\section{The octahedral calculation}\label{alg:section}

In this section we calculate the three homology groups in the octahedron
which are accessible by face reductions, together with the maps between
them. Exactness then expresses $\dim\J(D)$ in terms of one remaining
map $h$. The edge operators impose further conditions on $h$, leaving
the three coefficients in \eqref{intro:reduction}.

All the calculations here are on homology, over $\F$. In particular,
we do not require a Floer complex with zero differential for any of
the three large webs $K_0,K_1,L_1$. When we say that these webs can be
reduced, we mean by bigon, triangle and square relations; this has
nothing to do with reducibility of a connection.

\subsection{The webs and the octahedron}

The local pictures are those of \cite[Figure~1]{KM-tri}, reproduced
with our notation in Figure~\ref{fig:octahedron}. Outside the displayed
disk every web agrees with the exterior $E$ in
Figure~\ref{fig:dodecahedron-exterior}. Thus a local picture specifies
an entire closed web, not just a tangle considered in isolation.

For the calculations we also fix vertex labels. Regard $D$ as the
planar embedding of the generalized Petersen graph $G(10,2)$.
Its vertices are $0,\ldots,19$, and its unordered edges are
\begin{equation}\label{alg:Dedges}
 \{i,i+1\},\quad \{i,10+i\},\quad
 \{10+i,10+(i+2)\},\qquad i\in\mathbb Z/10.
\end{equation}
Only the indices within each set of ten vertices are reduced modulo ten.
Choose the edge joining $0$ to $1$. Delete its two endpoints and
the five edges incident to them. The remaining graph has 25 edges
and four vertices of valence two, labelled $(9,10,2,11)$ in cyclic
order. To regard it as the exterior $E$, attach a short arc from
each of these vertices to the boundary of the deleted disk. The
boundary points are northwest, northeast, southeast and southwest,
respectively. We use the same four labels for these attaching points.
The short arcs are suppressed in the following edge-list notation:
adding an edge between two of these labels means joining the
corresponding arcs through the disk. The three webs needed for the
calculation are
\begin{align}
 K_0&=E+\{9,10\}+\{2,11\},\nonumber\\
 K_1&=E+\{9,11\}+\{10,2\},\label{alg:webs}\\
 L_1&=E+\{20,21\}+\{20,9\}+\{20,11\}
                    +\{21,10\}+\{21,2\}.\nonumber
\end{align}
These are embedded using the planar exterior of $D$. To specify the
embedding for the supporting calculations, take the twelve simple
five-cycles of \eqref{alg:Dedges} as the faces. Orient the
lexicographically first cycle positively and orient the others so
that adjacent faces traverse their common edge in opposite directions.
This records the cyclic order of edges at each vertex, up to the
initial choice of orientation. The full lists are in
\texttt{support/finite/graph\_labels.json}.

Put
\[
 A=\J(K_0),\quad B=\J(K_1),\quad C=\J(L_1),\quad
 W=\J(K_2),\quad F=\J(L'_2),\quad M=\J(D).
\]
Here $D=L_0$. In $K_2$, the northwest--southeast arc passes below
the northeast--southwest arc, as drawn in the figures. The crossing
is a projection of two disjoint arcs in $\mathbb R^3$, not a vertex
of the web. The pertinent exact triangles of
\cite{KM-tri} are
\begin{align}
 A\xrightarrow{a}W\xrightarrow{r}C\xrightarrow{s}A,&\qquad
 W\xrightarrow{b}B\xrightarrow{t}M\xrightarrow{q}W,
 \label{alg:triangles}\\
 A\xrightarrow{\gamma}B\xrightarrow{\lambda}F\xrightarrow{\kappa}A,&\qquad
 C\xrightarrow{\eta}F\xrightarrow{\zeta}M\xrightarrow{\xi}C.
 \nonumber
\end{align}
The identities needed below are
\begin{equation}\label{alg:oct-identities}
 ba=\gamma,\qquad \kappa\eta=s,\qquad \lambda b=\eta r.
\end{equation}
The map $\gamma$ comes from the saddle cobordism between the two
smoothings. The foam defining $s$ has this saddle together with a
disk along the internal edge of $L_1$ that disappears. A dot on an
edge outside the disk can be moved through any of these cobordisms.
Consequently all the maps commute with the edge operators on $E$.

\input{sections/octahedron}

We collect these commuting edge operators into an algebra $R$.
For each of the 25 full edges of $E$, take one generator, represented
geometrically by a dot on that edge in the product cobordism.
The algebra is commutative.  At a trivalent exterior vertex, with incident
variables $x,y,z$, impose
\begin{equation}\label{alg:vertex-relations}
 x+y+z=0,\qquad xy+xz+yz=0,\qquad xyz=0.
\end{equation}
At a port the third variable means the sum of the two full-edge variables
incident to that vertex.  Impose the quadratic and cubic relations there
as well.  Each edge variable satisfies $x^3=0$.  These are the actual edge
relations of \cite{KM-web}; the algebra is not replaced by its image in
$\End(A\oplus B\oplus C)$.  Eliminating the 14 independent linear
relations leaves eleven generators.  A presentation uses their 55
commutators, the 18 quadratic and 18 cubic vertex relations, and their
eleven cubes.  The cube of any dependent edge follows from
\eqref{alg:vertex-relations}: if $z=x+y$, then
$x^2+xy+y^2=xy(x+y)=0$ implies $x^3=y^3=z^3=0$.

We now isolate the information about the two unknown groups $W,F$
which is needed to recover $M$. Set
\begin{equation}\label{alg:subquotients}
 V=\ker s,\quad U=\coker\gamma,\quad P=\coker s,\quad
 Z=\ker\gamma,\quad Q=Z/\im s.
\end{equation}
If $v\in V=\ker s$, exactness gives a lift $w\in W$ with $r(w)=v$.
Apply $b$ and pass to $U=B/\im\gamma$. This defines the secondary map
$h:V\to U$ by
\begin{equation}\label{alg:h}
 h(v)=[b(w)].
\end{equation}
Two choices differ by $a(x)$, and $ba(x)=\gamma(x)$, so this is well-defined.
The same observation with each exterior dot proves that $h$ is $R$-linear.

\begin{proposition}\label{alg:rank-formula}
For the octahedron \eqref{alg:triangles},
\[
 \dim M=\dim A+\dim B+\dim C
               -2\rk s-2\rk\gamma-2\rk h.
\]
\end{proposition}
\begin{proof}
Exactness gives
$\dim W=\dim A+\dim C-2\rk s$.
The subspace $\im\gamma$ lies in $\im b$ by $ba=\gamma$, and the image
of $\im b$ in $B/\im\gamma$ is precisely $\im h$.  Thus
$\rk b=\rk\gamma+\rk h$.  Apply the dimension identity for the second
triangle in \eqref{alg:triangles}.
\end{proof}

\subsection{Bases defined by foams}

We next explain the bases used to calculate the maps $s$ and $\gamma$.
A foam $I$ from the empty web to $K$ gives a class
$\J(I)(1)\in\J(K)$. The elementary face relations provide such
classes by reversing a sequence of reductions of $K$ to small webs.
For a bigon there are two creation foams, distinguished by a dot;
for a square the two resolutions give two collections of classes.
A triangle has a single reduction. Repeating the process for
$K_0,K_1,L_1$ ends at theta webs.

There is also a direct way to check that the resulting classes form
bases. Reflect a creation foam $I$ in the cobordism coordinate to
obtain $\overline I$, now a cobordism from $K$ to the empty web.
It defines a functional $\J(K)\to\F$. Joining $I_i$ to
$\overline I_j$ produces a closed dotted foam, whose invariant is
the pairing of the two classes. We will calculate these pairings
using the foam relations of \cite{KM-web}. Thus the matrix input
comes from maps in $\J$ itself; no comparison with a combinatorial
foam theory is assumed.

Here are the ordering conventions which specify the classes uniquely.
Give every graph the lexicographic ordering of
its unordered edge list.  Find a cycle by depth-first search, starting at
the least vertex and taking incident edges in this order.  Search first
for a bigon, then a triangle, then a square.  Use the standard creation
foams for the inverse reductions.  A new vertex produced by triangle
collapse receives the least unused label greater than every current label.
For a square with cyclically ordered exterior legs $0,1,2,3$, the two
branches are ordered $(01,23)$ and $(03,12)$.  The search is repeated in
each branch, in depth-first order.  For these three webs every selected
cycle is facial in the inherited rotation system.  This is a finite
check, recorded with the full reduction trees.

A bigon contributes the undotted creation and the creation with a dot on
its first cycle edge. At a terminal theta web use the six foams
\[
 x^iy^j\theta_-,\qquad 0\leq i\leq2,\quad0\leq j\leq1,
\]
where $\theta_-$ is the standard foam from the empty web to the
theta web, and $x,y$ place dots on its first two facets at the
outgoing edges. These are the ``half-foams'' used below: each has
one empty end, and becomes a closed foam after composition with a
reflected half-foam. Several examples are drawn in
Section~\ref{alg:foam-pictures}. Exponents and branches
are ordered lexicographically; the complete basis records give the edge
and stage of every dot.  Write the resulting classes as
$A_1,\ldots,A_{48}$, $B_1,\ldots,B_{36}$ and
$C_1,\ldots,C_{72}$.  The dimension formulas for bigons, triangles and
squares in \cite[Propositions 6.5--6.8]{KM-web} give the same dimensions.
Ignoring dots, the branch sizes are
\begin{equation}\label{alg:branch-sizes}
 A:(24,6,6,12),\qquad B:(12,12,12),\qquad
 C:(12,12,12,12,12,12).
\end{equation}
The assertion that these particular half-foams are bases can also be checked
by their nonsingular Gram matrices, as follows.

For $X=A,B,C$, let $I_i^X$ denote the creation foam representing
the $i$th class in the corresponding group. Its reflection
$\overline I_i^X$ has the opposite direction. The pairing matrix and
the matrices obtained by inserting $s$ or $\gamma$ between two
such foams are
\begin{align*}
 G_X(j,i)&=\J(\overline I_j^X I_i^X),\\
 P_s(j,i)&=\J(\overline I_j^A s I_i^C),&
 P_\gamma(j,i)&=\J(\overline I_j^B\gamma I_i^A).
\end{align*}
We compose from right to left in these formulas. For example,
$P_s(j,i)$ is obtained by creating $C_i$, applying the foam for $s$,
and closing with the reflection of the foam for $A_j$.
The ordinary product Hopf marking stays outside all these moves.
A \emph{movie} describes the resulting foam by its successive
three-dimensional slices. There are 97 movies after the dots have
been forgotten. Different dot placements give the remaining matrix
entries without introducing further underlying foams.

The closed foams can be evaluated by removing pairs of tetrahedral
points and then applying neck-cutting. The following form of
neck-cutting is convenient for this calculation.
Suppose a facet closes to an embedded disk $\Delta$, with boundary traversing
one seam circle once and with two distinct other incident facets.  Put $d$
dots on $\Delta$, and let $F_{a,b}$ be the foam obtained by deleting
$\Delta$ and its seam, capping the two other facets by parallel copies of
$\Delta$, and putting $a,b$ dots on the new caps.  Embedded neck-cutting
and the theta evaluation give
\begin{equation}\label{alg:disk-peel}
 \J(F)=\sum_{u,v=0}^2\Theta(d,u,v)\J(F_{2-u,2-v}),
\end{equation}
where $\Theta(d,u,v)=1$ exactly when $(d,u,v)$ is a permutation of
$(0,1,2)$.  Indeed, two small disjoint push-offs of $\Delta$ supply the
framed surgery disks in the adjacent sheets.  Neck-cutting separates a
standard theta foam from $F_{2-u,2-v}$.  This proves
\eqref{alg:disk-peel} by \cite[Propositions 5.6 and 6.1]{KM-web}.
No compression disk inferred solely from an abstract surface is used.

The supporting record specifies the facets of each movie after
separating them along the seams, their Euler characteristics, the
order in which their boundaries traverse the seams, and the
locations of the dots throughout the reductions.
A nonbipartite seam graph gives zero by
\cite[Corollary 8.5]{KM-web}, with dots covered by the dotted form of
Proposition 8.2, or by dot-to-handle replacement.
Otherwise use the embedded cancellation of two tetrahedral points along
a seam, \cite[Proposition 6.7]{KM-web}.  Its local boundary is the triangular
prism of Figure 11.  Follow the three seam-leg pairings and the bands on the
opposite corners.  After cancellation the certificate checks trivial sheet
monodromy and three distinct facets at every seam circle.  A connected
facet with Euler characteristic one and one boundary component is then an
embedded disk to which \eqref{alg:disk-peel} applies.  Repetition leaves
spheres in $\mathbb R^3$, evaluated by
\cite[Proposition 5.3]{KM-web}.  Dots are retained on tracked facets at
every step.

\begin{center}
\begin{tabular}{lrrr}
\hline
Family & Movies & Nonbipartite & Reduced to spheres\\\hline
$\overline I_j^A I_i^A$ &16&6&10\\
$\overline I_j^B I_i^B$ &9&2&7\\
$\overline I_j^C I_i^C$ &36&6&30\\
$\overline I_j^A s I_i^C$ &24&7&17\\
$\overline I_j^B\gamma I_i^A$ &12&9&3\\\hline
Total &97&30&67\\\hline
\end{tabular}
\end{center}

The full list of movies and local replacements is supplied with the
draft. These geometric calculations are distinct from the subsequent
linear algebra: checking a matrix rank does not establish that its
entries are the invariants of the specified embedded foams.
The evaluations use the listed relations for $\J$ and do not assume
a general equality between $J^\flat$ and $\J$.

\begin{proposition}[Finite pairing calculation]\label{alg:finite-input}
The specified half-foams have nonsingular Gram matrices of ranks
$48,36,72$.  In these bases, the capped evaluations and the exterior dot
evaluations give the matrices supplied with this draft, with
\begin{equation}\label{alg:input-ranks}
 \dim(A,B,C)=(48,36,72),\qquad
 \rk s=35,\qquad \rk\gamma=11.
\end{equation}
\end{proposition}
\begin{proof}[Computational proof and its scope]
Construct the 97 movies by stacking the specified reduction half-foams
and, when appropriate, the saddle or unzip.  The certificate applies the
preceding embedded moves, and evaluates their decorations by
\eqref{alg:disk-peel} and the sphere rule.  Nonsingularity of $G_X$,
together with the independently known dimensions, proves basis
completeness.  The matrices of the actual maps are consequently
\[
 s=G_A^{-1}P_s,\qquad \gamma=G_B^{-1}P_\gamma.
\]
Adding a dot at an exterior edge computes its action by the same method.
The full binary matrices and the move-by-move certificate are supporting
data.  The independently written checker accompanying this draft verifies
\eqref{alg:input-ranks}, the vertex relations and all subquotient and
intertwining identities from these matrices.  It does not constitute a
second implementation of the geometric movie evaluator.
\end{proof}

It follows already that $\dim W=50$ and
\begin{equation}\label{alg:64minus}
 \dim\J(D)=64-2\rk h.
\end{equation}
The dimensions of $(P,Z,Q,U,V)$ are $(13,37,2,25,37)$.

\subsection{The two extension constraints}

Commuting with the edge operators is not the only condition on $h$.
It must occur in the actual octahedron, in which the unknown groups
$W$ and $F$ are extensions of known $R$-modules. We spell out the
conditions for these extensions to exist. This turns the remaining
restriction on $h$ into a linear calculation, without trying to
construct $W$ or $F$ geometrically.

The exact sequences
\begin{align}
 0\longrightarrow Q\longrightarrow P\xrightarrow{\bar\gamma}B
       \longrightarrow U\longrightarrow0,&\label{alg:yoneda-bottom}\\
 0\longrightarrow V\longrightarrow C\xrightarrow{s_Z}Z
       \longrightarrow Q\longrightarrow0&\label{alg:yoneda-top}
\end{align}
define classes $e\in\operatorname{Ext}^2_R(U,Q)$ and
$f\in\operatorname{Ext}^2_R(Q,V)$.  The actual octahedron implies
$h^*e=0$ and $h_*f=0$. The following equations express these two
conditions directly in terms of the edge-operator matrices.

Choose vector-space splittings of the short exact sequences furnished by
\eqref{alg:triangles}.  For $0\to P\to W\to V\to0$, an exterior
generator acts by
\[
 \begin{pmatrix}P_e&\omega_e\\0&V_e\end{pmatrix},
 \qquad b=(\bar\gamma,f_0).
\]
The condition that $b$ be $R$-linear and induce $h$ is
\begin{equation}\label{alg:bottom-equations}
 B_e f_0+f_0V_e=\bar\gamma\omega_e,\qquad \pi f_0=h,
\end{equation}
where $\pi:B\to U$ is the quotient.  In addition the upper triangular
matrices must obey every relation of $R$.
For $0\to U\xrightarrow{\lambda}F\xrightarrow{\kappa}Z\to0$,
write the action and $\eta$ as
\[
 \begin{pmatrix}U_e&\omega'_e\\0&Z_e\end{pmatrix},
 \qquad \eta=\begin{pmatrix}g_0\\s_Z\end{pmatrix}.
\]
Then
\begin{equation}\label{alg:top-equations}
 U_e g_0+g_0C_e=\omega'_e s_Z,\qquad g_0i_V=h,
\end{equation}
where $i_V:V\hookrightarrow C$ is the inclusion.  The second equality
follows from $\eta r=\lambda b$.
These equations prove necessity of both extension constraints without
asserting that their solutions are realizable by geometric octahedra.

For completeness, the relation equations are linear, including cubic
relations.  If a word $w=e_1\cdots e_k$ acts on an extension of $Y$ by
$X$, its upper-right block is
\begin{equation}\label{alg:linear-relations}
 \sum_{j=1}^k X_{e_1}\cdots X_{e_{j-1}}
       \omega_{e_j}Y_{e_{j+1}}\cdots Y_{e_k}.
\end{equation}
For a polynomial relation, sum these expressions over its words.  There is
exactly one unknown block in each term.  Thus
\eqref{alg:bottom-equations} and \eqref{alg:top-equations}, together with
\eqref{alg:linear-relations}, are finite binary linear systems.

\begin{proposition}[Computed obstruction space]\label{alg:alternating}
There are an injection $\iota:\F^3\to U$ and a surjection
$\rho:V\to\F^3$, specified in the supporting data, such that the actual
map $h$ has the form
\begin{equation}\label{alg:H}
 h=\iota H\rho,\qquad
 H=\begin{pmatrix}0&\alpha&\chi\\
                  \alpha&0&\beta\\
                  \chi&\beta&0\end{pmatrix}.
\end{equation}
In particular $\rk h$ is zero or two, and $\dim\J(D)$ is 64 or 60.
\end{proposition}
\begin{proof}[Exact finite verification]
First solve all equations $U_eh=hV_e$ on the 925 entries of $h$; their
solution space has dimension 48.  Next solve the two full extension systems,
retaining all entries as unknowns and eliminating extension variables
before the entries of $h$.  No grading or restriction to homogeneous
extensions is imposed.  For the eleven-generator presentation above the
results are
\begin{center}
\begin{tabular}{lrrrr}
\hline
System & Variables & Rank & Constraints on $h$ & Allowed $h$ dimension\\\hline
\eqref{alg:bottom-equations} &7548&6928&922&3\\
\eqref{alg:top-equations} &12900&11868&922&3\\\hline
\end{tabular}
\end{center}
The two projected three-dimensional subspaces agree.  Their union with
the three coefficient matrices of $\iota H\rho$ still has dimension
three.  The checks also give $\rk\iota=\rk\rho=3$, proving the asserted
normal form for every admissible $h$.  The independently written program
\texttt{support/finite/check\_finite.py} constructs these ungraded systems
directly from the stored edge matrices.  It imports no code from the
original evaluator and uses a different set of eleven independent edges.
Its outputs include the newly computed bases of both projected spaces.

Finally, a nonzero alternating $3\times3$ matrix over $\F$ has rank two:
a nonzero off-diagonal entry gives a nonsingular $2\times2$ minor, whereas
its determinant vanishes.  Injectivity of $\iota$ and surjectivity of
$\rho$ preserve rank.  Apply \eqref{alg:64minus}.
\end{proof}

\subsection{Extracting the three coefficients}

To use gauge theory to calculate $h$, we need to express its three
coefficients by specified foams at the incoming and outgoing ends.
The following choices do this. They fix the coordinates in the
normal form \eqref{alg:H}, as well as the dots which will enter the
index calculation. Put
\begin{align*}
 c_1&=C_{14}+C_{19},\\
 c_2&=C_4+C_{40}+C_{65}+C_{68},\\
 c_3&=C_2+C_{14}+C_{15}+C_{27}+C_{38}+C_{50}+C_{51}+C_{67}.
\end{align*}
These satisfy $s(c_j)=0$ and $\rho(c_j)=e_j$.  Here and below we identify
$V$ with its inclusion in $C$. Reflecting the foam which represents
$B_i$ gives a cobordism from $K_1$ to the empty web. Let
$\operatorname{cap}(B_i):B\to\F$ be the resulting functional.
Its value on $B_j$ is $G_B(i,j)$, so its matrix is the $i$th row
of the pairing matrix, rather than the $i$th coordinate covector.
Define
\begin{align*}
 \ell_1&=\operatorname{cap}(B_2)+\operatorname{cap}(B_7),\\
 \ell_2&=\operatorname{cap}(B_{33}),\\
 \ell_3&=\operatorname{cap}(B_2)+\operatorname{cap}(B_{15})
           +\operatorname{cap}(B_{19})+\operatorname{cap}(B_{26}).
\end{align*}
Each $\ell_i$ annihilates $\im\gamma$ and so descends to $U$.
The supporting matrices verify $\ell_i\iota=e_i^T$; equivalently this can
be checked against the three quotient lifts
\begin{align*}
 x_1&=B_{11}+B_{18},\\
 x_2&=B_4+B_8+B_9+B_{16}+B_{17}+B_{21}+B_{28},\\
 x_3&=B_6+B_{11},
\end{align*}
for which $[x_j]=\iota(e_j)$ and $\ell_i(x_j)=\delta_{ij}$.
Consequently only two sources and two caps are needed:
\begin{equation}\label{alg:scalar-tests}
 \alpha=\ell_2h(c_1),\qquad
 \beta=\ell_2h(c_3),\qquad
 \chi=\ell_1h(c_3).
\end{equation}
Every summand of $c_1,c_3$ has one dot; the cap $B_{33}$ has two dots,
and each of $B_2,B_7$ has one dot.  Thus the three geometric expressions
have respectively three, three and two dots.  All dot locations are
specified by the stage and edge labels in the basis data.

For the last coefficient, it is essential to retain the two caps
together. It is their sum
$\operatorname{cap}(B_2)+\operatorname{cap}(B_7)$ that annihilates
$\im\gamma$; this is not asserted for either summand separately.
The source and cap identities, with the caps interpreted as the
pairing-matrix rows just described, are checked by the independent
linear-algebra calculation.

The classes $c_j$ are represented by the corresponding sums of actual
relative foam cycles for any compatible geometric Floer data.  The equality
$s(c_j)=0$ in this section means that their images are boundaries, not
that the image chain is literally zero. Sections~\ref{loc:section}
and~\ref{op:section} construct primitives for those boundaries and
identify the resulting secondary operation with \eqref{alg:h};
Section~\ref{en:section} supplies their simultaneous analytic control.

\input{sections/half_foam_pictures}