\section{Introduction}\label{intro:section}

A web is a trivalent graph embedded in $\mathbb R^3$, with the local
model of three coplanar rays at each vertex. A Tait coloring assigns one
of three colors to each edge, with three distinct colors at each vertex.
Colors are labelled: two colorings related by a permutation of the
colors are counted separately.

The instanton homology $\J(K)$ associates a finite-dimensional vector
space over $\F$ to a web $K$. It is defined in \cite{KM-web} using
$SO(3)$ orbifold connections. The local isotropy group at an edge
acts on the fiber by a nontrivial involution; at a vertex, the
isotropy representation is the standard Klein four-group in $SO(3)$. A disjoint Hopf link, placed
near infinity and equipped with marking data, removes automorphisms
from the configuration space. We use this marked theory throughout.

A foam is a cobordism between webs, with three sheets meeting along
seams and with isolated tetrahedral points allowed. It induces a map
on $\J$. Foams may also carry dots on their facets; each dot imposes
a degree-one cohomological constraint in the moduli-space count.
The elementary relations between these maps make it possible to
calculate $\J$ for many planar webs. In particular, a bigon, triangle
or square face can be removed by the relations in
\cite[Section~6]{KM-web}.

For planar webs the conjecture of \cite{KM-web} is
\[
 \dim_{\F}\J(K)=\#\{\text{Tait colorings of }K\}.
\]
The planar dodecahedron $D$ is the first test case to which none of
these three face reductions applies. It has sixty Tait colorings.
The deformation constructed in \cite{KM-def} gives the lower bound
$\dim\J(D)\geq60$: its generic rank is the Tait count, and the
undeformed dimension is at least this rank
\cite[Theorem~1.2 and Corollary~1.3]{KM-def}.

Choose an edge of $D$ and consider the local replacements used in the
octahedral diagram of \cite{KM-tri}. The webs agree outside a disk
containing this edge and its two endpoints. We denote this common
exterior by $E$; it is shown in Figure~\ref{fig:dodecahedron-exterior}.
Three of the resulting closed webs, denoted $K_0,K_1,L_1$, can be
reduced by the elementary face relations. Their homology groups and
the maps between them can therefore be calculated. The six local
pictures, together with the maps in the octahedron, are drawn in
Figure~\ref{fig:octahedron}.

\input{figures/dodecahedron_exterior}

The octahedron does not determine $\J(D)$ from these calculations
alone. There remains a map $h$ between subquotients of the known
groups. It is obtained by lifting through one map in an exact
triangle and then applying another; its definition is given in
\eqref{alg:h}. The calculation in Section~\ref{alg:section} gives
\begin{equation}\label{intro:reduction}
 \dim\J(D)=64-2\rk h,\qquad
 h=\iota
 \begin{pmatrix}0&\alpha&\chi\\\alpha&0&\beta\\\chi&\beta&0\end{pmatrix}
 \rho,
\end{equation}
where $\iota$ is injective and $\rho$ is surjective. Thus the unknown
part of the calculation consists of three coefficients in $\F$.
A nonzero alternating $3\times3$ matrix over $\F$ has rank two, so
the two remaining possibilities are $60$ and $64$. The upper bound
here is $64$, correcting the value $70$ reported in the earlier papers.

\begin{proposedtheorem}[Dodecahedron dimension]\label{intro:main}
For the planar dodecahedron web $D$, the three coefficients in
\eqref{intro:reduction} vanish:
\[
 \alpha=\beta=\chi=0,\qquad \dim_{\F}\J(D)=64.
\]
\end{proposedtheorem}

The proposed conclusion would give a counterexample to the planar
Tait-coloring conjecture. The word \emph{proposed} records the status
of the argument: the complete proof has not received independent
expert verification. In particular, the finite calculation leaving
$60$ or $64$ must be distinguished from the local perturbation,
excision and index arguments used here to select $64$. A proof of
the simultaneous choice of geometric data is included as
Theorem~\ref{en:simultaneous-data}, and the argument is assembled in
Section~\ref{conclusion:section}.

\subsection{Outline of the proof}

We first explain how the finite calculation is related to the geometry.
A foam from the empty web to a web $K$ determines an element of
$\J(K)$, since $\J(\varnothing)=\F$. Reversing the cobordism gives
a foam from $K$ to the empty web, and hence a linear functional on
$\J(K)$. Composing these two foams gives a closed foam whose invariant
is the value of the functional on the element. This is the pairing
used to verify bases and compute matrices in Section~\ref{alg:section}.
The reflection is in the cobordism coordinate; it is not an additional
operation on the planar web.

For $K_0,K_1,L_1$, repeated face reductions supply collections of
such foams. Evaluating their pairings gives dimensions $48,36,72$
and gives the two map ranks $35,11$ needed for the octahedron.
Dots on edges of the common exterior act on all the groups and
commute with all the maps. The resulting module relations, together
with the two extensions in the octahedron, constrain $h$ to the
form in \eqref{intro:reduction}. Specified incoming foams and
outgoing foams then extract its three coefficients. No identification
of $\J$ with a combinatorial foam homology is used.

To determine these coefficients we need more than relations on
homology. The definition of a secondary map involves choosing a
primitive for a boundary in a Floer complex. An arbitrary algebraic
choice of primitive would give no control over the topology of the
four-manifolds on which the relevant moduli spaces are defined. We instead
construct the primitive by counting instantons over specified
one-parameter families of metrics and perturbations.

The construction begins with the small webs that occur in the local
foam relations. In Section~\ref{lm:section} we choose perturbations
for which their actual Floer complexes have zero differential.
A relation between relative homology classes on one of these webs
is then an equality between the corresponding relative chains.
Section~\ref{loc:section} inserts these local relations into the
larger cobordisms and constructs the required chain homotopies by
stretching necks and applying excision. Combining the homotopies
along the sequence of face reductions gives a single primitive for
both source cycles that we need. Section~\ref{op:section} compares
this construction with the secondary map $h$, using the family
which proves the exact triangle for the three $L$-webs.

One point in excision requires particular care. Cutting the composite
of the Hopf-excision cobordism and its reverse produces two torus
ends. Each has two determinant-one lifts of its flat connection,
but the two lifts cannot be quotiented independently. The marking
data identify pairs by the simultaneous interchange
$(i,j)\mapsto(i+1,j+1)$, with indices in $\mathbb Z/2$.
There are consequently two joint states,
\[
 \mathcal T_\Delta=\F\mathfrak a\oplus\F\mathfrak b,
 \qquad \mathfrak a=[00]=[11],\quad \mathfrak b=[01]=[10].
\]
Both the flat cylinder and the pair of caps select the matching state
$\mathfrak a$, with multiplicity one. Section~\ref{loc:torus-geometry}
explains the geometry and marking data behind this assertion.

The final step is an index and action argument. Compose each incoming
foam, one-parameter operation and outgoing foam to obtain a capped
four-dimensional configuration $(X,\Sigma)$. Section~\ref{geo:section}
calculates the part of its index independent of the instanton action:
\[
 I_{\mathrm{top}}
 =-\tfrac32(\chi(X)+\sigma(X))+
   \chi(\Sigma)+\tfrac12\nu(\Sigma)-\tfrac12\tau(\Sigma)=\tfrac52.
\]
Here $\nu(\Sigma)$ is the foam self-intersection number and
$\tau(\Sigma)$ is the number of tetrahedral points. Including the
one family parameter and the dot constraints, a zero-dimensional
moduli space would have action $-1/16$ in the three-dot case or
$-3/16$ in the two-dot case.

For an unperturbed anti-self-dual connection the action is
nonnegative. With holonomy perturbations it need not be, but the
energy identity gives a lower bound $\kappa\geq-\delta$, where
$\delta$ bounds the total error introduced by the perturbations.
We must make this error less than $1/16$ for all the cobordisms and
families just described, uniformly as their necks become long.
Section~\ref{en:section} constructs compatible regular data with
this property. Both required negative action values are then
excluded, so the three counts vanish.

\subsection{Conventions}

All vector spaces are over $\F$. In Section~\ref{alg:section},
$A,B,C,W$ denote homology groups. In the later sections the same
letters denote geometric Floer complexes for the same webs, with
homology $HA,HB,HC,HW$. We retain the symbols for cobordism maps
when passing to homology, specifying the level when it matters.
Matrices act on column vectors. Basis subscripts are one-based;
arrays and branch indices in the supporting code are zero-based
unless otherwise stated.

All action comparisons retain the relative bundle class. The normal
number $\nu$ is the self-intersection of
\cite[Definition~2.5]{KM-web}, not the full relative Euler number of
the squared normal bundle. We use no absolute integer grading on
general $\J$. The auxiliary degree labels in the finite data are
not such a grading, and the obstruction calculation does not use them.

We take the theory of marked bifold connections, elliptic gauge fixing
and ordinary Floer compactification from the cited papers. The
additional local perturbations, families and index calculations are
developed here. The framed arcs and the specified augmentation of the
terminal web are part of the geometric data. An isomorphism between
homology groups does not identify their boundary webs: for example,
the two composites in the triangle relation have theta and prism
boundaries, respectively.

The supporting files contain both the exact linear algebra and the
lists of foam moves used to obtain its input. The former can check
ranks and identities once the matrices have been supplied; the latter
record the geometric calculation of those matrices. Neither replaces
a proof of transversality or gluing. Appendix~\ref{app:computation}
describes how to reproduce the calculations and the scope of the
checks.