the $^c$ structure $$ satisfying
\[
\langle c_1(\frt),S^2\rangle=s\cdot m.
\]
We are most interested in the $^c$ structures corresponding to $s=±1$, which are the $^c$ structures with maximal square. Note that the composition of the natural cobordisms from $S_2n^3(K)$ to $S^3_2n+m(K)$ (which has an extra boundary component $L(m,1)$) and the cobordism from $S^3_2n+m(K)$ to $S^3$ is $W'_2n(K)# D(-m,1)$. We identify $^c$ structures on $W_2n'(K)#D(-m,1)$ as pairs $#[s]$ where $∈^c(W'_2n(K))$ and $s∈2+1$.

We recall the notation of Ozsv\'{a}th and Szab\'{o} for $^c$ structures on $W_2n'(K)$. They write $_s$ and $_s$ for the $^c$ structures which satisfy
\[
\langle c_1(\frx_s),\Sigma\rangle +2n=2s\quad \text{and} \quad \langle c_1(\fry_s),\Sigma\rangle -2n=2s,
\]

There are also analogous $^c$ structures on $W_2n+m'(K)$, for which we write $_s$ and $_s$. The $^c$ structure $_s$ satisfies $⟨c_1(_s),Σ⟩+2n+m=2s$ and $_s$ is similar. We write
\[
G_{\frX_s},G_{\frY_s}\colon \ve{\CF}^-(S_{2n+m}^3(K))\to \ve{\CF}^-(S^3)
\]
for the corresponding cobordism maps.

If $s∈$, we may define a projection map $Π_s^δ (S^3)→^δ(S^3)$ by reading off only the component $T^s$. Using this map, we may form a hypercube
\[
\begin{tikzcd}
\CF^\delta(S^3_{2n}(K),[s])
\ar[d]
\ar[dr,dashed, "k"]
\ar[r, "F"]
& \tilde{\CF}{}^\delta(S^3)
\ar[d, "M"]
\ar[r, "\Pi_s M"]
\ar[d, "\id"]
\\
\CF^\delta(S^3_{2n+m}(K))
\ar[r, "G"]
\ar[r, "\Pi_s"]
\end{tikzcd}
\]
If we compress the above hyperbox, we obtain the following diagram:
\[
\begin{tikzcd}[column sep=3cm]
\CF^\delta(S^3_{2n}(K),[s])
\ar[d]
\ar[dr,dashed, "\Pi_s k"]
\ar[r, "{\CF(W'_{2n},\frx_s)}"]
\CF^\delta(S^3)
\ar[d, "\id"]
\\
\CF^\delta(S^3_{2n+m}(K))
\ar[r, "G_{\frX_s}"]
\CF^\delta(S^3)
\end{tikzcd}
\]
If we instead use $Π_s+2n+m$, we obtain a hypercube relating the maps $(W_2n',_s)$ and $G__s$.

The next step of the knot surgery formula is to construct a hypercube of the following form:
\begin{equation}
\begin{tikzcd}[column sep=1.5cm] \CF^\delta(S_{2n+m}^3(K)) \ar[r, "G_{\frX_s}+G_{\frY_s}"] \ar[d, "\Gamma"] \ar[dr,dashed,"j"]& \underline{\CF}^\delta(S^3) \ar[d, "\theta_w"]\\
\bA^\delta (K) \ar[r, "v+h_{2n}"]& \bB^\delta (K)
\end{tikzcd}
\label{eq:cone-cobordism->AB}
\end{equation}
The map $Γ$ is a holomorphic triangle counting map. The map $θ_w$ is the map for trivializing the twisted coefficients defining $^δ(S^3)$.

The above hypercube is built by gluing together hypercubes with the following shape:
\[
\begin{tikzcd} \CF^\delta(S_{2n+m}^3(K)) \ar[r, "G_{\frX_s}"] \ar[d, "\Gamma"]& \underline{\CF}^\delta(S^3) \ar[d, "\theta_w"]\\
\bA^\delta (K) \ar[r, "v"]& \bB^\delta (K)
\end{tikzcd}
\quad\text{and}\quad
\begin{tikzcd} \CF^\delta(S_{2n+m}^3(K)) \ar[r, "G_{\frY_s}"] \ar[d, "\Gamma"] \ar[dr,dashed,"j"]& \underline{\CF}^\delta(S^3) \ar[d, "\theta_w"]\\
\bA^\delta (K)\ar[r, "h_{2n}"]& \bB^\delta (K)
\end{tikzcd}
\]

We may combine the above hypercubes into a hyperbox of the following shape:
\begin{equation}
\begin{tikzcd}[column sep=3cm,labels=description]
\CF^\delta(S^3_{2n}(K),[s])
\ar[d, "{\CF^\delta(W_{2n,m})}"]
\ar[dr, "k"]
\ar[drr,dashed, "k_{\frx_s\#[- 1]}"]
\ar[rr, "{\CF(W_{2n}',\frx_s)}"]
& \CF^\delta(S^3)\ar[d, "\id"]\\
\bigg(\CF^\delta(S^3_{2n+m}(K))
\ar[d, "\Gamma"]
\ar[dr,"j"]
\ar[r, "G"]
\ar[rr, bend right=10, "G_{\frX_s}\Pi_{[s]}"]
&\underline{\CF}^\delta(S^3) \ar[d, "\theta_w"]\bigg)
\CF^\delta(S^3)\ar[d,"\theta_w"]
\\
\bigg(\bA^\delta(K)\ar[r, "v+h_{2n}"]
\ar[rr, bend right=10, "v\Pi^A_s"]
\bB^\delta(K)\bigg)& B_n^\delta(K)
\end{tikzcd}
\label{eq:non-involutive-cobordism-comp-1}
\end{equation}
In the above, $Π_[s]$ denotes projection of $^δ(S^3_2n+m(K)$ onto the $^c$ structure identified with $[s]∈/(2n+m)$.
Also, we view each parenthesized group as being a single point in the set $(1,2){0,1}×{0,1,2}$.

Symmetrically, there is another hypercube of the following form:
\begin{equation}
\begin{tikzcd}[column sep=3cm,labels=description]
\CF^\delta(S^3_{2n}(K),[s])
\ar[d, "{\CF(W_{2n,m})}"]
\ar[dr, "k"]
\ar[drr,dashed, "k_{\fry_s\#[1]}"]
\ar[rr, "{\CF(W_{2n}',\fry_s)}"]
& \CF^\delta(S^3)\ar[d, "\id"]
\\
\bigg(\CF^\delta(S^3_{2n+m}(K))
\ar[d, "\Gamma"]
\ar[dr,"j"]
\ar[drr,dashed, "\Pi_{s+2n+m}^B j"]
\ar[r, "G"]
\ar[rr, bend right=10, "G_{\frY_s}\Pi_{[s]}"]
&\underline{\CF}^\delta(S^3) \ar[d, "\theta_w"]\bigg)
& \CF^\delta(S^3)\ar[d,"\theta_w"] \\
\bigg(\bA^\delta(K)\ar[r, "v+h_{2n}"] \ar[rr, bend right=10, "h_{2n}\Pi^A_{s}"]& \bB^\delta(K)\bigg)& B_n^\delta(K)
\end{tikzcd}
\label{eq:non-involutive-cobordism-comp-2}
\end{equation}

Equations~\eqref{eq:non-involutive-cobordism-comp-1} and~\eqref{eq:non-involutive-cobordism-comp-2} can be used to compute the cobordism maps for $W_2n'(K)$, very similarly to Ozsv\'{a}th and Szab\'{o}'s \cite{OSIntegerSurgeries}*{Theorem~1.1}.

In particular, the fundamental objects to construct are the hypercubes in Equations~\eqref{eq:surgery-quasi-iso-main} and~\eqref{eq:surgery-quasi-iso-pre-quotient}. The involutive extension of Equation~\eqref{eq:surgery-quasi-iso-main} is one of the main theorems of \cite{HHSZExact}. We now describe how to extend Equation~\eqref{eq:surgery-quasi-iso-pre-quotient} to an involutive hypercube:
\begin{prop}\label{prop:tilde-hypercubes} There is a hypercube of the form:
\[
\begin{tikzcd}
\ve{\CFI}^-(S^3_{2n}(K))\ar[d] \ar[dr,dashed, "k"] \ar[r, "F"]& \tilde{\ve{\CFI}}{}^-(S^3)\ar[d, "M"]\\
\ve{\CFI}^-(S^3_{2n+m}(K))\ar[r, "G"]& \tilde{\ve{\CFI}}{}^-(S^3).
\end{tikzcd}
\]
Furthermore, $G$ and $F$ are the involutive cobordism maps summed over all $\Spin^c$ structures (as defined in this paper) and $M$ is the map $\sum_{s\in 2\Z+1} U^{m(s^2-1)/8}T^{m(s+1)/2}\cdot \id+EQ$. The map $E$ decomposes into a sum
\[
E=\sum_{s\in 2\Z+1} E_s
\]
where each $E_s$ is of homogeneous grading $-m(s^2-1)/4+1$.
\end{prop}

\begin{proof}The proof is mostly by a straightforward reinterpretation of the techniques of \cite{HHSZExact}. There is one point where our description in \cite{HHSZExact} is not convenient for our present purpose, which is in the first subcube of the central hypercube. We recall the Heegaard quintuple $(\gs'',\gs',\gs,\ds,\ds')$ used to define the first central hypercube. There is a natural hypercube, obtained by pairing two hypercubes (in \cite{HHSZExact}, we referred to this hypercube as the first \emph{auxiliary} hypercube). Over $\bF[T]$, this hypercube takes the following form:
\[
\begin{tikzcd}[
column sep={2cm,between origins},
row sep=1cm,
fill opacity=.7,
text opacity=1,
\ve{\CF}^-(\gs,\ds)
\ar[dr, "f_{\g\to \g'}^{\dt}"]
\ar[ddd, "f_{\g}^{\dt\to \dt'}"]
\ar[rr, "f_{\g\to \g''}^{\dt}"]
\ar[dddrr,dashed, "h_{\g\to \g''}^{\dt\to \dt'}"]
\ar[drrr,dashed, "h_{\g\to \g'\to \g''}^{\dt}"]
&[.7 cm]
\ar[dr, "M"]
\ar[ddd,"f_{\g''}^{\dt\to \dt'}"]
&[.7 cm]
\\
&[.7 cm]
\ve{\CF}^-(\gs',\ds)
\ar[rr,line width=2mm,dash,color=white,opacity=.7]
\ar[rr, "f_{\g'\to \g''}^{\dt}"]
&[.7 cm]
\tilde{\ve{\CF}}{}^-(\gs'',\ds)
\ar[ddd,"f_{\g''}^{\dt\to \dt'}"]
\\
\\
\tilde{\ve{\CF}}{}^-(\gs,\ds')
\ar[dr, "f_{\g\to \g'}^{\dt'}"]
\ar[rr, "f_{\g\to \g''}^{ \dt'}", pos=.4]
\ar[drrr,dashed, "h_{\g\to\g'\to \g''}^{\dt'}"]
&[.7 cm]\,&
\tilde{\ve{\CF}}{}^-(\gs'',\ds')
\ar[dr, "M"]
\\
& [.7 cm]
\tilde{\ve{\CF}}{}^-(\gs',\ds')
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu,"f_{\g'}^{\dt\to \dt'}"]
\ar[rr, "f_{\g'\to \g''}^{\dt'}"]
\ar[from=uuuul,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuuul,dashed, "h_{\g\to \g'}^{\dt\to \dt'}"]
&[.7 cm]
\tilde{ \ve{\CF}}{}^-(\gs'',\ds')
\ar[from=uuull,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuull,"h_{\g'\to \g''}^{\dt\to \dt'}", dashed, pos=.3]
\end{tikzcd}
\]
The length 3 map (not shown) is the pentagon counting map $p_{\g\to \g'\to \g''}^{\dt\to \dt'}$.

We recall that the diagram $(\gs',\ds'')$ admits a 3-handle map to a diagram $\cH_0$, and that both $(\gs,\ds')$ and $(\gs',\ds')$ are genus 1 stabilizations of $\cH_0$. Furthermore, the maps $f_{\g\to \g''}^{\dt'}$ and $f_{\g'\to \g''}^{\dt'}$ are the maps for surgering on a 0-framed unknot. Since the relevant Heegaard triples are both stabilizations, both of these maps admit left inverses, which are a 3-handle map followed by a stabilization. Write $\sigma$ for the stabilization map which has $(\gs,\ds')$ as its codomain, and write $\sigma'$ for the stabilization map which has $(\gs',\ds')$ as its codomain.

As a replacement for the diagram $\cC_{\textrm{cen}}^{(1)}$ in \cite{HHSZExact}, we build the following diagram:
\[
\begin{tikzcd}[
column sep={2.5cm,between origins},
row sep=1.4cm,
fill opacity=.7,
text opacity=1,
\ve{\CF}^-(\gs,\ds)
\ar[dr, "f_{\g\to \g'}^{\dt}"]
\ar[ddd, "\id"]
\ar[rr, "f_{\g\to \g''}^{\dt}"]
\ar[dddrr,dashed, "\sigma F_3 h_{\g\to \g''}^{\dt\to \dt'}"]
&[.7 cm]
\ar[dr, "M"]
\ar[ddd,"\sigma F_3 f_{\g''}^{\dt\to \dt'}"]
\ar[ddddr,dashed, "\sigma' F_3 h_1 \sigma F_3 f_{\g''}^{\dt\to \dt'}"]
&[.7 cm]
\\
&[.7 cm]
\ve{\CF}^-(\gs',\ds)
\ar[rr,line width=2mm,dash,color=white,opacity=.7]
\ar[rr, "f_{\g'\to \g''}^{\dt}"]
&[.7 cm]
\tilde{\ve{\CF}}{}^-(\gs'',\ds)
\ar[ddd,"\sigma' F_3 f_{\g''}^{\dt\to \dt'}"]
\ar[from=lllu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=lllu,dashed, "h_{\g\to \g'\to \g''}^{\dt}"]
\\
\\
\ve{\CF}^-(\gs,\ds)
\ar[dr, "f_{\g\to \g'}^{\dt}"]
\ar[rr, "f_{\g}^{\dt\to \dt'}", pos=.4]
\ar[drrr,dashed, "h_{\g\to \g'}^{\dt\to \dt'}"]
&[.7 cm]\,&
\tilde{\ve{\CF}}{}^-(\gs,\ds')
\ar[dr, "f_{\g\to \g'}^{\dt'}"]
\\
& [.7 cm]
\ve{\CF}^-(\gs',\ds)
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu,"\id"]
\ar[rr, "f_{\g'}^{\dt\to \dt'}"]
&[.7 cm]
\tilde{ \ve{\CF}}{}^-(\gs',\ds')
\ar[from=uuull,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuull,"\sigma' F_3 h_{\g'\to \g''}^{\dt \to \dt'}", dashed, pos=.3]
\end{tikzcd}
\]
The length 3 map (not shown) is $\sigma' F_3 p_{\g\to \g'\to \g''}^{\dt\to \dt'}+\sigma'F_3 h_1 \sigma F_3 h_{\g\to \g''}^{\dt\to \dt'}$. Following the notation of \cite{HHSZExact}, the map $h_1$ denotes $h_{\g\to \g'\to \g''}^{\dt'}$. The hypercube relations are straightforward to verify. Compare \cite{HHSZExact}*{Lemma~16.4}.

The hypercubes $\cC_{\mathrm{cen}}^{(2)}$ and $\cC_{\mathrm{cen}}^{(3)}$ are more straightforward to modify, and we leave the details to the reader. Upon compressing and stacking these hypercubes, we get the diagram in the statement. It remains to verify the listed claims. The identification of $J$ and $G$ as cobordism maps is straightforward, and we leave the details to the reader. Compare \cite{HHSZExact}*{Lemma~16.14}.

We now consider the map labeled $M$ in the statement, the length one component of this map (i.e. the non-$Q$ component) may be identified with $\sum_{s\in 2\Z+1} U^{m(s^2-1)/8}T^{m(s+1)/2}$ by the lattice point counting argument in \cite{OSIntegerSurgeries}*{Section~3}. It remains to identify the $Q$-term. The $Q$-term is essentially the same as the map $E$ considered in \cite{HHSZExact}*{Section~19}. Therein, we showed that $E$ is null-homotopic over $\bF[U,T]/(T^m-1)$. Our argument used at several places that the coefficients were in $\bF[U,T]/(T^m-1)$ instead of $\bF[U,T]$, so we give an alternate argument that is sufficient for our present purposes.

The first observation is that the map $E$ decomposes over $\Spin^c(D(-m,1))$, which we have already identified with $2\Z+1$. The grading change formula is also a routine consequence of this identification, and the standard grading change formulas for the cobordism maps in Heegaard Floer homology \cite{OSIntersectionForms}.
\end{proof}

\begin{rem}
We write $M=M_0+Q M_1$. Since $\CF^-(S^3)\simeq \bF[U]$ is supported only in even gradings, and $M_1$ may be viewed as a chain map from $\ve{\CF}^-(S^3)$ to itself which shifts the mod 2 grading by 1, we may conclude that $M_1\simeq 0$. In particular, by adding a term to the length 3 map of the hypercube, we may assume that $M_1=0$. Furthermore, fixing $\delta$ and letting $m$ be sufficiently large, we see that $E^\delta$ is homogeneously graded, ao the null-homotopy may also be taken to be homogeneously graded.
\end{rem}

In \cite{HHSZExact}*{Section~21}, we also described how to extend the hypercube in Equation~\eqref{eq:cone-cobordism->AB} into the involutive setting. Using this, together with Proposition~\ref{prop:tilde-hypercubes}, we may construct the involutive analog of the hyperboxes in Equation~\eqref{eq:non-involutive-cobordism-comp-1} and~\eqref{eq:non-involutive-cobordism-comp-2}. We construct this hyperbox so that the top face has the hyperbox from Equation~\eqref{eq:non-involutive-cobordism-comp-1}, which involves the $^c$ structures $_s$ and $_s$ with $s=n$. There is an important subtlety in that the bottom face of this hyperbox will be the hypercube from Equation~\eqref{eq:non-involutive-cobordism-comp-2}, involves the $^c$ structures $_-n$ and $_-n$. (Note that $_-n=_n$).

The front face of this hyperbox will compress to have the diagonal map $H_2nṽ$, where $H_2n$ is the lenth 2 map from the involutive mapping cone of \cite{HHSZExact}.

We add one more level to this hyperbox so that the top and bottom levels faces coincide. We add this level to the bottom of the hyperbox. We build this new level as a pair of hypercubes. The first hypercube has the following shape:
\[
\begin{tikzcd}[
column sep={2cm,between origins},
row sep=.8cm,
fill opacity=1,
text opacity=1,
\CF^\delta(S_{2n}^3(K),[n])
\ar[dr, "{\CF(W_{2n,2n+m})}"]
\ar[drr, "k"]
\ar[ddd, "\id"]
\ar[rrr, "{\CF(W_{2n}',\fry_{-n})}"]
\ar[drrrr,dashed, "k_{\fry_{-n}\#[1]}"]
&[.7 cm]&\CF^\delta(S^3)
\ar[rd, "\id"]
\ar[ddd,"\id", pos=.65]
&[.7 cm]
\\
&[.7 cm]
\bigg(\CF^\delta(S_{2n+m}(K))
\ar[r]
\ar[rrr,bend right=12,line width=2mm,dash,color=white,opacity=.7]
\ar[rrr,bend right=12, "G_{\frY_{-n}}\Pi_{[-n]}"]
\underline{\CF}^\delta(S^3)\bigg)
&[.7 cm]
\CF^\delta(S^3)
\\
\\
\CF^\delta(S_{2n}^3(K),[n])
\ar[dr, "{\CF(W_{2n,2n+m})}"]
\ar[rrr, "{\CF(W_{2n}',\frx_n)}"]
\ar[drr,"k"]
\ar[drrrr,dashed, pos=.4, "k_{\frx_n\#[-1]} "]
&[.7 cm]\,&&
\CF^\delta(S^3)
\ar[dr, "\id"]
\\
& [.7 cm]
\bigg(\CF^\delta(S_{2n+m}^3(K))
\ar[r]
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu,"\id"]
\ar[rrr, bend right=12, "G_{\frX_n}\Pi_{[n]}"]
\underline{\CF}^\delta(S^3)\bigg)
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu,"\id"]
&[.7 cm]
\CF^\delta(S^3)
\ar[from=uuu, "\id"]
\ar[from=lluuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=lluuu, dashed,"\Pi_{n}"]
\end{tikzcd}
\]
In the above $Π_n$ denotes projection onto the $B_n$-summand of $^-(S^3)$. The hypercube relations are straightforward.

We finally have one additional hypercube, as follows:
\[
\begin{tikzcd}[
column sep={3cm,between origins},
row sep=.8cm,
fill opacity=1,
text opacity=1,
\bigg(\CF^\delta(S^3_{2n+m}(K))
\ar[dr, "\Gamma"]
\ar[ddd, "\id"]
\ar[r]
\ar[rrr, bend left=12, "G_{\frY_{-n}}\Pi_{[-n]}"]
\ar[dr, "\theta_w"]
\ar[ddd,"\id", shift right=2mm, pos=.6]
\ar[dddrr,dashed, "\Pi_n", pos=.6]
\ar[rd, "\theta_w"]
\ar[ddd,"\id"]
\ar[r]
\ar[rrr,bend right=12,line width=2mm,dash,color=white,opacity=.7]
\ar[rrr, bend right=12, "h \Pi^A_{-n}"]
\bB^\delta(K)\bigg)
\ar[from=ull,crossing over,"j"]
\ar[from=ullll, crossing over, "\Pi^B_{n} j",dashed]
\\
\\
\bigg(\CF^\delta(S_{2n+m}^3(K))
\ar[dr, "\Gamma"]
\ar[r]
\ar[drr, "j"]
\ar[rrr,bend left=12, "G_{\frX_n} \Pi_{[n]}"]
\underline{\CF}^\delta(S^3)\bigg)
\ar[dr,"\theta_w"]
\,&&
\CF^\delta(S^3)
\ar[dr, "\theta_w"]
\\
& [.7 cm]
\bigg(\bA^\delta(K)
\ar[r]
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu,"\id", pos=.4]
\ar[rrr, bend right=12, "v\Pi^A_n"]
\bB^\delta(K)\bigg)
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu,"\id"]
&[.7 cm]
\ar[from=uuu, "\id"]
\ar[from=lluuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=lluuu, dashed,"\Pi^B_n", pos=.4]
\end{tikzcd}
\]
The above is easily check to be a hypercube of chain complexes, so the proof is complete.

\section{The $\SO(3)$-twist} \label{sec:twist}

In this section, we show that the diffeomorphism map associated to $∈π_1((3))/2$ acts on $(Y)$ by $+ QΦ$, proving Theorem~\ref{thm:twist-map}. Our argument will actually identify the map with $QΦι$, though this is homotopic to $QΦ$ on $(Y)$. In Section~\ref{sec:twisting-knots-links}, we will further consider an analogous twisting diffeomorphism map on involutive knot Floer homology, and show that it is trivial.

Before beginning with the computation, we recall several topological perspectives about $$. The first is as a Dehn twist along a small $S^2$ enclosing the basepoint. A regular neighborhood of such a sphere is $S^2×I$. We recall that $(S^2)≃O(3)$ so the generator of $π_1((3))$ gives a diffeomorphism of $S^2×I$ which is the identity on the boundary.

There is another perspective on $$, is as follows. Write $^+(Y,w)$ for the orientation preserving diffeomorphisms $ϕ$ such that $ϕ(w)=w$; likewise, write $^f(Y,w)$ for the orientation preserving diffeomorphisms $ϕ$ such that $ϕ(w)=w$ and $d_w ϕ=𝕀$. If we fix a trivialization of $T_w Y$, we obtain  a Serre fibration
\[
\Diff^f(Y,w)\to \Diff^+(Y,w)\to \SO(3),
\]
and hence a map $π_1((3))→π_0(^f(Y,w))$.

\subsection{Heegaard diagrams for the twist map}

We first describe the Heegaard diagrams used to define $_*$. Recall that in non-involutive Heegaard Floer homology, if $ϕΣ→Σ$ is a diffeomorphism, then the action of $ϕ_*$ on Heegaard Floer homology is induced by composing the maps in the following diagram:
\[
\begin{tikzcd}
\CF(\Sigma,\as,\bs)
\ar[r, "f_{\a\to \phi(\a)}^{\b\to \phi(\b)}"]&
\CF(\Sigma,\phi^{-1}(\as), \phi^{-1}(\bs))\ar[r, "T_\phi"]&
\CF(\Sigma,\as,\bs)
\end{tikzcd}
\]
where $f_→ϕ()^→̱ϕ()̱$ is a composition of holomorphic triangle maps and change of almost complex structure maps, and $T_ϕ$ is the tautological map on intersection points. The diffeomorphism maps on involutive Heegaard Floer homology are defined similarly, except that we replace $(Σ,,)$ with a one dimensional hyperbox (whose maps of non-zero length correspond to the involution), and we replace $f_→ϕ()^→̱ϕ()̱$ and $T_ϕ$ with morphisms of hyperboxes.

This has the following instantiation in our present setting. Let $δ$ be a set of doubling arcs (with boundary on $w$) and $δ̃$ be a set obtained by applying a boundary Dehn twist in a loop which encircles $w$. Let $$ and $$ be their corresponding doubled curves on $Σ# Σ̅$. Then $_*$ is obtained by compressing the following diagram (where we are omitting an extra column corresponding to the tautological portion of the diffeomorphism map):
\begin{equation}
\begin{tikzcd}
\CF(\Sigma,\as,\bs)
\ar[r]
\ar[d]&
\CF(\Sigma,\as,\bs)
\ar[d]
\\
\CF(\Sigma\# \bar \Sigma,\as \bar \bs, \bs \bar \bs)
\ar[r]
\ar[dr,dashed]
\ar[d]&
\CF(\Sigma\# \bar \Sigma,\as \bar \bs, \bs \bar \bs)
\ar[d]
\\
\CF(\Sigma\# \bar \Sigma,\as \bar \bs, \Ds)
\ar[r]
\ar[d]
\ar[dr,dashed]
\CF(\Sigma\# \bar \Sigma,\as \bar \bs, \tilde{\Ds})
\ar[d]
\\
\CF(\Sigma\# \bar \Sigma,\as \bar \bs, \as\bar \as)
\ar[r]
\ar[d]&
\CF(\Sigma\# \bar \Sigma,\as \bar\bs, \as \bar \as)
\ar[d]
\\
\CF( \bar \Sigma,\bar \bs, \bar \as)
\ar[r]
\CF(\bar \Sigma,\bar \bs, \bar \as)
\end{tikzcd}
\label{eq:Dehn-twist-map}
\end{equation}
The relevant Heegaard diagrams are shown in Figure~\ref{fig:twist_fig1}.

\begin{figure}[ht]
\input{twist_fig1.pdf_tex}
\caption{Top row, the doubling arcs $\boldsymbol{\delta}$ (left) and $\tilde{\boldsymbol{\delta}}$ (right). On the bottom row, we show the doubled diagrams, and indicate how $\tilde{\Ds}$ is obtained from $\Ds$ by moving the basepoint in a loop around the connected sum tube.}
\label{fig:twist_fig1}
\end{figure}

\begin{rem} Since $\tw_*$ can be interpreted in terms moving the basepoint around the connected sum neck on $\Sigma\# \bar \Sigma$, one might expect that the formula for the hypercube $\pi_1$-action from \cite{ZemBordered}*{Section~13} would show that $\tw_*$ is null-homotopic as a morphism of 1-dimensional hypercubes. That is, if we view $\g_*$ as a 2-dimensional diagram below, then there is a 3-dimensional hypercube taking the form shown on the right:
\[
\g_*=
\begin{tikzcd}[row sep=3cm]
\CF(Y)
\ar[r, "\g_*"]
\ar[dr,dashed, "\g_*"]
\ar[d, "\iota"]
& \CF(Y)
\ar[d, "\iota"]
\\
\CF(Y) \ar[r, "\g_*"] & \CF(Y)
\end{tikzcd}
\begin{tikzcd}[
column sep={1.3cm,between origins},
row sep=.8cm,
fill opacity=1,
text opacity=1,
\CF(Y)
\ar[dr, "\id"]
\ar[ddd, "\iota"]
\ar[rr, "\g_*"]
\ar[dddrr,dashed, "\g_*"]
\ar[ddddrrr,dotted]
&[.7 cm]
\ar[rd, "\id"]
\ar[ddd,"\iota"]
&[.7 cm]
\\
&[.7 cm]
\CF(Y)
\ar[rr,line width=2mm,dash,color=white,opacity=.7]
\ar[rr, "\id"]
&[.7 cm]
\CF(Y)
\ar[ddd,"\iota"]
\ar[from=ulll,line width=2mm,dash,color=white,opacity=.7]
\ar[from=ulll,dashed]
\\
\\
\CF(Y)
\ar[dr, "\id"]
\ar[rr, "\gamma_*"]
\ar[drrr,dashed]
&[.7 cm]\,&
\CF(Y)
\ar[dr, "\id"]
\\
& [.7 cm]
\CF(Y)
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu,"\iota"]
\ar[rr, "\id"]
% 	\ar[from=uuuul,line width=2mm,dash,color=white,opacity=.7]
% 	\ar[from=uuuul,dashed]
&[.7 cm]
\CF(Y)
\end{tikzcd}.
\]
We observe however that the techniques of \cite{ZemBordered} do not guarantee that the 3-dimensional hypercube so constructed has the same maps on the top and bottom faces of the hypercube. (This may seem surprising since the length one components of $\g_*$ are the identity on the nose). Concretely, the homotopy on the top face involves maps related to relative homology actions on stabilized diagrams, which are in general not preserved by the involution. In particular, such a hypercube does not give us an $\bF[U,Q]/Q^2$-equivariant homotopy $\id\simeq \tw_*$ on $\CFI(Y)$.
\end{rem}

\subsection{Degenerating connected sums}

Our argument depends on a somewhat complicated neck stretching argument. Our strategy is adapted from work of Ozsv\'{a}th and Szab\'{o} in the setting of bordered knot Floer homology \cite{OSBorderedHFK}. We have explored some of these ideas further in \cite{HHSZExact} and \cite{ZemBordered}.

We focus first on the case of two ordinary Heegaard diagrams $=(Σ,,,w)$ and $'=(Σ',',,',w')$, and we take the connected sum at $w$ and $w'$ (deleting one of the basepoints).

We first degenerate the connected sum tubes by letting the neck length approach $∞$. We obtain maps which count \emph{perfectly matched} moduli spaces. We write $_∧(# ')$ for the resulting chain complex.

There are homotopy equivalences
\[
\Psi^{\Sigma<\Sigma'}, \Psi^{\Sigma'<\Sigma}\colon \CF_{\wedge}(\cH\# \cH')\to \CF(\cH)\otimes_{\bF[U]} \CF(\cH').
\]

The map $Ψ^Σ<Σ'$ counts holomorphic disk pairs $(u,u')$ of expected dimension $0$ such that
\begin{enumerate}
\item For $[0,1]\times (-\infty,0)$, the punctures of $u$ and $u'$ consist of perfectly matched pairs; that is, each puncture of $u$ is paired with a puncture of $u'$ which has identical projection to $[0,1]\times \R$ (and vice-versa).
\item There are an even number of $u$ and $u'$ punctures on $[0,1]\times \{0\}$, the \emph{special line}. These punctures all have distinct projections to $[0,1]\times \{0\}$, and alternate between $u$ and $u'$. The left-most puncture is from $u$.
\item In $[0,1]\times (0,\infty)$, $u$ and $u'$ may have any number of punctures, and there is no constraint.
\end{enumerate}
The map $Ψ^Σ'<Σ$ is defined similarly, except that the left-most puncture on the special line is from $u'$.

\begin{prop} The maps $\Psi^{\Sigma<\Sigma'}$ and $\Psi^{\Sigma'<\Sigma}$ are chain maps. Furthermore, they are homotopy equivalences, with homotopy inverses given by the maps which count curves in $0$-dimensional moduli spaces with matching conditions in the regions above and below $[0,1]\times \{0\}$ reversed.
\end{prop}

A proof, modeled on Ozsv\'{a}th and Szab\'{o}'s work \cite{OSBorderedHFK}, may be found in \cite{ZemBordered}*{Section~11.2}. An extension of the connected sum maps $Ψ^Σ<Σ'$ to the setting of connected sums of hypercubes (under certain topological assumptions) is also proven in \cite{ZemBordered}*{Proposition~11.8}.

\subsection{Setting up the computation}

In Section~\ref{sec:naturality-knots-and-links} we considered several stabilized models of the involution where we used multiple connected sum tubes, but a single basepoint. To perform our computation, we will join $Σ$ and $Σ̅$ using a tube at $w$ and a tube at another point $p$.  Considering this stabilized model will allow us to localize the computation.

\begin{figure}[ht]
\input{twist_fig5.pdf_tex}
\caption{A once stabilized model of the involution. On the top, we show the doubling arcs on $\Sigma$. On the bottom left, we have an overview of the two tubes. On the bottom right, we zoom in on the $w$-tube region.}
\label{twist_fig:5}
\end{figure}

Write $c_p$ for the meridian of the $p$-tube. Note that we can get a valid Heegaard diagram by replacing $c_p$ with a meridian of the other tube. There are two natural choices, $c_w$ and $c̅_w$, depending on which side of $w$ we put the meridian.

We recall that the stabilized model of the involution from Section~\ref{sec:knots} takes the following form:
\[
F_3^{\a c_p, \a c_p}\circ f_{\a c_p \bar \b}^{\Dt\to (\a  c_p\bar \a)}\circ f_{\a  c_p\bar \b}^{(\b c_p \bar \b)\to \Dt}\circ  F_1^{c_p\bar \b, c_p \bar \b}.
\]

We now modify the sequence of curves we use in this Heegaard diagram, by breaking the sequence
\[
\bs c_p \bar \bs\to \Ds\to \as c_p \bar \as
\]
into a longer sequence. This will make some holomorphic curve counts easier.

We pick sets of doubled curves $_1,…, _n$, containing $2g+1$ curves each, on $Σ# Σ̅$ such that the following hold:
\begin{enumerate}
\item $\Ds_i=\Ds_i^0 \cup D$, where $\Ds_i^0$ are doubled curves on the unstabilized diagram, obtained by doubling arcs with two endpoints at $w$. Here $D$ is a single curve which obtained by doubling an arc which connects $p$ to $w$.
\item $(\Sigma \# \bar \Sigma,\bs \bar \bs,\Ds_1^0)$ is algebraically rigid. (Compare \cite{HHSZExact}*{Definition~6.5}).
\item $(\Sigma \# \bar \Sigma,\as \bar \as,\Ds_n^0)$ is algebraically rigid.
\item Each $\Ds_{i+1}^0$ is obtained from $\Ds_i^0$ by a small isotopy or a simple arcslide of the doubling arcs on $\Sigma\setminus N(p)$, not crossing over the endpoint corresponding to $D$. (In particular each $(\Sigma\# \bar \Sigma, \Ds_i^0,\Ds_{i+1}^0,w)$ is algebraically rigid). Note that this implies that $\Ds_{i+1}^0$ is obtained from $\Ds_i^0$ by an elementary handleslide.
\end{enumerate}

We will use the following model of the involution:
\[
F_3^{\a c_p, \a c_p}
\circ
\Psi_{(\a \bar c_w \bar \b)\to (\a c_p \bar \b)}^{(\a \bar c_w \bar \a)\to (\a c_p \bar \a)}
\circ
f_{\a \bar c_w \bar \b}^{\Dt_n\to (\a \bar c_w \bar \a)}
\circ \cdots\circ
f_{\a \bar c_w \bar \b}^{\Dt_1\to \Dt_2}
\circ
f_{\a \bar c_w \bar \b}^{(\b \bar c_w \bar \b)\to \Dt_1}
\circ
\Psi_{(\a  c_p \bar \b)\to (\a \bar c_w \bar \b )}^{(\b c_p \bar \b)\to (\b \bar c_w\bar \b)}
\circ
F_1^{c_p\bar \b, c_p \bar \b}
\]

The motivation for using this model is that most of the holomorphic curve counts in the above diagram occur on a connected sum of the ordinary doubled diagram for $Y$ on $Σ#Σ̅$ and a genus 1 diagram where the diffeomorphism acts non-trivially. The genus 1 region is a neighborhood of $D∪c_w$; we call it $𝕋$.

We now write $_i$ for $(_i)$. If we write $_i=_i^0∪D$, then $_i=_i^0∪D̃$, where $D̃$ is obtained from $D$ by applying the twist in the torus region.

Our argument will proceed  as follows. We will view each of the holomorphic polygon counts as occurring on a connected sum of $Σ# Σ̅$ and $$. We will build a 3-dimensional hyperbox whose front and back faces are expanded versions of the hyperbox in Equation~\eqref{eq:Dehn-twist-map}. The new dimension corresponds to the map $Ψ^Σ# Σ̅<$. The back face will consist of the perfectly matched complexes. The front face will consist of the trivially matched complexes with some extra terms. The extra terms correspond to the $QΦ$ summand of $_*$.

\subsection{Initial and final hypercubes}

In this section, we extend the following hypercubes (with holomorphic curves counted with perfect matching) into a third dimension corresponding to the map $Ψ^Σ# Σ̅<$:
\[
\begin{tikzcd} \CF(\as,\bs)
\ar[r, "\id"]
\ar[d, "F_1^{c_p \bar \b,c_p \bar \b}"]
\CF(\as, \bs)
\ar[d, "F_1^{c_p \bar \b,c_p \bar \b}"]
\\
\CF(\as c_p \bar \bs)
\ar[r, "\id"]&
\CF(\as c_p \bar \bs)
\end{tikzcd}
\quad \begin{tikzcd}
\CF(\as \bar c_w \bar \as,\as \bar c_w \bar \as)
\ar[r, "\id"]
\ar[d]
\CF(\as \bar c_w \bar \as,\as \bar c_w \bar \as)
\ar[d]
\\
\CF(\as c_p \bar \as,\as c_p \bar \as)
\ar[r, "\id"]
\ar[d, "F_3^{\a c_p, \a c_p}"]&
\CF(\as c_p \bar \as,\as c_p \bar \as)
\ar[d, "F_3^{\a c_p, \a c_p}"]
\\
\CF(\bar \bs, \bar \as) \ar[r, "\id" ] &\CF(\bar \bs, \bar \as).
\end{tikzcd}
\]

In the above, there is no diagonal term because the diffeomorphism $$ fixes all of the attaching curves on the Heegaard diagram.

\begin{lem}\label{lem:1-handle-c_w-commutation} If we stretch the almost complex structure sufficiently on the connected sum tubes, we have the equality
\[
\Psi_{(\a  c_p \bar \b)\to (\a \bar c_w \bar \b )}^{(\b c_p \bar \b)\to (\b \bar c_w\bar \b)}\circ  F_1^{c_p \bar \b, c_p \bar \b}=F_1^{\bar c_w \bar \b, \bar c_w \bar \b}.
\]
\end{lem}
\begin{proof} This follows from our stabilization result for holomorphic triangles in Proposition~\ref{prop:multi-stabilization-counts}. Topologically, it is sufficient to observe that on $\bar \Sigma$, there is a sequence of handleslides of $c_p$ over the other $\bar \bs$ curves (not crossing over $w$) which takes $c_p$ to $\bar c_w$ (i.e. handleslide $c_p$ over each $\bar \bs$ curve twice).
\end{proof}

We now investigate the next terms in the sequence. In Figure~\ref{twist_fig:7}, we draw the diagram $(, c̅_w,D, D̃, c̅_w',w)$.

\begin{figure}[ht]
\input{twist_fig7.pdf_tex}
\caption{The special genus 1 region and the diagram $(\bT^2, \bar {c}_w, D, \tilde{D}, \bar{c}_w',w)$. The dashed circle is the connected sum point. Also shown is a holomorphic quadrilateral that will play a role in our argument.}
\label{twist_fig:7}
\end{figure}

Next, we observe
\[
F_3^{\a c_p, \a c_p}=F_3^{c_p,c_p}\circ F_3^{\a,\a}.
\]
We note that after surgering out the $$ curves, the curves $c_p$ become isotopic to $c_w$. Hence we identify
\[
F_3^{c_p,c_p}\circ F_3^{\a,\a}=F_3^{c_w,c_w}\circ F_3^{\a,\a}.
\]
In a similar manner to Lemma~\ref{lem:1-handle-c_w-commutation}, we have the strict commutation
\begin{equation}
F_3^{\a,\a}\circ\Psi_{(\a \bar c_w \bar \b)\to (\a c_p \bar \b)}^{(\a \bar c_w \bar \a)\to (\a c_p \bar \a)}
\Psi_{(\bar c_w \bar \b)\to (c_w\bar \b)}^{(\bar c_w \bar \a)\to (c_w \bar \a)}\circ F_3^{\a,\a}.\label{eq:commute-3-handle-c-w-c-w'}
\end{equation}

We now prove a simple index bound for the diagram $(,c̅_w,  c_w)$. We view this diagram as having both a basepoint $w∈$, as well as a connected sum point $p∈$. Note that these do not coincide. See Figure~\ref{twist_fig:8}.

\begin{figure}[ht]
\input{twist_fig8.pdf_tex}
\caption{The diagram $(\bT, \bar c_w,  c_w,w)$. The dashed circle is the connected sum point $p$. }
\label{twist_fig:8}
\end{figure}

\begin{lem}
\label{lem:index-bound}
Suppose that $\phi\in \pi_2(\theta^+, \theta)$ is a class of disks on $(\bT,\bar c_w, c_w,w)$.
\begin{enumerate}
\item Then $\mu(\phi)=2n_w(\phi)+\gr(\theta^+,\theta)$.
\item If $D(\phi)\ge 0$, then $\mu(\phi) \ge n_p(\phi)$.
\end{enumerate}
\end{lem}
\begin{proof}The first claim is the Maslov grading formula.

We now consider the second claim. Consider first the case that $\theta=\theta^+$. Then any class $\phi=\ve{e}_{\theta}+k\cdot P+n\cdot [\bT]$, where $P$ is the periodic domain class which has $n_w=0$ and $n_p=1$. We observe $\mu(\phi)=2n$, and $n_p(\phi)=n+k$. Also, $n$, $n-k$ and $n+k$ are multiplicities of this disk class, so all must be nonnegative. This implies, in particular, that $n\ge |k|$. Ergo,
\[
\mu(\phi)=2n\ge n+k= n_p(\phi),
\]
as claimed.

We now consider the case that $\theta = \theta^-$. In this case, write $\phi_0$ for the bigon which has $n_p=1$ and write $\phi=\phi_0+k\cdot  P+n\cdot [\bT]$ for a general class. In this case, $n$, $n-k$ and $n+k+1$ are multiplicities on the diagram. Hence, $n\ge k$.  We have
\[
\mu(\phi)=1+2n\ge 1+n+k= n_p(\phi),
\]
completing the proof.
\end{proof}

We now perform a key computation:

\begin{lem}\label{lem:pair}  We have the relation
\[
F_3^{ c_w, c_w}\circ \Psi^{(\bar c_w \bar \a)\to (c_w \bar \a)}_{(\bar c_w \bar \b)\to (c_w \bar \b)}\simeq\left(\begin{array}{l} \xs\otimes \theta^-\mapsto \xs\\
\xs\otimes \theta^+\mapsto \Phi_w^{\bar\Sigma}(\xs)
\end{array} \right)\circ \Psi^{\bar\Sigma<\bT}
\]
where $\Phi_w^{\bar \Sigma}$ is the standard basepoint action on $\CF(\bar \Sigma, \bar \bs, \bar \as)$, usually denoted $\Phi_w$.
\end{lem}\label{lem:cw-cwbar}
\begin{proof} The map $\Psi^{(\bar c_w \bar \a)\to (c_w \bar \a)}_{(\bar c_w \bar \b)\to (c_w \bar \b)}$ is a composition of two triangle maps. The diagram $(\Sigma,\bar c_w, c_w, w)$ is shown in Figure~\ref{twist_fig:8}. The transition map may be viewed as handlesliding $\bar c_w$ over the connected sum point.

We first claim that index bound which implies that the 3-handle map is well-defined implies also that
\begin{equation}
F_3^{c_w,c_w}=F_3^{c_w,c_w}\circ \Psi^{\bar\Sigma<\bT}
\label{eq:hypercube-3-handle-preliminary}.
\end{equation}
To see this, note that $\Psi^{\bar\Sigma<\bT}$ counts holomorphic disk-pairs $(u_{\Sigma}, u_{\bT})$ on $(\bar\Sigma, \bar \bs, \bar \as)\# (\bT,c_w,c_w)$ which have a special line matching condition. Suppose that $u_{\Sigma}$ has $S$ marked points below the special line and $M$ marked points along the special line. The matching condition implies that $u_{\bT}$ has the same number of marked points below and on the special lines, respectively. If $\phi_{\bT}\in \pi_2(\theta,\theta')$ is the class of $u_{\bT}$, we observe that
\[
\mu(\phi_{\bT})=2n_w(\phi_{\bT})+\gr(\theta,\theta')\ge 2M+2S+\gr(\theta,\theta').
\]
In the composition $F_3^{c_w,c_w}\circ \Psi^{\bar\Sigma<\bT}$, only curves with $\theta'=\theta^-$ contribute. Therefore, the above shows that
\[
\mu(\phi_{\bT})\ge 2M+2S.
\]
By assumption, $\Psi^{\bar\Sigma<\bT}$ counts 0-dimensional moduli spaces, so we must have
\[
0=\mu(\phi_\Sigma)+\mu(\phi_{\bT})-2S-2M\ge \mu(\phi_\Sigma).
\]
By transversality, if $\phi_\Sigma$ admits a holomorphic representative, then $\mu(\phi_\Sigma)\ge 0$ with equality if and only if $\phi_\Sigma$ is a constant class. Hence, $F_3^{c_w,c_w}\circ \Psi^{\bar\Sigma<\bT}$ counts only curves which represent the constant class on $\Sigma$. It is easy to see by considering the diagram $(\bT,c_w,c_w)$ that this also implies that these classes are also constant on $\bT$. Hence, Equation~\eqref{eq:hypercube-3-handle-preliminary} follows.

Therefore, we may replace $F_3^{c_w,c_w}$ in the left side of the equation in the statement with $F_3^{c_w,c_w}\circ \Psi^{\bar\Sigma<\bT}$. We now consider the 1-parameter moduli spaces which would naturally be used to commute $\Psi^{\bar\Sigma<\bT}$ past both of the triangle maps used to define $\Psi^{(\bar c_w \bar \a)\to (c_w \bar \a)}_{(\bar c_w \bar \b)\to (c_w \bar \b)}$. These are the 1-dimensional moduli spaces of holomorphic triangles which have a special line which has an even number of marked points, which are $(\bar\Sigma)<\bT$ matched. Here, we view the triangle $\Delta$ as $[0,1]\times \R$ with the puncture $\{1\}\times \{0\}$ removed. The special lines are of the form $[0,1]\times \{s\}$, $s\in \R$.

In these moduli spaces, there are ends corresponding to the composition
$\Psi^{(\bar c_w \bar \a)\to (c_w \bar \a)}_{(\bar c_w \bar \b)\to (c_w \bar \b)}\circ\Psi^{\bar\Sigma<\bT}$, as well as ends corresponding to a chain homotopy. There is also the possibility of an additional end where a holomorphic disk forms at the same height as the special line, and some the punctures along the special line may also degenerate into a holomorphic disk. We claim that these ends are constrained to a broken curve containing the following curves:
\begin{enumerate}[label=($u$-\arabic*), ref=$u$-\arabic*]
\item An index one disk on $(\bT, \bar c_w,  c_w)$ which has a single marked point, which is from $\bT$. \item A trivial disk on $(\bar\Sigma, \bar \bs, \bar \bs)$.
\item \label{triangle-3} A pair of triangles on $(\bT,\bar c_w,  c_w, c_w)\wedge (\bar\Sigma, \bar \as, \bar \bs,\bar \bs)$ which have marked points at the same height as the $\bar \bs$-$\bar \bs$ vertex of the triangle. Furthermore, $\bar\Sigma$ and $\bT$ marked points alternate along this line, and both the left-most and right-most punctures are from $\Sigma$.
\end{enumerate}
See Figure~\ref{twist_fig:9}. We say that triangle pair satisfying~\eqref{triangle-3} is \emph{$(\bar\Sigma<\bT<\cdots<\bar\Sigma)$-matched}.

To establish that these are the only ends, we consider a potential degeneration where the special line on the triangle $\Delta$ has the same height as the $\bar \bs$-$\bar \bs$ puncture of $\Delta$.

We consider a limiting curve, and write $\psi^l_{\bar \Sigma}, \psi^l_\bT$ for the homology classes of the holomorphic triangles, and we write $\phi^r_{\bar \Sigma}$ and $\phi^r_{\bT}$ for the classes of the holomorphic disks that break off. We suppose that there are $n^r_{\bar \Sigma}$ and $n^r_\bT$ marked points of the disk classes along the special line, and $S^r=S^r_{\bar \Sigma}=S^r_{\bT}$ marked points below the special line (i.e. perfectly matched). The expected dimension of the disk class with this matching condition is
\begin{equation}
\mu(\phi^r_{\bar\Sigma})+\mu(\phi^r_{\bT})-n^r_{\bar \Sigma}-n^r_\bT+1-2 S^r.
\label{eq:expected-dimension-monster-disk}
\end{equation}
The above quantity must be 1 in a generic degeneration.
On the other hand, since $\phi^r_{\bar \Sigma}$ is a disk class on $(\bar\Sigma, \bar \bs, \bar \bs)$, and the connected sum point is the basepoint, we have
\[
\mu(\phi^r_{\bar\Sigma})\ge 2n_p(\phi^r_{\bar\Sigma}) \ge 2(n^r_{\bar\Sigma}+S^r)
\]
by the absolute grading formula. Similarly, Lemma~\ref{lem:index-bound} implies that
\[
\mu(\phi^r_{\bT})\ge n^r_\bT+S^r.
\]
We observe that the expected dimension in Equation~\eqref{eq:expected-dimension-monster-disk} is at least
\[
n_{\bar\Sigma}^r+1+ S^r.
\]
Hence $S^r=n_{\bar\Sigma}^r=0$ for a generic degeneration.
Since $|n_{\bT}^r-n_{\bar\Sigma}^r|\le 1$, we must have $n_\bT^r\in \{0,1\}$.

We therefore conclude that $\mu(\phi_{\bar\Sigma}^r)=n_p(\phi_{\bar\Sigma}^r)=0$ and $\mu(\phi_\bT^r)=1$, and $n_{\bT}^r=1$, for a generic degeneration.

From this broken curve, we may extend the moduli space, by considering the $(\bar\Sigma<\bT<\dots<\bar\Sigma)$-matched moduli space space of triangles where the special line has height below the $\bar \bs$-$\bar \bs$ boundary puncture of the triangle $\Delta$.

There is a natural map
\[
\Psi^{\bar\Sigma<\bT<\cdots<\bar\Sigma}\colon \CF_\wedge(\bar \bs c_w,\bar \as \bar c_w)\to \CF_{\otimes}( \bar \bs c_w,\bar \as \bar c_w)
\]
which counts curve pairs of expected dimension 0 with the following matchings:
\begin{enumerate}
\item Punctures along the special line are $(\bar\Sigma<\bT<\dots< \bar\Sigma)$-matched.
\item Punctures below the special line are perfectly matched.
\item Punctures above the special line are trivially matched.
\end{enumerate}
We claim that
\[
\Psi^{\bar\Sigma<\bT<\cdots<\bar\Sigma}\simeq (\id\otimes \Phi_w^{\bar\Sigma})\circ  \Psi^{\bar\Sigma<\bT}.
\]
This is proven by splitting the special line into two special lines. The higher special line contains a single $\bar\Sigma$-marked point. The lower special line is $\bar\Sigma<\bT$ matched. Other marked points are perfectly matched below the lower special line, and trivially matched above it. (I.e. there is no change in matching as we cross the $\bar\Sigma$-matched line).

\begin{figure}[ht]
\input{twist_fig10.pdf_tex}
\caption{A schematic of the homotopy $\Psi^{\bar\Sigma<\bT<\dots<\bar\Sigma}\simeq (\id\otimes \Phi^{\bar\Sigma}_w)\circ \Psi^{\bar\Sigma<\bT}$.}
\label{twist_fig:10}
\end{figure}

We count the ends of these 1-parameter moduli spaces with two special lines. The only non-canceling ends are as follows:
\begin{enumerate}
\item An index 1 disk breaking off while the special lines have finite, non-zero distance, giving a chain homotopy
\item  The vertical distance between the two canceling ends approaches $0$ or $\infty$. The distance $0$ end contributes $\Psi^{\bar\Sigma<\bT<\dots<\bar\Sigma}$ while the distance $\infty$ end corresponds to the composition $(\id\otimes \Phi_w^{\bar \Sigma})\circ \Psi^{\bar\Sigma<\bT}$.
\end{enumerate}
Counting ends as above yields
\[
\Psi^{\bar\Sigma<\bT<\dots<\bar\Sigma}\simeq (\id\otimes\Phi_w^{\bar \Sigma}) \circ \Psi^{\bar\Sigma<\bT}.
\]

Finally, we put all the pieces together. When counting triangles with trivial matching conditions, it is straightforward to see that
\[
\Psi^{(\bar c_w \bar \a)\to (c_w \bar \a)}_{(\bar c_w \bar \b)\to (c_w \bar \b)}=\id\otimes \id,
\]
on the level of intersection points.

Counting the ends of the above moduli spaces, we have  ends which contribute the map
\[F_3^{ c_w,  c_w}\circ \Psi^{(\bar c_w \bar \a)\to (c_w \bar \a)}_{(\bar c_w \bar \b)\to (c_w \bar \b)} \circ \Psi^{\bar\Sigma<\bT}.
\]
This is equal to
\[\left(\begin{array}{l} \xs\otimes \theta^-\mapsto \xs\\
\xs\otimes \theta^+\mapsto 0
\end{array} \right)\circ \Psi^{\bar\Sigma<\bT}
\]
Additionally, we have the contribution corresponding to extending the bigon degeneration. This gives and extra contribution of $\Psi^{\bar\Sigma<\bT}$ post-composed with
\[
\xs\otimes \theta^+\mapsto \Phi_w^{\bar\Sigma}(\xs).
\]
Summing this with the previous contribution gives the main statement.
\end{proof}

\begin{figure}[ht]
\input{twist_fig9.pdf_tex}
\caption{Canceling moduli space ends in Lemma~\ref{lem:cw-cwbar}. Squares denote special inputs of the holomorphic polygon maps. The curves in the square indicate 1-dimensional moduli spaces, and the arrows labeled with $\d$ indicate codimension 1 degenerations. Solid dots are $\bar\Sigma$-punctures and open dots are $\bT$-punctures. }
\label{twist_fig:9}
\end{figure}

\begin{rem} Note that the proof of Equation~\eqref{eq:hypercube-3-handle-preliminary} adapts easily to show that
\[
F_3^{\a,\a} \circ  \Psi^{\Sigma\# \bar \Sigma<\bT}=\Psi^{\bar \Sigma<\bT}\circ F_3^{\a,\a}.
\]
\end{rem}

\subsection{Central hypercubes}

In this section, we describe how to commute $\Psi^{\Sigma\# \bar \Sigma<\bT}$ past the levels of $\tw_*$ which involve the quadrilateral maps.
The main result of this section is the following:

\begin{lem}
\label{lem:tensor-product-hyerpcube-middle}The map $\Psi^{\Sigma\# \bar \Sigma<\bT}$ extends to a homotopy equivalence of hyperboxes from the perfectly matched hyperbox
\[
\begin{tikzcd}
\CF(\as \bar c_w \bar \bs, \bs \bar c_w \bar \bs)
\ar[r]
\ar[d]
\ar[dr,dashed]
&\CF(\as \bar c_w \bar \bs, \bs \bar c_w \bar \bs)
\ar[d]
\\
\CF(\as \bar c_w \bar \bs, \bs D \bar \bs)
\ar[r]
\ar[d]
\ar[dr,dashed]
&\CF(\as \bar c_w \bar \bs, \bs \tilde D \bar \bs)
\ar[d]
\\
\CF(\as\bar c_w \bar \bs, \Ds_1)
\ar[r]
\ar[d]
\ar[dr,dashed]
\CF(\as\bar c_w \bar \bs, \tilde{\Ds}_1)
\ar[d]
\\
\vdots\ar[d]\ar[dr,dashed]& \vdots \ar[d]\\
\CF(\as\bar c_w \bar \bs, \Ds_n)
\ar[r]
\ar[d]
\ar[dr,dashed]
\CF(\as\bar c_w \bar \bs, \tilde{\Ds}_n)
\ar[d]
\\
\CF(\as \bar c_w \bar \bs, \as D \bar \as)
\ar[r]\ar[d]
\ar[dr,dashed]
&\CF(\as \bar c_w \bar \bs, \as \tilde D \bar \as)
\ar[d]
\\
\CF(\as \bar c_w \bar \bs, \as \bar c_w \bar \as)
\ar[r]
&\CF(\as \bar c_w \bar \bs, \as \bar c_w \bar \as)
\end{tikzcd}
\]
to its trivially matched counterpart.
\end{lem}
\begin{proof}
We observe that all of the attaching curves which appear in the statement are disjoint from the connected sum tube which connects $\bar\Sigma$ and $\bT^2$. In fact, we may describe the entire
hyperbox using a connected sum operation on hypercubes of attaching curves. This connected sum operation on hypercubes of attaching curves is a small modification of the one considered by Lipshitz, Ozsv\'{a}th and Thurston in \cite{LOTDoubleBranchedII}*{Section~3.5} (see \cite{ZemBordered}*{Section~9.5} for our present notation). If $\cL_{\b}=(\bs_\veps,\Theta_{\veps,\veps'})_{\veps\in \bE_n}$ and $\cL_{\b'}=(\bs'_{\nu}, \Theta_{\nu,\nu'})_{\nu\in \bE_m}$ are two hypercubes of attaching curves on Heegaard surfaces $\Sigma$ and $\Sigma'$, then $\cL_{\b}\# \cL_{\b'}$ is a hypercube of attaching curves of dimension $n+m$. The curves of $\cL_{\b}\# \cL_{\b'}$ are defined as
\[
\cB_{(\veps,\nu)}=\bs_{\veps}\cup \bs'_{\nu}
\]
with small translations of curves taken to achieve admissibility.  In our present setting, if the connected sum of $\Sigma$ and $\Sigma'$ is taken at either $w$ or $w'$ (deleting this basepoint), then the curves $\cB_{(\veps,\nu)}$ will also form an admissible diagram which is algebraically rigid, so we take the length 1 Floer chains of $\cL_{\b}\# \cL_{\b'}$ to be the top degree generators, and all higher length chains to vanish.

In the hyperbox in the statement of the lemma, each constituent hypercube of chain complexes is obtained by pairing the 0-dimensional hypercube $\as \bar c_w \bar \bs$ with a 2-dimensional hypercube of beta attaching curves. We view $\as \bar c_w \bar \bs$ as the connected sum of 0-dimensional hypercubes $\as \bar \bs$ with $\bar c_w$. In the top-most and bottom-most hypercubes, the beta hypercube of attaching curves are connected sums of a 0-dimensional hypercube on $\Sigma\# \bar \Sigma$ with a 2-dimensional hypercube on $\bT^2$. The remaining beta hypercubes are connected sums of a 1-dimensional hypercube on $\Sigma\# \bar \Sigma$ and a 1-dimensional hypercube on $\bT^2$.

Work of the last author \cite{ZemBordered}*{Section~11.3} constructs, under suitable hypotheses, a homotopy equivalence between the hyperbox in the lemma statement (i.e. the hyperbox obtained by counting curve pairs with perfect matching) and the hyperbox obtained by counting curve pairs with trivial matching. The precise statement may be found in \cite{ZemBordered}*{Proposition~11.8} and involves the following somewhat technical definitions:
\begin{enumerate}
\item The alpha and beta hypercubes of attaching curves must be \emph{algebraically rigid} and consist of handleslide equivalent attaching curves.
\item \label{item:technical-point} A technical condition about the placement of the connected sum points and the basepoint must be satisfied. See \cite{ZemBordered}*{Definition~11.4}. We say that a hypercube of algebraically rigid, handleslide equivalent attaching curves is \emph{graded} by the connected sum point $p$ if for all nonnegative classes $\phi\in \pi_2(\Theta_{\veps_1,\veps_2},\dots, \Theta_{\veps_{n-1},\veps_n})$ (where $\Theta_{\veps_i,\veps_{i+1}}$ are chains from the hypercube), we have
\[
\mu(\phi)\ge 2n_p(\phi).
\]
\end{enumerate}

We recall the statement of \cite{ZemBordered}*{Proposition~11.8}. Suppose that $\Sigma$ and $\Sigma'$ are Heegaard surfaces with distinguished points $p\in \Sigma$ and $p'\in \Sigma'$, and $\cL_{\a}$ and $\cL_{\b}$ are algebraically rigid hypercubes on $\Sigma$ and $\cL_{\a'}$ and $\cL_{\b'}$ are algebraically rigid hypercubes on $\Sigma'$. If $\cL_{\a}$ is graded by $p$ and $\cL_{\b'}$ is graded by $p'$, then the map
\[
\Psi^{\Sigma'<\Sigma}\colon \CF_{\wedge}(\cL_{\a}\# \cL_{\a'}, \cL_{\b}\# \cL_{\b'})\to  \CF_{\otimes}(\cL_{\a}\# \cL_{\a'}, \cL_{\b}\# \cL_{\b'})
\]
is a homotopy equivalence. Here $\CF_{\wedge}$ has differential counting perfectly matched curve pairs, while $\CF_{\otimes}$ counts trivially matched curve pairs.

We now return to the original setting of the present lemma. By construction, each hypercube is algebraically rigid. Furthermore, the alpha hypercubes $\as\bar \bs$ and $\bar c_w$ are vacuously graded by the connected sum point since they are 0-dimensional. All of the hypercubes on $\Sigma\# \bar \Sigma$ are graded by the connected sum point since the connected sum point is adjacent to the basepoint and the hypercubes are algebraically rigid. The beta hypercubes on $\bT^2$ are not all graded by the basepoint, but all of the alpha hypercubes which lie on $\bT^2$ are. The only difference in our present situation and \cite{ZemBordered} is that the hypercubes of attaching curves do not all consist of handleslide equivalent attaching curves. For example, $\bs\bar \bs$ is not handleslide equivalent to $\Ds_1^0$. Nonetheless, the proof of \cite{ZemBordered}*{Proposition~11.8} still goes through since Heegaard diagram formed from a pair of curves in the diagram represents an algebraically rigid diagram for a connected sum of $S^1\times S^2$'s and the grading assumption in \eqref{item:technical-point} is satisfied. Hence, the same argument as in \cite{ZemBordered}*{Proposition~11.8} provides the homotopy equivalence in the statement.
\end{proof}

Next, we view the above hyperbox as consisting of attaching curves on the disjoint union of $\Sigma\# \bar \Sigma$ and $\bT$, instead of the connected sum of these two diagrams. Note that compression commutes with pairing up to homotopy, so we may instead compress the hyperbox of attaching curves:
\begin{equation}
\begin{tikzcd}[row sep=.4cm, column sep=1.4cm]
\bs \bar c_w \bar \bs\ar[r]\ar[d] &\bs \bar c_w \bar \bs\ar[d]\\
\bs D \bar \bs \ar[r]\ar[d]& \bs \tilde D \bar \bs\ar[d]\\
\Ds_1\ar[r]\ar[d] &\tilde{\Ds}_1 \ar[d]\\
\vdots\ar[d]& \vdots\ar[d]\\
\Ds_n\ar[r]\ar[d] &\tilde{\Ds}_n \ar[d]\\
\as D \bar \as\ar[r]\ar[d]& \as \tilde D \bar \as\ar[d]\\
\as\bar c_w \bar \as\ar[r] &\as \bar c_w \bar \as
\end{tikzcd}\label{eq:expanded-hyperbox-tw-map}
\end{equation}
Note that we have not defined compression of hyperboxes of attaching curves. Nonetheless, the function-composition approach from Section~\ref{sec:hypercubes} generalizes to this setting with minimal care.
In our present case this amounts to viewing each of the $n+2$  levels in the above diagram
as a twisted complex in the Fukaya category. Each pair of adjacent levels in the diagram determines a morphism between 1-dimensional hypercubes of attaching curves. A pair of morphisms between hypercubes of attaching curves may be composed, so we compose iteratively all of these morphisms. Note that this is not the same as applying the ordinary $A_\infty$-composition map $\mu_{n+2}$. Rather, it is the result of applying $\mu_2$ repeatedly.

\begin{lem}
\label{lem:compressed-Fuk-hyperbox}
The compression of the diagram in Equation~\eqref{eq:expanded-hyperbox-tw-map} takes the following form:
\[
\begin{tikzcd}[row sep=.6cm, column sep=1.4cm]
\bs \bar c_w \bar \bs\ar[r, "1"]\ar[d, "\ve{X}|\theta^-"] \ar[dr, "\ve{X}'|\theta^+"] &\bs \bar c_w \bar \bs\ar[d,"\ve{X}|\theta^-"]\\
\as\bar c_w \bar \as\ar[r, "1"] &\as \bar c_w \bar \as
\end{tikzcd}
\]
Here,  $\ve{X}$ is the iterated composition of the vertical arrows on $\Sigma\# \bar \Sigma$, and $\theta^{\pm}$ are the intersection points of $\bar c_w\cap \bar c_w'$. Furthermore, the cycle $\ve{X}'$ is homologous to $\ve{X}$.
\end{lem}
\begin{proof} We view the diagram as being obtained by composing $n+1$ morphisms between 1-dimensional hypercubes of Lagrangians. Viewing this diagram as a sequence of morphisms $\theta_{n+1},\dots, \theta_1$, we form the composition as
\[
\mu_2(\theta_{n+1},\dots, \mu_2(\theta_2,\theta_1)).
\]
We first claim that except for the final composition, there are no diagonal terms in any partial composition. To see this, consider the first composition; the same argument will apply to all later compositions except for the final. There are three holomorphic quadrilateral counts which contribute to the diagonal term, corresponding to the following quadrilaterals:
\begin{enumerate}
\item $(\bs \bar c_w \bar \bs, \bs \bar c_w \bar \bs, \bs \tilde D \bar \bs, \tilde \Ds_1)$.
\item $(\bs \bar c_w \bar \bs, \bs D \bar \bs, \bs \tilde D \bar \bs, \tilde \Ds_1)$.
\item $(\bs \bar c_w \bar \bs, \bs D \bar \bs, \Ds_1, \tilde \Ds_1)$.
\end{enumerate}
%\item $(\bs \bar c_w \bar \bs,\bs \bar c_w \bar \bs, \tilde \Ds_1, \tilde \Ds_2)$,
%\item $(\bs \bar c_w \bar \bs,\Ds_1, \tilde \Ds_1, \tilde \Ds_2)$,
%\item $(\bs \bar c_w \bar \bs,\Ds_1, \Ds_2, \tilde \Ds_2)$
Each of these Heegaard quadruples decomposes as a connected sum where both factors have one set of attaching curves which is a small translation of an adjacent set. Since we are counting curves using trivial matchings, we use \cite{HHSZExact}*{Proposition~11.5} to see that no holomorphic quadrilaterals contribute. Similarly, using a model count on the genus 1 diagram, we observe that the two triangle compositions in the vertical direction coincide.

Therefore the final composition is of a diagram of the following form:
\begin{equation}
\begin{tikzcd}
\bs \bar c_w \bar \bs\ar[r]\ar[d, "\ve{X}|\theta"] &\bs \bar c_w \bar \bs\ar[d, "\ve{X}|\theta"]\\
\as D \bar \as\ar[r]\ar[d] &\as \tilde D \bar \as \ar[d]\\
\as\bar c_w \bar \as\ar[r] &\as \bar c_w \bar \as
\end{tikzcd}
\label{eq:hyperbox-lagrangians-to-be-compressed}
\end{equation}
where $\theta$ denotes the intersection point of $\bar c_w\cap D$ or $\bar c_w \cap \tilde{D}$.

As a first step, we consider the length 1 maps in the compression of Equation~\eqref{eq:hyperbox-lagrangians-to-be-compressed}. The length one chain on the left side of the compression is given by counting triangles on $(\Sigma\# \bar \Sigma, \bs \bar \as, \as \bar \as, \as \bar \as)$ and $(\bT^2, \bar c_w, D, \bar c_w)$. There is no matching condition, so we count triangles on the two diagrams separately and see that the triangle maps applied to $(\ve{X}|\theta, \Theta^+| \theta)$ have output $\ve{X}|\theta^-$. The computation on $\Sigma\# \bar \Sigma$ follows from nearest-point triangle counts, and the computation on $\bT^2$ is a direct and straightforward triangle count.

We now consider the length 2 map of the compression. The small translate theorems \cite{HHSZExact}*{Propositions~11.1 and 11.5} prohibit holomorphic quadrilaterals except on
\[
(\Sigma\# \bar \Sigma,\bs\bar \bs, \as \bar \as, \as \bar \as, \as \bar \as)\#(\bT^2,\bar c_w,D, \tilde D, \bar c_w).
\]
Since the diagram on $\Sigma\# \bar \Sigma$ has several adjacent sets of attaching curves which are small translates of one another, the only curves counted consist of pairs $(u,u')$ where $u$ is an index 0 quadrilateral on $\Sigma\# \bar \Sigma$ and $u'$ is an index $-1$ quadrilateral on $\bT^2$. Furthermore, the two quadrilaterals share the same almost complex structure parameter (viewing the moduli space of rectangles as $\R$). There is a unique such curve $u'$, shown in Figure~\ref{twist_fig:7}. Write $\ve{X}'|\theta^-$ for the output.

To see that $\ve{X}'$ is homologous to $\ve{X}$ we deform the matching, by instead requiring the quadrilaterals $u$ and $u'$ to have almost $\ev_{\Box}(u)-\ev_{\Box}(u')=t$, where $\ev_{\Box}$ denotes the evaluation map to the moduli space of complex rectangles, identified with $\R$. We then send $t\to \infty$. The effect is to change $\ve{X}'$ by a boundary. This shows that $\ve{X}'$ is homologous to the composition
\[
\mu_2(\mu_2(\ve{X}, \Theta_{\a\bar \a, \a \bar \a}), \Theta_{\a\bar \a, \a \bar \a}).
\]
%\mu_2(\Theta_{\a \bar c_w \bar \a,\a \bar c_w \bar \a}, \mu_2(\Theta_{\a \bar c_w \bar \a, \Dt_n}, \ve{X})):=\mu_2(\Theta_{\a \bar c_w \bar \a,\a \bar c_w \bar \a},\ve{X}),
which we identify with $\ve{X}$ via the nearest point map. This completes the proof.
\end{proof}

%If we pair the above with $\as \bar c_w \bar \bs$, and then compose with the 1-handle maps, as well as the map from Lemma~\ref{lem:pair}, we obtain the hypercube
%\begin{tikzcd}\CF(\as, \bs)\ar[r, "\id"]\ar[dr, "\Phi_w \iota"] \ar[d, "\iota"] & \CF(\as, \bs)\ar[d, "\iota"]\\
%\CF(\bar \bs, \bar \as)\ar[r, "\id"] & \CF(\bar \bs, \bar \as)

\subsection{Proof of Theorem~\ref{thm:twist-map}}

We now complete our proof of the computation of the map $\tw_*$:
\begin{proof}[Proof of Theorem~\ref{thm:twist-map}] It remains to put
the pieces together to finish the proof. We build a 3-dimensional hyperbox to realize the homotopy. The back face is the map $\tw_*$, computed using perfect matching between $\Sigma\# \bar \Sigma$ and $\bT^2$.

The front face is defined very similarly, using the trivial (i.e. tensor product) matchings, with a modification of one cube. In place of
\[
\begin{tikzcd}[column sep=2cm, row sep=1.5cm, labels=description]
\CF_\wedge( \bar c_w \bar \bs,\bar c_w \bar \as)
\ar[r,"\id"]
\ar[d, "F_3^{c_w,c_w}\Psi^{(\bar c_w \bar \a)\to (c_w \bar \a)}_{(\bar c_w \bar \b)\to (c_w \bar \b)}"]&
\CF_\wedge(\bar c_w \bar \bs,\bar c_w \bar \as)
\ar[d, "F_3^{c_w,c_w}\Psi^{(\bar c_w \bar \a)\to (c_w \bar \a)}_{(\bar c_w \bar \b)\to (c_w \bar \b)}"]
\\
\CF(\bar \bs, \bar \as)\ar[r, "\id"]
& \CF(\bar \bs, \bar \as)
\end{tikzcd}
\]
we have
\[
\begin{tikzcd}[column sep=1.5cm, row sep=1cm, labels=description]
\CF_\otimes(\bar c_w \bar \bs,\bar c_w \bar \as)
\ar[r,"\id"]
\ar[d, "\Psi'"]&
\CF_\otimes(\bar c_w \bar \bs,\bar c_w \bar \as )
\ar[d, "\Psi'"]
\\
\CF(\bar \bs, \bar \as   )\ar[r, "\id"]
& \CF(\bar \bs, \bar \as)
\end{tikzcd}
\]
\[
\Psi'=\left(\begin{array}{l} \xs\otimes \theta^-\mapsto \xs\\
\xs\otimes \theta^+\mapsto \Phi_w(\xs)
\end{array} \right)\circ \Psi^{\bar\Sigma<\bT}
\]
Lemmas~\ref{lem:1-handle-c_w-commutation}, \ref{lem:tensor-product-hyerpcube-middle} and Equation~\eqref{eq:hypercube-3-handle-preliminary} show that the hyperbox relations are satisfied for appropriate choices of length 3 maps on the interior of the cube.

Next, we compress the front face of the hypercube described above. We pair the hypercube from Lemma~\ref{lem:compressed-Fuk-hyperbox} with $\as \bar c_w \bar \bs$ and then stack with the 2-dimensional hypercube
\[
\begin{tikzcd}\CF(\bar\Sigma,\bar \bs, \bar \as)\ar[r, "\id"]\ar[d, "\eta"]& \CF(\bar \Sigma, \bar \bs, \bar \as) \ar[d, "\eta"]\\
\CF(\Sigma,\as,\bs)\ar[r, "\id"]& \CF(\Sigma,\as,\bs)
\end{tikzcd}
\] The so-constructed cube has the form
\[
\begin{tikzcd}\CF(\as, \bs)\ar[r, "\id"]\ar[dr, "\Phi_w \iota"] \ar[d, "\iota"] & \CF(\as, \bs)\ar[d, "\iota"]\\
\CF(\as, \bs)\ar[r, "\id"] & \CF(\as, \bs),
\end{tikzcd}
\]
completing the proof.
\end{proof}

\subsection{Knots and links}\label{subsec:twist-knots}

\label{sec:twisting-knots-links}

If $K$ is a knot, then \emph{a-priori} there is an action $\tw^\mu_*$ corresponding to rotating a neighborhood of $K$ in the meridianal direction. (Rotation in the longitudinal direction is well known to induce an interesting automorphism by work of Sarkar \cite{SarkarMovingBasepoints}). Despite the interesting potential for a new map, we have the following:

\begin{prop}
The map $\tw^{\mu}_*$ is chain homotopic to $\id$ on $\cCFKI(K)$. The same holds for links.
\end{prop}
\begin{proof}We consider the stabilized model of the involution. For knots and links, the doubling arcs $\ds$ have endpoints at a point $p$ which is not one of the basepoints of the knot. Hence, applying $\tw^{\mu}$ to each Heegaard diagram has no effect on any of the attaching curves.
\end{proof}

\begin{figure}[h]
\input{twist_fig11.pdf_tex}
\caption{A doubled knot diagram (for the unknot). Meridianal twisting preserves the diagram. }
\end{figure}

\begin{rem} We can also recover the above result on the non-stabilized doubling models of the knot involution, considered in Section~\ref{sec:doubling-knots} and \cite{HHSZExact}*{Section~2.2}, though the argument is more subtle. Here, we double at one of the basepoints (say $w\in \Sigma$), with the $\ds$ curves having endpoints at $w$. In this model, after attaching 1-handles, we move $w$ to the position of $\bar z$. The map $\tw^\mu$ still changes the curves $\Ds$, but since there is no basepoint in the tube region (unlike in the case of closed 3-manifolds), the effect on the level of diagrams is to act by a surface diffeomorphism which is isotopic to the identity relative to the basepoints on $\Sigma\# \bar \Sigma$.
\end{rem}

\bibliographystyle{custom}
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% \bib, bibdiv, biblist are defined by the amsrefs package.
\begin{bibdiv}
\begin{biblist}

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\bib{HMZConnectedSum}{article}{
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\bib{JMConcordance}{article}{
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pages={3623\ndash 3673},

\bib{JTNaturality}{unpublished}{
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author={Zemke, Ian},
title={Naturality and mapping class groups in {H}eegaard {F}loer
note={e-print, \url{arXiv:1210.4996}},

\bib{JCob}{article}{
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\bib{JZContactHandles}{article}{
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title={Contact handles, duality, and sutured {F}loer homology},
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\bib{KirbyCalculus}{article}{
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\bib{LinExact}{article}{
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\bib{LOTDoubleBranchedII}{article}{
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author={Ozsv\'{a}th, Peter~S.},
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\bib{LOTBordered}{article}{
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title={Bordered {H}eegaard {F}loer homology},
journal={Mem. Amer. Math. Soc.},

\bib{MOIntegerSurgery}{unpublished}{
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author={Ozsv{\'a}th, Peter~S.},
title={Heegaard {F}loer homology and surgeries on links},
note={e-print, \url{arXiv:1011.1317}},

\bib{Ni-homological}{article}{
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title={Homological actions on sutured {F}loer homology},
journal={Math. Res. Lett.},
pages={1177\ndash 1197},

\bib{OSIntersectionForms}{article}{
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four-manifolds with boundary},
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pages={58\ndash 116},

\bib{OSDisks}{article}{
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author={Szab\'o, Zolt\'an},
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journal={Ann. of Math. (2)},
pages={1027\ndash 1158},

\bib{OSProperties}{article}{
author={Ozsv{\'a}th, Peter~S.},
author={Szab{\'o}, Zolt{\'a}n},
title={Holomorphic disks and three-manifold invariants: properties and
journal={Ann. of Math. (2)},
pages={1159\ndash 1245},

\bib{OSTriangles}{article}{
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author={Szab\'o, Zolt\'an},
title={Holomorphic triangles and invariants for smooth four-manifolds},
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pages={326\ndash 400},

\bib{OSLinks}{article}{
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{A}lexander polynomial},
journal={Algebr. Geom. Topol.},
pages={615\ndash 692},

\bib{OSIntegerSurgeries}{article}{
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author={Szab\'{o}, Zolt\'{a}n},
title={Knot {F}loer homology and integer surgeries},
journal={Algebr. Geom. Topol.},
pages={101\ndash 153},

\bib{OSBorderedHFK}{unpublished}{
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author={Szab\'{o}, Zolt\'{a}n},
title={Algebras with matchings and knot {F}loer homology},
note={e-print \url{arXiv:1912.01657}},

\bib{RasmussenKnots}{thesis}{
author={Rasmussen, Jacob},
title={{F}loer homology and knot complements},
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\bib{SarkarMaslov}{article}{
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\bib{TNConcordance}{unpublished}{
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title={Concordance maps in $\mathit{HFK}^-$},
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\bib{ZemGraphTQFT}{unpublished}{
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title={Graph cobordisms and {H}eegaard {F}loer homology},
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\bib{ZemQuasi}{article}{
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pages={3461\ndash 3518},

\bib{ZemDuality}{unpublished}{
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title={Duality and mapping tori in {H}eegaard {F}loer homology},
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\bib{ZemConnectedSums}{article}{
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title={Connected sums and involutive knot {F}loer homology},
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\bib{ZemAbsoluteGradings}{article}{
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\bib{ZemBordered}{unpublished}{
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\bib{ZemHatGraphTQFT}{article}{
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\end{biblist}
\end{bibdiv}

\end{document}