\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{booktabs,longtable,array,graphicx}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,positioning}
\usepackage{enumitem}
\usepackage{microtype}
\usepackage[colorlinks=true,linkcolor=black,citecolor=black,urlcolor=blue,
 pdfauthor={},pdftitle={The dodecahedron web: integrated referee draft, version 5}]{hyperref}
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposedtheorem}[theorem]{Proposed theorem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\F}{\mathbb F_2}
\newcommand{\J}{J^\sharp}
\newcommand{\CF}{CF^\sharp}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\rk}{rank}
\DeclareMathOperator{\coker}{coker}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\Ext}{Ext}
\setlist[enumerate]{itemsep=3pt,topsep=5pt}
\setlength{\emergencystretch}{1.5em}
\setcounter{tocdepth}{1}
\input{figures/foam_styles}
\title{The dodecahedron web and its instanton homology}
\author{P. B. Kronheimer and T. S. Mrowka}
\date{November 2026}
\begin{document}
\maketitle

\input{sections/foreword}


\begin{abstract}
For the planar dodecahedron web $D$, the octahedral diagram of instanton
homology reduces the possible values of $\dim_{\F}\J(D)$ to $60$ and
$64$. The remaining choice is determined by three coefficients of a
secondary map. We give a proposed proof that all three vanish and that
the dimension is $64$. We express the coefficients as counts of instantons
on specified foam cobordisms with one-parameter families of metrics.
The index calculation requires these instantons to have negative action.
An estimate uniform under neck stretching excludes them when the
perturbations are sufficiently small. The construction includes local
Floer complexes with zero differential, the marking calculation for
excision, and a simultaneous choice of regular perturbations compatible
with the families. Exact matrices and the supporting foam calculations
are provided separately.
\end{abstract}

% \begin{center}
% \fbox{\begin{minipage}{0.91\textwidth}\small
% \textbf{Status.} This manuscript presents a proposed proof for expert review.
% The argument for the simultaneous choice of geometric data is included
% in Theorem~\ref{en:simultaneous-data}. The dimension-$64$ conclusion,
% including the local perturbation and index calculations on which it
% depends, remains provisional pending independent mathematical review.
% \end{minipage}}
% \end{center}
\begingroup\small
\tableofcontents
\endgroup
\clearpage


\input{sections/introduction}
\input{sections/finite}
\input{sections/local_models}
\input{sections/localization}
\input{sections/operation}
\input{sections/geometry}
\input{sections/energy}
\input{sections/conclusion}
\appendix
\input{sections/computation}
% \input{sections/integration}

\clearpage
\begin{thebibliography}{ISST}
\item[]
References to section, theorem, proposition, equation and figure numbers
in [KM1]--[KM5] use the arXiv versions linked below, rather than the
journal versions.
\medskip

\bibitem[KM1]{KM-web}
P.~B. Kronheimer and T.~S. Mrowka,
\emph{Tait colorings, and an instanton homology for webs and foams},
J. Eur. Math. Soc. \textbf{21} (2019), no.~1, 55--119.
\href{https://doi.org/10.4171/JEMS/831}{doi:10.4171/JEMS/831}.
\url{https://arxiv.org/abs/1508.07205v1}.

\bibitem[KM2]{KM-tri}
P.~B. Kronheimer and T.~S. Mrowka,
\emph{Exact triangles for $SO(3)$ instanton homology of webs},
J. Topol. \textbf{9} (2016), no.~3, 774--796.
\href{https://doi.org/10.1112/jtopol/jtw010}{doi:10.1112/jtopol/jtw010}.
\url{https://arxiv.org/abs/1508.07207v1}.

\bibitem[KM3]{KM-def}
P.~B. Kronheimer and T.~S. Mrowka,
\emph{A deformation of instanton homology for webs},
Geom. Topol. \textbf{23} (2019), no.~3, 1491--1547.
\href{https://doi.org/10.2140/gt.2019.23.1491}{doi:10.2140/gt.2019.23.1491}.
\url{https://arxiv.org/abs/1710.05002v2}.

\bibitem[KM4]{KM-unknot}
P.~B. Kronheimer and T.~S. Mrowka,
\emph{Khovanov homology is an unknot-detector},
Publ. Math. IH\'ES \textbf{113} (2011), 97--208.
\href{https://doi.org/10.1007/s10240-010-0030-y}{doi:10.1007/s10240-010-0030-y}.
\url{https://arxiv.org/abs/1005.4346v1}.

\bibitem[KM5]{KM-yaft}
P.~B. Kronheimer and T.~S. Mrowka,
\emph{Knot homology groups from instantons},
J. Topol. \textbf{4} (2011), no.~4, 835--918.
\href{https://doi.org/10.1112/jtopol/jtr024}{doi:10.1112/jtopol/jtr024}.
\url{https://arxiv.org/abs/0806.1053v2}.

\bibitem[ISST]{ISST}
H.~Imori, T.~Sano, K.~Sato and M.~Taniguchi,
\emph{Cobordism maps in Khovanov homology and singular instanton homology I},
arXiv:2505.12095v2 (2026).
\url{https://arxiv.org/abs/2505.12095v2}.
\end{thebibliography}
\end{document}