[\cite{wagner}]
	\label{wagner-thm}
	If a graph $G$ is maximal $K_5$-minor-free, then $G$ can be built by a sequence of $2$-sums and
	$3$-sums, without deleting edges, starting from maximal planar graphs and copies of $M_8$.

\subsection{Definitions and Examples}
Correspondence coloring generalizes list coloring, and was introduced to resolve a longstanding open
question \cite{DP} on the list-chromatic number of certain planar graphs.  Recall that a list assignment
$L$ for a graph $G$ gives to each vertex $v$ of $G$ a set $L(v)$ of allowable colors.  An \emph{$L$-coloring}
is a proper coloring $\vph$ such that $\vph(v)\in L(v)$ for all vertices $v$.  If $|L(v)|=t$ for all $v$,
then $L$ is called a $t$-assignment and an \emph{$L$-packing}
consists of $L$-colorings $\vph_1,\ldots,\vph_t$ such that $\vph_i(v)\ne \vph_j(v)$ for all vertices $v$
and all distinct $i,j\in[t]$.  That is, the colors $\vph_i(v)$ partition $L(v)$ for each vertex $v$.

	For a graph $G$, a \emph{correspondence $t$-cover} $\MM$ assigns
	to each $v\in V(G)$ a set of allowable colors $\{1_v,\ldots,t_v\}$ and to each edge $vw\in E(G)$
	a matching between $\{1_v,\ldots,t_v\}$ and $\{1_w,\ldots,t_w\}$. An $\MM$-coloring $\vph$
	picks for each vertex $v$ a color $\vph(v)$ (from the set $\{1_v,\ldots,t_v\}$) such that for
	each edge $vw\in E(G)$ the colors $\vph(v),\vph(w)$ are not matched to each other.
	(It is straightforward to verify that correspondence coloring generalizes list coloring.)
	Two $\MM$-colorings $\vph_1,\vph_2$ of $G$ are \emph{disjoint} if $\vph_1(v)\ne\vph_2(v)$ for all
	$v\in V(G)$.  An $\MM$-packing for $G$ consists of disjoint $\MM$-colorings $\vph_1,\ldots, \vph_t$.

	The \emph{list-chromatic number}, $\chil(G)$, and \emph{list packing number},
	$\chisl(G)$, are the minimum values of $t$ such that for every $t$-assignment $G$ admits,
	respectively, a list coloring and a list packing.  Analogously, for correspondence $t$-covers, we
	define the \emph{correspondence chromatic number}, $\chic(G)$, and the \emph{correspondence packing
	number}, $\chisc(G)$. List packing and correspondence packing were introduced in~\cite{CCvBDK}, where
	the authors discussed connections with other areas of discrete math.  These parameters have since
	been studied in numerous papers~\cite{BCK25,CCvB,CCvBDK2,CCvBZ,CH23,CDKH24,CSr,KMMP22,KM25,M23}.

	As a warmup with these definitions, we provide a few examples.  Note that always $\chi(G)\le \chil(G)
	\le \chic(G)\le \chisc(G)$; and also $\chil(G)\le \chisl(G)\le \chisc(G)$.  For a graph class $\G$
	and a graph parameter $f$, let $f(\G):=\max_{G\in \G}f(G)$.  Let $\F$ denote the class of forests,
	$\C$ the class of cycles, and $\C_e$ the class of cycles of even length.
	So $2=\chi(\F)=\chil(\F)=\chisl(\F)=\chic(\F)=\chisc(\F)$.  The lower bound is trivial, and the
	upper bound follows from \Cref{ext-lem} below.  It is well-known (and easy to check) that
	$\chil(\C_e) = 2 < 3 = \chil(\C)$.  But the reader may find the following more surpising:
	$\chisl(\C_e)=\chisl(\C)=3$, $\chic(\C_e)=\chic(\C)=3$, and $\chisc(\C_e)=\chisc(\C)=4$.
	In the first case, the upper bound requires a bit of work.  But in the second and third cases,
	the upper bounds hold because the graphs are $2$-degenerate; in
	particular, the last of these follows from \Cref{ext-lem} below.  The lower bounds on both
	$\chic(\C_e)$ and $\chisc(\C_e)$ come from constructing a cover that violates a necessary
	parity-type invariant needed to find the desired coloring or packing.
	For more details on these lower bounds, see \cite{CCvBDK2} or the introduction of~\cite{CSr}.

\subsection{Main Result}
The authors of \cite{CCvBDK} proposed the problem of determining $\chisc(\G)$ for various graph classes
$\G$.  Let $\K_r$ denote the class of all graphs that are $K_r$-minor-free.  It is easy to check that
$\chisc(\K_2)=1$, $\chisc(\K_3)=2$, and $\chisc(\K_4)=4$; the final upper bound follows from Lemma~\ref{ext-lem}
below because all graphs in $\K_4$ are 2-degenerate.  So the first open case is $\chisc(\K_5)$.
The best known upper bound is $\chisc(\K_5)\le 8$.  This follows from \cite[Theorem~1.7]{CCvBZ} because every
$K_5$-minor-free graph has maximum average degree less than 6.  We cautiously believe that this upper bound
can be improved as follows.

\begin{conj}\label{conj1}
	If $G$ is $K_5$-minor-free and $\MM$ is a correspondence 6-cover, then $G$ has 6 disjoint
	$\MM$-colorings.  That is, $\chisc(\K_5)=6$.
\end{conj}

Dvo\v{r}\'{a}k, Norin, and Postle~\cite{DNP19} called a graph \emph{weighted $\epsilon$-flexible} with
respect to a list-assignment $L$ if there exists a probability distribution on the $L$-colorings $\phi$
of $G$ such that $\mathbb{P}(\phi(v)=c) \ge \epsilon$ for every vertex $v$ and color $c\in L(v)$.
In other words, weighted $\epsilon$-flexibility ensures that at each vertex each color has a decent chance
to appear in the random list-coloring.  Dvo\v{r}\'{a}k, Norin, and Postle  proved that there is an absolute
constant $\epsilon>0$ such that all planar graphs are weighted $\epsilon$-flexible with respect to every
$7$-assignment, and they posed as an open problem to show the same for $6$-assignments or even
$5$-assignments.  While this problem remains wide open, for $7$-assignments the value of $\epsilon$ has been
improved considerably from $7^{-36}$ to $2^{-7}$ by~\cite{KMMP22} and~\cite{CCvB}.
As detailed in~\cite{CCvBZ} and~\cite{KMMP22}, every graph $G$ is weighted $\frac{1}{\chisl(G)}$-flexible
with respect to every $\chisl(G)$-assignment. Thus a proof of Conjecture~\ref{conj1}  will make significant
progress: it will imply that every planar graph (which is $K_5$-minor-free) is weighted
$\frac{1}{6}$-flexible with respect to every $6$-assignment.
\bigskip

The main goal of this note is to prove the following result, in support of the above conjecture.

\begin{mainthm}
	\label{thm1}
	Let $G$ be a $K_5$-minor-free graph.  (i) For every correspondence 6-cover $\MM$ of $G$,
	there exist disjoint $\MM$-colorings $\vph_1,\vph_2,\vph_3$.  That is, for all $v\in V(G)$
	the colors $\vph_1(v),\vph_2(v),\vph_3(v)$ are pairwise distinct.
	(ii) But the analogous statement is false if we replace 6-cover with 5-cover.
\end{mainthm}

The fact that $K_5$-minor-free graphs are $5$-degenerate easily implies that they have list-chromatic number
at most $6$, which is  close to the optimal bound $5$. However, for the correspondence packing number this
approach performs quite poorly.  As discussed above, $\chisc(\C) \le 4$ due to the $2$-degeneracy of cycles.
More generally, by \Cref{ext-lem} we have $\chisc(G)\leq 2d$ for every $d$-degenerate graph $G$, and this is
sharp \cite[Prop. 24]{CCvBDK} for every $d \ge 1$, due to large complete bipartite graphs. Therefore a vanilla
degeneracy argument can merely yield $\chisc(\K_5) \le  10$, while it is known that $\chisc(\K_5)\le 8$. This
indicates that establishing Conjecture~\ref{conj1} is nontrivial. On the lower bound side, note that the
second part of our Main Theorem implies that $\chisc(\K_5)\ge 6$.
\bigskip

\subsection{Proof overview}
When constructing disjoint list colorings and correspondence colorings, it is often convenient
to proceed by induction.  Given a graph $G$, a correspondence $t$-cover $\MM$ of $G$
and disjoint partial $\MM$-colorings $\vph^0_1,\ldots,\vph^0_s$, for each vertex $v$
that is not yet colored we seek a color $\vph_j(v)$ to extend $\vph^0_j$ to $v$.
We write $L_j(v)$ for the subset of $[t]$ that is not yet forbidden from use as
$\vph_j(v)$, due to the colors already used on $N(v)$ and the matchings $\MM$.
The following lemma, from~\cite{CCvBDK}, is generally useful for handling vertices of low degree.
Its proof is instructive, so we include it.

\begin{lemA}[\cite{CCvBDK}]
	\label[lem]{ext-lem}
	Let $G$ be a graph with a correspondence $t$-cover $\MM$ and fix $v\in V(G)$.
	If $d(v)\le t/2$, then any disjoint $\MM$-colorings $\vph_1',\ldots,\vph_s'$
	of $G-v$ extend to disjoint $\MM$-colorings $\vph_1,\ldots,\vph_s$ of $G$.
\end{lemA}
\begin{proof}
	We build an auxiliary bipartite graph $\B$ with $1,\ldots,t$ as the vertices of the first
	part, and $\vph'_1,\ldots,\vph'_s$ as the vertices of the second part, and $i\vph'_j\in E(\B)$
	if $i\in L_j(v)$.  To extend the $\MM$-colorings $\vph_j'$, we seek a matching $M_{\B}$
	saturating the second part.  For this we use Hall's Theorem.

	Note that $d_{\B}(i)\ge s-d(v)$ and $d_{\B}(\vph'_j)\ge t-d(v)$, for all $i\in[t]$
	and $j\in[s]$.  Consider $S\subseteq\{\vph_1',\ldots,\vph_s'\}$.  Clearly
	$|N_{\B}(S)|\ge t-d(v)\ge t/2$.  So if $|S|\le t/2$, then $|N_{\B}(S)|\ge |S|$
	as desired.  Otherwise, $|S|>t/2\ge d(v)$. So $|S|+d_{\B}(i) \ge |S|+s-d(v) > s$ because
	$|S|> d(v)$.  So by Pigeonhole $i\in N_{\B}(S)$ for all $i\in [t]$.  That is, $|N_{\B}(S)| =
	t \ge |S|$, as desired.  Thus, by Hall's Theorem we have the desired matching $M_{\B}$ in $\B$.
\end{proof}

By definition, each matching in $\MM$ need not be perfect.  But to prove the results in this paper, it
suffices to consider the case that all matchings are perfect; if not, then we can grow them greedily.
(In what follows, we implicitly assume that all matchings in $\MM$ are perfect.)

An \emph{$r$-sum} of graphs $G_1$ and $G_2$ is formed from the disjoint union $G_1+G_2$ by specifying
an $r$-clique $x_1,\ldots,x_r$ in $G_1$ and an $r$-clique $y_1,\ldots,y_r$ in $G_2$, identifying $x_i$ with
$y_i$ for each $i\in[r]$, and possibly deleting some edges of the resulting $r$-clique.  The \emph{Wagner
graph} $M_8$ (also called the \emph{M\"{o}bius 8-ladder}) is formed from the $8$-cycle $z_1\cdots z_8$ by
adding the 4 chords $z_iz_{i+4}$ for each $i\in [4]$.
Wagner~\cite{wagner} characterized the class of maximal $K_5$-minor-free graphs as follows.

\subsection{Main Result}
The authors of \cite{CCvBDK} proposed the problem of determining $\chisc(\G)$ for various graph classes
$\G$. Let $\K_r$ denote the class of all graphs that are $K_r$-minor-free. It is easy to check that
$\chisc(\K_2)=1$, $\chisc(\K_3)=2$, and $\chisc(\K_4)=4$; the final upper bound follows from Lemma~\ref{ext-lem}
below because all graphs in $\K_4$ are 2-degenerate. So the first open case is $\chisc(\K_5)$.
The best known upper bound is $\chisc(\K_5)\le 8$. This follows from \cite[Theorem~1.7]{CCvBZ} because every
$K_5$-minor-free graph has maximum average degree less than 6. We cautiously believe that this upper bound
can be improved as follows.

Dvo\v{r}\'{a}k, Norin, and Postle~\cite{DNP19} called a graph \emph{weighted $\epsilon$-flexible} with
respect to a list-assignment $L$ if there exists a probability distribution on the $L$-colorings $\phi$
of $G$ such that $\mathbb{P}(\phi(v)=c) \ge \epsilon$ for every vertex $v$ and color $c\in L(v)$.
In other words, weighted $\epsilon$-flexibility ensures that at each vertex each color has a decent chance
to appear in the random list-coloring. Dvo\v{r}\'{a}k, Norin, and Postle proved that there is an absolute
constant $\epsilon>0$ such that all planar graphs are weighted $\epsilon$-flexible with respect to every
$7$-assignment, and they posed as an open problem to show the same for $6$-assignments or even
$5$-assignments. While this problem remains wide open, for $7$-assignments the value of $\epsilon$ has been
improved considerably from $7^{-36}$ to $2^{-7}$ by~\cite{KMMP22} and~\cite{CCvB}.
As detailed in~\cite{CCvBZ} and~\cite{KMMP22}, every graph $G$ is weighted $\frac{1}{\chisl(G)}$-flexible
with respect to every $\chisl(G)$-assignment. Thus a proof of Conjecture~\ref{conj1} will make significant
progress: it will imply that every planar graph (which is $K_5$-minor-free) is weighted
$\frac{1}{6}$-flexible with respect to every $6$-assignment.
\bigskip

The fact that $K_5$-minor-free graphs are $5$-degenerate easily implies that they have list-chromatic number
at most $6$, which is close to the optimal bound $5$. However, for the correspondence packing number this
approach performs quite poorly. As discussed above, $\chisc(\C) \le 4$ due to the $2$-degeneracy of cycles.
More generally, by \Cref{ext-lem} we have $\chisc(G)\leq 2d$ for every $d$-degenerate graph $G$, and this is
sharp \cite[Prop. 24]{CCvBDK} for every $d \ge 1$, due to large complete bipartite graphs. Therefore a vanilla
degeneracy argument can merely yield $\chisc(\K_5) \le 10$, while it is known that $\chisc(\K_5)\le 8$. This
indicates that establishing Conjecture~\ref{conj1} is nontrivial. On the lower bound side, note that the
second part of our Main Theorem implies that $\chisc(\K_5)\ge 6$.
\bigskip

\begin{lemA}[\cite{CCvBDK}]
 \label[lem]{ext-lem}
 Let $G$ be a graph with a correspondence $t$-cover $\MM$ and fix $v\in V(G)$.
 If $d(v)\le t/2$, then any disjoint $\MM$-colorings $\vph_1',\ldots,\vph_s'$
 of $G-v$ extend to disjoint $\MM$-colorings $\vph_1,\ldots,\vph_s$ of $G$.
\end{lemA}
\begin{proof}
 We build an auxiliary bipartite graph $\B$ with $1,\ldots,t$ as the vertices of the first
 part, and $\vph'_1,\ldots,\vph'_s$ as the vertices of the second part, and $i\vph'_j\in E(\B)$
 if $i\in L_j(v)$. To extend the $\MM$-colorings $\vph_j'$, we seek a matching $M_{\B}$
 saturating the second part. For this we use Hall's Theorem.

By definition, each matching in $\MM$ need not be perfect. But to prove the results in this paper, it
suffices to consider the case that all matchings are perfect; if not, then we can grow them greedily.
(In what follows, we implicitly assume that all matchings in $\MM$ are perfect.)

An \emph{$r$-sum} of graphs $G_1$ and $G_2$ is formed from the disjoint union $G_1+G_2$ by specifying
an $r$-clique $x_1,\ldots,x_r$ in $G_1$ and an $r$-clique $y_1,\ldots,y_r$ in $G_2$, identifying $x_i$ with
$y_i$ for each $i\in[r]$, and possibly deleting some edges of the resulting $r$-clique. The \emph{Wagner
graph} $M_8$ (also called the \emph{M\"{o}bius 8-ladder}) is formed from the $8$-cycle $z_1\cdots z_8$ by
adding the 4 chords $z_iz_{i+4}$ for each $i\in [4]$.
Wagner~\cite{wagner} characterized the class of maximal $K_5$-minor-free graphs as follows.

\begin{conj}\label{conj1}
	If $G$ is $K_5$-minor-free and $\MM$ is a correspondence 6-cover, then $G$ has 6 disjoint
	$\MM$-colorings.  That is, $\chisc(\K_5)=6$.
\end{conj}

\subsection{Main Result}
The authors of \cite{CCvBDK} proposed the problem of determining $\chisc(\G)$ for various graph classes
$\G$. Let $\K_r$ denote the class of all graphs that are $K_r$-minor-free. It is easy to check that
$\chisc(\K_2)=1$, $\chisc(\K_3)=2$, and $\chisc(\K_4)=4$; the final upper bound follows from Lemma~\ref{ext-lem}
below because all graphs in $\K_4$ are 2-degenerate. So the first open case is $\chisc(\K_5)$.
The best known upper bound is $\chisc(\K_5)\le 8$. This follows from \cite[Theorem~1.7]{CCvBZ} because every
$K_5$-minor-free graph has maximum average degree less than 6. We cautiously believe that this upper bound
can be improved as follows.

Dvo\v{r}\'{a}k, Norin, and Postle~\cite{DNP19} called a graph \emph{weighted $\epsilon$-flexible} with
respect to a list-assignment $L$ if there exists a probability distribution on the $L$-colorings $\phi$
of $G$ such that $\mathbb{P}(\phi(v)=c) \ge \epsilon$ for every vertex $v$ and color $c\in L(v)$.
In other words, weighted $\epsilon$-flexibility ensures that at each vertex each color has a decent chance
to appear in the random list-coloring. Dvo\v{r}\'{a}k, Norin, and Postle proved that there is an absolute
constant $\epsilon>0$ such that all planar graphs are weighted $\epsilon$-flexible with respect to every
$7$-assignment, and they posed as an open problem to show the same for $6$-assignments or even
$5$-assignments. While this problem remains wide open, for $7$-assignments the value of $\epsilon$ has been
improved considerably from $7^{-36}$ to $2^{-7}$ by~\cite{KMMP22} and~\cite{CCvB}.
As detailed in~\cite{CCvBZ} and~\cite{KMMP22}, every graph $G$ is weighted $\frac{1}{\chisl(G)}$-flexible
with respect to every $\chisl(G)$-assignment. Thus a proof of Conjecture~\ref{conj1} will make significant
progress: it will imply that every planar graph (which is $K_5$-minor-free) is weighted
$\frac{1}{6}$-flexible with respect to every $6$-assignment.
\bigskip

\begin{lemA}[\cite{CCvBDK}]
 \label[lem]{ext-lem}
 Let $G$ be a graph with a correspondence $t$-cover $\MM$ and fix $v\in V(G)$.
 If $d(v)\le t/2$, then any disjoint $\MM$-colorings $\vph_1',\ldots,\vph_s'$
 of $G-v$ extend to disjoint $\MM$-colorings $\vph_1,\ldots,\vph_s$ of $G$.
\end{lemA}
\begin{proof}
 We build an auxiliary bipartite graph $\B$ with $1,\ldots,t$ as the vertices of the first
 part, and $\vph'_1,\ldots,\vph'_s$ as the vertices of the second part, and $i\vph'_j\in E(\B)$
 if $i\in L_j(v)$. To extend the $\MM$-colorings $\vph_j'$, we seek a matching $M_{\B}$
 saturating the second part. For this we use Hall's Theorem.

An \emph{$r$-sum} of graphs $G_1$ and $G_2$ is formed from the disjoint union $G_1+G_2$ by specifying
an $r$-clique $x_1,\ldots,x_r$ in $G_1$ and an $r$-clique $y_1,\ldots,y_r$ in $G_2$, identifying $x_i$ with
$y_i$ for each $i\in[r]$, and possibly deleting some edges of the resulting $r$-clique. The \emph{Wagner
graph} $M_8$ (also called the \emph{M\"{o}bius 8-ladder}) is formed from the $8$-cycle $z_1\cdots z_8$ by
adding the 4 chords $z_iz_{i+4}$ for each $i\in [4]$.
Wagner~\cite{wagner} characterized the class of maximal $K_5$-minor-free graphs as follows.

So to prove our Main Theorem, we would like to reduce to the case of planar graphs and copies of
 $M_8$. However, we must ensure that our disjoint $\MM$-colorings $\vph_1,\vph_2,\vph_3$
 agree on the copies of $K_2$ and $K_3$ where we perform our 2-sums and 3-sums. This motivates the
 following slightly stronger lemmas. We handle the copies of $M_8$ first, even for 6 disjoint
 $\MM$-colorings, since this step is easy.

In this section we present proofs of our results. We first prove part (i) of our Main Theorem, by using
\Cref{M8-lem}, \Cref{planar-lem}, and Theorem~\ref{wagner-thm} (Wagner's characterization of
$K_5$-minor-free graphs). That is, given a $K_5$-minor-free graph
$G$ and a correspondence 6-cover $\MM$, we show how to construct disjoint $\MM$-colorings
$\vph_1,\vph_2,\vph_3$. Next, we prove \Cref{planar-lem} using our Key Lemma; and after this we prove our
Key Lemma via a ``Thomassen-style'' induction proof. Finally, we conclude this section by proving part (ii)
of our Main Theorem. That is, we construct a $K_5$-minor-free graph $G$ and a correspondence 6-cover
$\MM$ of $G$ that do not admit 3 disjoint $\MM$-colorings.

By \Cref{wagner-thm}, $G$ can be constructed by a sequence of 2-sums and 3-sums starting from
 copies of $M_8$ and planar graphs. Furthermore, by possibly reordering the sequence, we can assume
 that for each 2-sum or 3-sum one of the graphs being summed is itself either a copy of $M_8$ or a
 planar graph. Fix such a construction sequence, and let $q$ be the sum of its
 numbers of 2-sums and 3-sums. We proceed by induction on $q$. If $q=0$, then $G$ is planar or a
 copy of $M_8$, so we are done by \Cref{M8-lem} or by \Cref{planar-lem}. To be precise, in the latter
 case, we should first fix a copy $C$ of $K_3$ and disjoint $\MM$-colorings of $C$. But this
 is easy by \Cref{ext-lem}. This completes the base case, $q=0$.

Now assume that $q\ge 1$. So $G$ is formed from a 2-sum or 3-sum of graphs $G_1$ and $G_2$,
 where $G_2$ is planar or a copy of $M_8$ and $G_1$ is formed from a sequence of 2-sums and
 3-sums of length $q-1$. By the induction hypothesis, $G_1$ has
 disjoint $\MM$-colorings $\vph_1',\vph_2',\vph_3'$. Let the copy of $K_2$ or $K_3$ in $G_2$,
 that is involved in the sum to form $G$, inherit from the corresponding vertices in $G_1$ three
 disjoint $\MM$-colorings $\vph_1^0,\vph_2^0,\vph_3^0$. By \Cref{M8-lem} or \Cref{planar-lem},
 we can extend $\vph_1^0,\vph_2^0,\vph_3^0$ to disjoint $\MM$-colorings $\vph_1'',\vph_2'',
 \vph_3''$ of $G_2$. (If $G_2$ is planar and only a $K_2$ is precolored, then we first extend
 the precoloring to a $K_3$, via \Cref{ext-lem}, before invoking \Cref{planar-lem}; this is possible
 because $G_2$ is \emph{maximal planar}, so every $K_2$ is contained in a $K_3$.) Since
 $\vph_j'$ and $\vph_j''$ agree on the vertices of $V(G_1)\cap V(G_2)$, for all $j\in[3]$,
 together they give our disjoint $\MM$-colorings $\vph_1,\vph_2,\vph_3$ of $G$.
\end{proof}

\begin{conj}\label{conj1}
	If $G$ is $K_5$-minor-free and $\MM$ is a correspondence 6-cover, then $G$ has 6 disjoint
	$\MM$-colorings.  That is, $\chisc(\K_5)=6$.
\end{conj}

So to prove our Main Theorem, we would like to reduce to the case of planar graphs and copies of
	$M_8$.  However, we must ensure that our disjoint $\MM$-colorings $\vph_1,\vph_2,\vph_3$
	agree on the copies of $K_2$ and $K_3$ where we perform our 2-sums and 3-sums.  This motivates the
	following slightly stronger lemmas.  We handle the copies of $M_8$ first, even for 6 disjoint
	$\MM$-colorings, since this step is easy.

\begin{lem}
	\label[lem]{M8-lem}
	Given a copy of $M_8$ and a correspondence 6-cover $\MM$ of $M_8$, there exist disjoint
	$\MM$-colorings $\vph_1,\ldots,\vph_6$.  Furthermore, if $\vph_1^0,\ldots,\vph_6^0$ are
	disjoint $\MM$-colorings of a specified $K_2$ in $M_8$, then we can choose
	$\vph_1,\ldots,\vph_6$ to extend $\vph_1^0,\ldots,\vph_6^0$.
\end{lem}
\begin{proof}
	We use induction on the number $n'$ of uncolored vertices.  The base case, $n'=0$, is trivial.
	And the induction step holds, because $M_8$ is 3-degenerate, by Lemma~\ref{ext-lem} with $s:=6$
	and $t:=6$.
\end{proof}

Now we consider planar graphs.

\begin{lem}
	\label[lem]{planar-lem}
	Let $G$ be a planar graph with a 6-correspondence cover $\MM$.  If $C$ is a $K_3$ in $G$,
	then each choice $\vph_1^0,\vph_2^0, \vph_3^0$ of disjoint $\MM$-colorings of $C$ extends to
	disjoint $\MM$-colorings $\vph_1,\vph_2,\vph_3$ of $G$.
\end{lem}

The key to the proof of Lemma~\ref{planar-lem} is proving a still stronger statement, given below in the Key Lemma, that more easily facilitates
proof by induction.  Our technique is known as a ``Thomassen-style'' proof, due to the strikingly short and
elegant proof of Carsten Thomassen~\cite{Thomassen-5choosable} that all planar graphs are 5-choosable.  This
method has since been applied many times~\cite{CCKPSZ, GZ, KT, LWZZ, thomassen-grotzsch2, WL,
zhu-4choosable}.  For a unified study of numerous such examples, see~\cite[Chapter 11]{GCM}.

\begin{keylem}
	Let $G$ be a plane near-triangulation, with a correspondence 6-cover $\MM$.  Denote the vertices
	on the outer face of $G$, in clockwise order, by $w_1,\ldots,w_n$.  There exist disjoint
	$\MM$-colorings $\vph_1,\vph_2,\vph_3$ that extend $\vph_1^0,\vph_2^0,\vph_3^0$ and that have
	$\vph_j(v)\in L_j(v)$ for all $j\in[3]$ if $\vph_1^0,\vph_2^0,\vph_3^0$ and $L_1,L_2,L_3$
	satisfy the following conditions
	\begin{enumerate}
		\item[(1)] $L_1(v)=L_2(v)=L_3(v)=[6]$
			for each vertex $v$ not on the outer face.
		\item[(2)] For each vertex $w_i$ with $i\in\{3,\ldots,n\}$ there exist distinct colors
			$c_1,\ldots,c_6\in [6]$ with
			$L_1(w_i)=\{c_1,c_2,c_3,c_4\}$ and
			$L_2(w_i)=\{c_1,c_2,c_5,c_6\}$ and
			$L_3(w_i)=\{c_3,c_4,c_5,c_6\}$.
		\item[(3)]
			$\vph_1^0,\vph_2^0,\vph_3^0$ are disjoint $\MM$-colorings of $G[\{w_1,w_2\}]$.
	\end{enumerate}
\end{keylem}

To prove the Main Theorem, the authors “would like to reduce to the case of planar graphs and copies of $M_8$” using Wagner’s decomposition, but they note they “must ensure that [the] disjoint $\MM$-colorings $\vph_1,\vph_2,\vph_3$ agree on the copies of $K_2$ and $K_3$ where we perform our 2-sums and 3-sums,” which “motivates the following slightly stronger lemmas.” They “handle the copies of $M_8$ first, even for 6 disjoint $\MM$-colorings, since this step is easy”: for $M_8$ they “use induction on the number $n'$ of uncolored vertices,” with base case “$n'=0$” trivial, and the induction step because “$M_8$ is 3-degenerate, by Lemma~\ref{ext-lem} with $s:=6$ and $t:=6$,” yielding disjoint colorings that can be chosen to extend prescribed disjoint colorings on a specified $K_2$.

For planar graphs, they aim to prove Lemma~\ref{planar-lem} (extending prescribed disjoint $\MM$-colorings on a $K_3$), and state that “the key to the proof of Lemma~\ref{planar-lem} is proving a still stronger statement, given below in the Key Lemma, that more easily facilitates proof by induction,” using a “Thomassen-style” inductive method on plane near-triangulations with specified outer-face structure and list conditions.

Question: Let an $r$-sum of graphs $G_1$ and $G_2$ be formed from their disjoint union by choosing an $r$-clique $x_1,\dots,x_r$ in $G_1$ and an $r$-clique $y_1,\dots,y_r$ in $G_2$, identifying $x_i$ with $y_i$ for each $i\in[r]$, and in general possibly deleting some edges of the resulting identified $r$-clique. Let $M_8$ denote the Wagner graph, obtained from the $8$-cycle $z_1\cdots z_8$ by adding the four chords $z_i z_{i+4}$ for $i\in[4]$. If $G$ is a maximal $K_5$-minor-free graph, which statement holds?

$G$ can be constructed from maximal planar graphs and copies of $M_8$ by a sequence of $2$-sums and $3$-sums, and no edges are deleted from the identified clique in any of these sums.

Choices:
B. $G$ can be constructed from maximal planar graphs and copies of $M_8$ by a sequence of $2$-sums and $3$-sums, where in each sum one may delete some edges of the identified clique.
C. $G$ can be constructed from maximal planar graphs and copies of $M_8$ by a sequence of $2$-sums and $3$-sums.
D. $G$ can be constructed from planar graphs and copies of $M_8$ by a sequence of $2$-sums and $3$-sums, and no edges are deleted from the identified clique in any of these sums.
E. $G$ can be constructed from maximal planar graphs and copies of $M_8$ by a sequence of clique-sums of order at most $3$, and no edges are deleted from the identified clique in any of these sums.

edge-deletion prohibition in Wagner decomposition

restriction to exactly 2-sums and 3-sums


\label{thm:main}
  Let $\dim X =2$. There is a unique $\hat T\in\PD{X}{\mathbb{R}^2}$ such that $\hat T(B_X)\supset B_{\mathbb{R}^2}$ and
  \begin{equation}
    \label{eq:tm=d2}
    \|\hat T\| = d_2(X).
  \end{equation}
  $\hat T$ is characterised by the following property:

  There are $x_i\in S_X$, $i=1,2$, with $x_1\neq \pm x_2$, and $y_i\in S_X$, $i=1,2$, with $y_1\neq \pm y_2$, such that:
  \begin{equation}
    \label{eq:x-distance}
    |\hat T x_i| = \|\hat T\|,\quad i=1,2,
  \end{equation}
  \begin{equation}
    \label{eq:y-contact}
    |\hat T y_i| = 1,\quad i=1,2,
  \end{equation}
  \begin{equation}
    \label{eq:x-between-y}
    y_1\in\mathrm{cone}\, \{x_1,x_2\},\quad y_2\in\mathrm{cone}\,\{-x_1,x_2\}.
  \end{equation}

\begin{thm}
  \label{thm:main}
  Let $\dim X =2$. There is a unique $\T\in\PD{X}{\R^2}$ such that $\T(B_X)\supset B_{\R^2}$ and
  \begin{equation}
    \label{eq:tm=d2}
    \|\T\| = d_2(X).
  \end{equation}
  $\T$ is characterised by the following property:

  There are $x_i\in S_X$, $i=1,2$, with $x_1\neq \pm x_2$, and $y_i\in S_X$, $i=1,2$, with $y_1\neq \pm y_2$, such that:
  \begin{equation}
    \label{eq:x-distance}
    |\T x_i| = \|\T\|,\quad i=1,2,
  \end{equation}
  \begin{equation}
    \label{eq:y-contact}
    |\T y_i| = 1,\quad i=1,2,
  \end{equation}
  \begin{equation}
    \label{eq:x-between-y}
    y_1\in\cone \{x_1,x_2\},\quad y_2\in\cone\{-x_1,x_2\}.
  \end{equation}
\end{thm}

Recall that the Banach-Mazur distance between the isomorphic normed spaces $X$ and $Y$ is denoted by
\begin{equation}
 \label{eq:def-bmd}
 d(X,Y) := \inf \{\|T\|\cdot\|T^{-1}\|:\ T:X\to Y\ \text{isomorphism}\},
\end{equation}
for details see e.g.\ \cite[Section 6.1.1]{pietsch}.

If $X$ and $Y$ are of one and the same finite dimension, then the minimum in \eqref{eq:def-bmd} is attained.

An important characteristic of a finite dimensional space is its distance to the Euclidean space. Here we study this characteristic in the simplest case: the plane.

So, for a 2-dimensional normed space $(X,\|\cdot\|)$
$$
 d_2(X) := d(X,\mathbb{R}^2),
$$
where $\mathbb{R}^2$ is considered with the standard Euclidean norm $|\cdot|$. It is easy to check that
\begin{equation}
 \label{eq:d2-inclusion}
 d_2(X) = \min \{\|T\|:\ T:X\to \mathbb{R}^2,\ T(B_X)\supset B_{\mathbb{R}^2}\},
\end{equation}
where $B_X$ and $S_X$ stand for the unit ball and the unit sphere of $X$ respectively. Indeed, $T(B_X)\supset B_{\mathbb{R}^2}$ is equivalent to $\|T^{-1}\| \le 1$, so we have that $d_2(X)$ is at most the right-hand side of \eqref{eq:d2-inclusion}. On the other hand, multiplying the operator $T$ by a real constant does not change $\|T\|\cdot\|T^{-1}\|$, so we may assume that $\|T^{-1}\| = 1$ in \eqref{eq:def-bmd}.

Using \textit{polar decomposition}, we can reduce \eqref{eq:d2-inclusion}. Note that if $U$ is in $\mathbf{O}(2)$ -- the group of isometries of $\mathbb{R}^2$, that is, $U^{-1} = U^t$ if considered as a matrix -- then $\|UT\| = \max_{x\in S_X} |UTx| = \max_{x\in S_X} |Tx| = \|T\|$, and similarly $\|T^{-1}U^{-1}\| = \|T^{-1}\|$, so we can quotient by $\mathbf{O}(2)$. Denote by $\PD{X}{\mathbb{R}^2}$ the operators from $X$ to $\mathbb{R}^2$ that are positive definite symmetric matrices in any basis. By \cite[Theorem~7.3.1, p.449]{hj} each isomorphism $T:X\to\mathbb{R}^2$ can be represented as $T=UT_1$, where $U\in\mathbf{O}(2)$ and $T_1\in\PD{X}{\mathbb{R}^2}$. Therefore,
\begin{equation}
 \label{eq:d2-reduction}
 d_2(X) = \min \{\|T\|:\ T\in\PD{X}{\mathbb{R}^2},\ T(B_X)\supset B_{\mathbb{R}^2}\}.
\end{equation}

We prove in a new way the following characterisation, which is essentially contained in \cite{ader,mory}.

\begin{equation}
  \label{eq:d2-inclusion}
  d_2(X) = \min \{\|T\|:\ T:X\to \R^2,\ T(B_X)\supset B_{\R^2}\},
\end{equation}

\begin{equation}
  \label{eq:def-bmd}
  d(X,Y) := \inf \{\|T\|\cdot\|T^{-1}\|:\ T:X\to Y\ \text{isomorphism}\},
\end{equation}

As an illustration, consider the following picture.

There are $u_i,v_i\in S_{\R^2}$, $i=1,2$, with $u_1\neq \pm u_2$, $v_1\neq \pm v_2$, $v_1\in\cone \{u_1,u_2\}$, $v_2\in\cone \{-u_1,u_2\}$, and
 \begin{equation}
 \label{eq:equiv-alternance}
 \|\tilde Tu_i\| = 1/\|\tilde T^{-1}\|,\quad \|\tilde T v_i\| = 1,\quad i=1,2.
 \end{equation}
 \end{pro}
 Let a co-ordinate system be fixed in $X$ in which
 \begin{equation}
 \label{eq:r-*}
 S_X = \{r(\varphi)(\cos\varphi,\sin\varphi):\ \varphi\in\R\},
 \end{equation}
 for a strictly positive, $\pi$-periodic and continuous $r:\R\to\R^+$.

In these fixed coordinates each $T\in\PD{\R^2}{X}$ is identified with a matrix
 $$
 T = \begin{pmatrix}
 a & b \\ b & c
 \end{pmatrix},\quad a,c>0,\quad \det T = ac-b^2>0.
 $$
 Obviously,
 $$
 x \in T(S_{\mathbb{R}^2}) \iff |T^{-1}x| = 1 \iff x^t T^{-2} x = 1.
 $$
 That is, $T(S_{\mathbb{R}^2})$ is the ellipse:
 $$
 a_1 x_1^2 + 2 b_1 x_1x_2 + c_1 x_2^2 = 1,
 $$
 where
 $$
 \begin{pmatrix} a_1 & b_1 \\ b_1 & c_1 \end{pmatrix} = T^{-2},
 $$
 so $a_1,c_1 > 0$ and $b_1^2 < a_1c_1$. In polar co-ordinates then $T(S_{\mathbb{R}^2}) = \{\rho(\varphi)(\cos\varphi,\sin\varphi):\ \varphi\in\R\}$, where
 \begin{equation}
 \label{eq:rho-trig}
 \rho^{-2}(\varphi) = a_2 + b_2\cos 2\varphi + c_2\sin 2\varphi,
 \end{equation}
 with $a_2 = (a_1+c_1)/2$, $b_2 = (a_1-c_1)/2$ and $c_2 = b_1$, so
 \begin{equation}
 \label{eq:ellipse-cond}
 a_2 > 0,\quad a_2^2 > b_2^2 + c_2^2.
 \end{equation}
 Usually this is expressed in terms of $\cos2(\varphi-\theta)$, where $\theta$ is the tilt angle. Let $\mathcal E$ be the family of all ellipses in polar co-ordinates, that is, all curves in the form \eqref{eq:rho-trig} for all values of the parameters $a_2,b_2,c_2$ which satisfy \eqref{eq:ellipse-cond}. It is clear that the cone \eqref{eq:ellipse-cond} parametrises one-to-one the ellipses, as well as $\PD{\R^2}{X}$, which we will use when establishing the uniqueness part of our main result. Let
 \begin{equation}
 \label{eq:C-def}
 \mathcal{C} := \{\log \rho:\ \rho\in\mathcal{E}\}.
 \end{equation}
 Let also, see \eqref{eq:r-*},
 $$
 f := \log r.
 $$
 Since $\|T\| \le 1$ is equivalent to $T(S_{\mathbb{R}^2})\subset B_X$, which in polar co-ordinates translates to
 $$
 \rho \le r \iff g := \log \rho \le f,
 $$
 the constraint in \eqref{eq:t-1-prob} becomes $g\in\mathcal{C}$, $g \le f$. For the objective, since $\rho(\varphi)(\cos\varphi,\sin\varphi) \in T(S_{\R^2})$, we have $|T^{-1}(\rho(\varphi)(\cos\varphi,\sin\varphi))| = 1$, hence $|T^{-1}(\cos\varphi,\sin\varphi)| = 1/\rho(\varphi)$. For $x = r(\varphi)(\cos\varphi,\sin\varphi) \in S_X$,
 $$
 |T^{-1}x| = \frac{r(\varphi)}{\rho(\varphi)} = \exp (f(\varphi)-g(\varphi)),
 $$
 and the problem \eqref{eq:t-1-prob} is equivalent to
 $$
 \begin{cases}
 \max_\varphi \exp (f(\varphi)-g(\varphi)) \to \min\\
 g\in\mathcal{C},\quad g \le f.
 \end{cases}
 $$
 Since the exponential is strictly monotonically increasing, Proposition~\ref{prop:equiv} is equivalent to Proposition~\ref{prop:C-cal-equiv} below. There, for a $\pi$-periodic function $h$
 $$
 \|h\|_\infty := \max_\varphi |h(\varphi)|.
 $$
 \begin{pro}
 \label{prop:C-cal-equiv}
 The optimisation problem
 \begin{equation}
 \label{eq:C-cal-equiv}
 \begin{cases}
 \|f - g\|_\infty \to \min\\
 g\in\mathcal{C},\quad g \le f
 \end{cases}
 \end{equation}
 has a unique solution $\bar g$ characterised by the following property:

We are ready to prove the uniqueness part of our main result.
 \begin{pro}
 \label{prop:unical}
 Let $\bar g\in \mathcal{C}$. Assume that there exist points $\varphi_i,\psi_i$, $i=1,2$, satisfying \eqref{eq:alternance-pts} and such that
 \begin{equation}
 \label{eq:alternance-pts-values-2}
 f(\varphi_i) - \bar g(\varphi_i) = \bar g(\psi_i) - f(\psi_i) =\|f-\bar g\|_\infty,\quad i=1,2.
 \end{equation}
 Then $\bar g$ is the unique solution to \eqref{eq:Ch}.
 \end{pro}
 \begin{proof}
 Let $\hat g \in \mathcal{C}$ be such that
 \begin{equation}
 \label{eq:51}
 \|f-\hat g\|_\infty \le \|f - \bar g\|_\infty.
 \end{equation}
 If we can prove that from this follows $\hat g = \bar g$ then obviously $\bar g$ is a solution to \eqref{eq:Ch} and, moreover, unique.

Assume that
 \begin{equation}
 \label{eq:71}
 \bar g \neq \hat g.
 \end{equation}
 Let $h := \bar g - \hat g$. We can write $h(\varphi_i) = (f-\hat g) (\varphi_i) - (f-\bar g) (\varphi_i)$. By \eqref{eq:alternance-pts-values-2} we have $(f-\bar g)(\varphi_i) = \|f-\bar g\|_\infty$, so $h(\varphi_i) = (f-\hat g)(\varphi_i) - \|f-\bar g\|_\infty \le \|f-\hat g\|_\infty - \|f-\bar g\|_\infty \le 0$ by \eqref{eq:51}. Thus $h(\varphi_i) \le 0$.
 Similarly, $h(\psi_i) = (f-\hat g)(\psi_i) + (\bar g -f)(\psi_i) = (f-\hat g)(\psi_i) + \|f-\bar g\|_\infty \ge \|f-\bar g\|_\infty - \|f-\hat g\|_\infty \ge 0$. We have shown that
 \begin{equation}
 \label{eq:72}
 h(\varphi_i) \le 0 \le h(\psi_i),\quad i=1,2.
 \end{equation}
 Consequently, there exist $\xi_i\in[\varphi_i,\psi_i]$, $i=1,2$ (in particular, $\xi_1<\xi_2$) such that $h(\xi_i) = 0$ for $i=1,2$. If there were a third point in $[\varphi_1,\varphi_1+\pi)$ at which $h=0$, then Proposition~\ref{prop:elipses-3-pts-equial} would have implied $\bar g = \hat g$, contradicting \eqref{eq:71}, so
 \begin{equation}
 \label{eq:73}
 h(\varphi) \neq 0,\quad \forall \varphi \in [\varphi_1,\varphi_1+\pi) \setminus \{\xi_1,\xi_2\}.
 \end{equation}
 Also, if $h'(\xi_i) = 0$ for some $i=1,2$, then again Proposition~\ref{prop:elipses-3-pts-equial} implies $\bar g = \hat g$, so
 \begin{equation}
 \label{eq:74}
 h'(\xi_i) \neq 0,\quad i=1,2.
 \end{equation}
 From \eqref{eq:73} and $\xi_1\le\psi_1<\varphi_2\le\xi_2$ it follows that $h(\varphi) \neq 0$ for all $\varphi\in(\psi_1,\varphi_2)$. Since $h$ is continuous, it has a constant sign on $(\psi_1,\varphi_2)$, so there are two possibilities:

Thus \eqref{eq:contra} is verified, but it contradicts \eqref{eq:alternance-*}. Therefore, there is $\varphi_2\in\Phi\cap(\psi_1,\psi_2)$ and $\varphi_i,\psi_i$ satisfy \eqref{eq:alternance-pts} and \eqref{eq:alternance-pts-values-2}.
 \end{proof}
 In view of the work so far the following completes our study.
 \begin{pro}
 \label{prop:final}
 The problem \eqref{eq:Ch} has a solution.
 \end{pro}
 \begin{proof}
 Here it is easier to work with the standard form of the ellipses in polar co-ordinates. Take $(a,b,c)\in K$, that is, $a>\sqrt{b^2+c^2}$, and set
 $$
 b':= \sqrt{b^2+c^2},\quad b = b'\cos 2\theta,\ c = b'\sin 2\theta.
 $$
 Then
 $$
 \mathfrak{R}(a,b,c) = a + b'\cos 2(\varphi-\theta).
 $$
 $K$ translates to
 $$
 K' := \{(a,b',\theta):\ a>b'\ge0,\ \theta\in[0,\pi]\},
 $$
 and
 $$
 \mathcal{C} = \{-(1/2)\log(a + b'\cos 2(\varphi-\theta)) : (a,b',\theta) \in K'\}.
 $$
 Let $(a_n,b_n',\theta_n)\in K'$ be such that for
 $$
 g_n(\varphi) := -\frac{1}{2}\log(a_n + b_n'\cos 2(\varphi-\theta_n))
 $$
 we have
 $$
 \lim_{n\to\infty} \|f-g_n\|_\infty = \mathfrak{d}.
 $$
 By taking a subsequence we may assume that $\theta_n\to\theta$, as $n\to\infty$. Since
 $$
 g_n(\theta_n+\pi/4) = -\frac{1}{2}\log a_n,
 $$
 and $\|g_n\|_\infty < \|f\|_\infty + \mathfrak{d} + 1$ for large enough $n$'s, the sequence $\{\log a_n\}_{n=1}^\infty$ is bounded, so there is $r>0$ such that
 $$
 r \le a_n \le 1/r,\quad\forall n\in\mathbb{N}.
 $$
 Therefore, we may assume that $a_n\to a >0$, as $n\to\infty$. We also assume that $b_n'\to b'\le a$, as $n\to\infty$. If $b'=a$ then
 $$
 g_n(\theta_n+\pi/2) = -\frac{1}{2}\log(a_n - b_n') \to \infty,\text{ as }n\to\infty,
 $$
 which is a contradiction. So, $(a,b',\theta)\in K'$ and by continuity it follows that the corresponding $g\in\mathcal{C}$ solves \eqref{eq:Ch}.
 \end{proof}

Here
$$
  \mathrm{cone}\, \{x,y\} := \{\alpha x + \beta y:\ \alpha,\beta \ge 0\}.
$$
Note that the condition characterising the optimal operator depends only on the operator itself and not on $d_2(X)$. This is to be expected, because the problem suitably stated is convex. Not strictly convex, however, so the uniqueness is not immediate.

As an illustration, consider the following picture.

\begin{center}
  \begin{tikzpicture}
    \draw[color=gray] (0,0) circle [radius = 2cm];
    \coordinate (y1) at (86:2);
    \coordinate (y1m) at (180+86:2);
lldraw (y1) circle (1pt);
    \node[above] at (y1) {$\hat Ty_2$};

    \coordinate (y2) at (26:2);
    \coordinate (y2m) at (180+26:2);
lldraw (y2) circle (1pt);
    \node[right] at (y2) {$\hat Ty_1$};

    \coordinate (x1) at (56:4/1.73205080757);
    \coordinate (x1m) at (180+56:4/1.73205080757);
lldraw (x1) circle (1pt);
    \node[right, yshift=1.6mm] at (x1) {$\hat Tx_2$};

    \draw[thick] (y1) -- (x1) -- (y2);
    \draw[thick] (y1m) -- (x1m) -- (y2m);

    \coordinate (a) at (-12:2);
    \coordinate (am) at (180-12:2);
    \coordinate (b) at (-42:4/1.73205080757);
    \coordinate (bm) at (180-42:4/1.73205080757);

    \draw[thick] (a) -- (b);
    \draw[thick] (am) -- (bm);
    \draw[thick] (y2) arc(26:-12:2);
    \draw[thick] (y2m) arc(180+26:180-12:2);

    \coordinate (u) at (272:2);
    \coordinate (um) at (92:2);
    \draw[thick] (y1m) arc(266:272:2);
    \draw[thick] (y1) arc(86:92:2);

    \coordinate (v) at (302:4/1.73205080757);
    \draw[thick] (u) -- (v);
    \coordinate (vm) at (122:4/1.73205080757);
    \draw[thick] (um) -- (vm);

    \draw[thick] (v) arc(302:318:4/1.73205080757);
    \draw[thick] (vm) arc(122:138:4/1.73205080757);

    \coordinate (x2) at (310:4/1.73205080757);
lldraw (x2) circle (1pt);
    \node[right] at (x2) {$\hat T x_1$};
  \end{tikzpicture}
\end{center}

From it we can also see one important, but immediate, corollary. Since one of the angles between $\hat T y_1$ and $\hat T y_2$, and between $\hat T y_2$ and $-\hat T y_1$ (say the one between $\hat T y_1$ and $\hat T y_2$ as on the picture) will be $\le \pi/2$; and since $\hat T x_2$ will be within the angle between the tangents to the unit circle at $\hat T y_1$ and $\hat T y_2$, due to convexity, the distance from $\hat T x_2$ to zero, that is $d_2(X)$, is at most $\sqrt{2}$. Therefore,
$$
  d_2(X) \le \sqrt{2}.
$$

Note that Theorem~\ref{thm:main} easily follows from
\begin{principle}[\cite{ader}]
  For the unique $\hat T$ there is no angle such that all points $x\in S_X$ with $|\hat T x| = \|\hat T\|$ are inside this angle and all points $y\in S_X$ with $|\hat T y| = 1$ are outside.
\end{principle}
Here, an angle is
$$
  \mathrm{cone}\, \{a,b\} \cup \mathrm{cone}\, \{-a,-b\}
$$
for some $a,b\in\mathbb{R}^2$.

The original work \cite{ader} states that there should be at most $4$ points realising Ader's Non-separation Principle, and for these $4$ points it is then clear that the alternance described in Theorem~\ref{thm:main} must be in place. By Maurey's result \cite{mory}, see also \cite{gru-ko}, the ellipse of minimal Banach-Mazur distance is unique, which yields the uniqueness part of Theorem~\ref{thm:main}.

Obviously, the problem can be re-stated more geometrically like: find the minimal $k\ge1$ such that there is an ellipse inscribed in $B_X$ whose homothetic image with ratio $k$ is circumscribed about $B_X$. We will actually be using this form below, but we mention it here to point out the relationship to the celebrated L\"owner--John ellipse, see e.g.\ \cite{john}, that is, the ellipse of maximal area inscribed in $B_X$. The L\"owner--John ellipsoid is extensively used for estimating the Banach-Mazur distance to Euclidean space, probably because Ader's result \cite{ader} had apparently been forgotten until Grundbacher and Kobos recently unearthed it, see \cite{gru-ko}.

However, if $B_X$ is not symmetric the estimate through the L\"owner--John ellipsoid can grow progressively worse with the dimension. Consider $\mathbb{R}^n$ and let $B_{X_n}$ be the convex hull of $B_{\mathbb{R}^n}$ and $\sqrt{n}e$, where $e\in S_{\mathbb{R}^n}$. Then $B_{\mathbb{R}^n}$ is the L\"owner--John ellipsoid for $B_{X_n}$, see \cite{rakov} for a nice and simple proof, so it gives only the generic estimate $\sqrt{n}$. On the other hand, by the rotational symmetry of $B_{X_n}$ around the axis through $e$, $d(X_n,\mathbb{R}^n)\le\sqrt{3}$.

Our interest in the specifically two-dimensional case comes from our previous works \cite{iv-tr,iv-pa-tr,nik1,nik2}. Of course, similar problems can also be considered in different metrics, like e.g.\ the Hausdorff, see \cite{kenderov,gru-ke,ke-ki}.

In conclusion, note that the convexity of $B_X$ will not be used at all in the proofs below, which is not surprising, as they go through Chebyshev approximation.

A proof route for Theorem~\ref{thm:main} is indicated via Ader’s Non-separation Principle. The text states that Theorem~\ref{thm:main} “easily follows from” the following: for the unique optimal $\hat T$, “there is no angle such that all points $x\in S_X$ with $|\hat T x|=\|\hat T\|$ are inside this angle and all points $y\in S_X$ with $|\hat T y|=1$ are outside,” where an “angle” is $\mathrm{cone}\{a,b\}\cup\mathrm{cone}\{-a,-b\}$. The original work \cite{ader} asserts there are “at most $4$ points” realizing this non-separation; “for these $4$ points it is then clear that the alternance described in Theorem~\ref{thm:main} must be in place,” yielding the configuration of two norming points $x_i$ with $|\hat T x_i|=\|\hat T\|$ and two contact points $y_i$ with $|\hat T y_i|=1$ arranged in the stated cone relations. The uniqueness part is attributed to Maurey’s result (see \cite{mory,gru-ko}) that “the ellipse of minimal Banach–Mazur distance is unique,” which “yields the uniqueness part of Theorem~\ref{thm:main}.”

A proof route for the main theorem is indicated via Ader’s Non-separation Principle. The text states that the main theorem “easily follows from” the following: for the unique optimal $\hat T$, “there is no angle such that all points $x\in S_X$ with $|\hat T x|=\|\hat T\|$ are inside this angle and all points $y\in S_X$ with $|\hat T y|=1$ are outside,” where an “angle” is $\mathrm{cone}\{a,b\}\cup\mathrm{cone}\{-a,-b\}$. The original work \cite{ader} asserts there are “at most $4$ points” realizing this non-separation; “for these $4$ points it is then clear that the alternance described in the main theorem must be in place,” yielding the configuration of two norming points $x_i$ with $|\hat T x_i|=\|\hat T\|$ and two contact points $y_i$ with $|\hat T y_i|=1$ arranged in the stated cone relations. The uniqueness part is attributed to Maurey’s result (see \cite{mory,gru-ko}) that “the ellipse of minimal Banach–Mazur distance is unique,” which “yields the uniqueness part of the main theorem.”

Question:
Let \(X\) be a 2-dimensional normed space, with unit ball \(B_X\) and unit sphere \(S_X\). Let \(\mathbb{R}^2\) be equipped with its Euclidean norm \(|\cdot|\) and unit ball \(B_{\mathbb{R}^2}\). For isomorphic normed spaces \(X\) and \(Y\), define the Banach--Mazur distance by
\[
 d(X,Y):=\inf\{\|T\|\,\|T^{-1}\|: T:X\to Y \text{ is an isomorphism}\},
\]
and set \(d_2(X):=d(X,\mathbb{R}^2)\). Let \(\mathrm{PD}(X,\mathbb{R}^2)\) denote the class of linear operators \(T:X\to\mathbb{R}^2\) represented by positive definite symmetric matrices, and for vectors \(u,v\) write \(\mathrm{cone}\{u,v\}=\{\alpha u+\beta v:\alpha,\beta\ge 0\}\). Under these assumptions, which existence statement holds?

There exists a unique operator \(\hat T\in \mathrm{PD}(X,\mathbb{R}^2)\) such that \(\hat T(B_X)\supset B_{\mathbb{R}^2}\) and \(\|\hat T\|=d_2(X)\). Moreover, this unique operator \(\hat T\) is characterized by the existence of points \(x_1,x_2\in S_X\) with \(x_1\neq \pm x_2\), and \(y_1,y_2\in S_X\) with \(y_1\neq \pm y_2\), such that for \(i=1,2\),
\[
|\hat T x_i|=\|\hat T\|,\qquad |\hat T y_i|=1,
\]
and
\[
y_1\in \mathrm{cone}\{x_1,x_2\},\qquad y_2\in \mathrm{cone}\{-x_1,x_2\}.
\]

Choices:
B.
There exists a unique operator \(\hat T\in \mathrm{PD}(X,\mathbb{R}^2)\) such that \(\hat T(B_X)\supset B_{\mathbb{R}^2}\) and \(\|\hat T\|=d_2(X)\). Moreover, this unique operator \(\hat T\) is characterized by the existence of points \(x_1,x_2\in S_X\) with \(x_1\neq \pm x_2\), and \(y_1,y_2\in S_X\) with \(y_1\neq \pm y_2\), such that for \(i=1,2\),
\[
|\hat T x_i|=\|\hat T\|,\qquad |\hat T y_i|=1,
\]
and
\[
y_1\in \mathrm{cone}\{x_1,x_2\},\qquad y_2\in \mathrm{cone}\{x_1,x_2\}.
\]
C. There exists an operator \(\hat T\in \mathrm{PD}(X,\mathbb{R}^2)\) such that \(\hat T(B_X)\supset B_{\mathbb{R}^2}\) and \(\|\hat T\|=d_2(X)\).
D.
For every operator \(T\in \mathrm{PD}(X,\mathbb{R}^2)\) satisfying \(T(B_X)\supset B_{\mathbb{R}^2}\) and \(\|T\|=d_2(X)\), there exist points \(x_1,x_2\in S_X\) with \(x_1\neq \pm x_2\), and \(y_1,y_2\in S_X\) with \(y_1\neq \pm y_2\), such that for \(i=1,2\),
\[
|T x_i|=\|T\|,\qquad |T y_i|=1,
\]
and
\[
y_1\in \mathrm{cone}\{x_1,x_2\},\qquad y_2\in \mathrm{cone}\{-x_1,x_2\}.
\]
E.
There exists a unique operator \(\hat T\in \mathrm{PD}(X,\mathbb{R}^2)\) such that \(\hat T(B_X)= B_{\mathbb{R}^2}\) and \(\|\hat T\|=d_2(X)\). Moreover, this unique operator \(\hat T\) is characterized by the existence of points \(x_1,x_2\in S_X\) with \(x_1\neq \pm x_2\), and \(y_1,y_2\in S_X\) with \(y_1\neq \pm y_2\), such that for \(i=1,2\),
\[
|\hat T x_i|=\|\hat T\|,\qquad |\hat T y_i|=1,
\]
and
\[
y_1\in \mathrm{cone}\{x_1,x_2\},\qquad y_2\in \mathrm{cone}\{-x_1,x_2\}.
\]

alternance cone placement of the second contact point

dropped uniqueness and the four-point alternance characterization

uniqueness of the optimal positive definite symmetric operator

containment \(\hat T(B_X)\supset B_{\mathbb{R}^2}\) strengthened to equality of images


For $3$-manifolds obtained by Dehn surgery on a $(1,1)$ knot in $S^3$, the implications
\[
(a)\Leftrightarrow(c)\quad\text{and}\quad (a)\Leftarrow(b)
\]
hold in the L-space conjecture.

A central theme in $3$-manifold topology is the relationship between taut foliations, left-orderability, and Heegaard Floer homology. The L-space conjecture \cite{boyer2013spaces,juhasz2015survey} predicts that the following are equivalent for an irreducible rational homology $3$-sphere $Y$:
\begin{enumerate}[label=(\alph*)]
    \item $Y$ is a non-L-space;
    \item $\pi_1(Y)$ is left-orderable;
    \item $Y$ admits a co-oriented taut foliation.
\end{enumerate}
In general, only the implication (c)$\Rightarrow$(a) \cite{ozsvath2004holomorphic,bowden2016approximating,kazez2017c0} is known.

Dehn surgery on knots in $S^3$ provides a natural source of both L-spaces and non-L-spaces. For a nontrivial knot $K\subset S^3$, the set of slopes yielding L-spaces \cite{kronheimer2007monopoles,rasmussen2017floer} is known to be one of the following three possibilities:
\begin{enumerate}
\item $[2g(K)-1,\infty)$, in which case $K$ is called a positive L-space knot;
\item $(-\infty,-2g(K)+1]$, in which case $K$ is called a negative L-space knot;
\item the empty set, in which case $K$ is called a non-L-space knot.
\end{enumerate}
Here $g(K)$ denotes the Seifert genus of $K$. Thus, the L-space interval is completely determined once we know which case $K$ falls into. In contrast, the corresponding interval for taut foliations or left-orderability is much less understood.

In this paper, we study the existence of taut foliations on non-L-spaces obtained by Dehn surgeries on L-space knots in $S^3$.

Most known positive L-space knots in $S^3$ arise as positive braid closures. For example, among the $632$ positive L-space knots in the SnapPy census with at most $9$ tetrahedra, $631$ are positive braid closures \cite{baker2024census}, and the remaining example $o9\_30634$ is the only known exception. Moreover, every positive $(1,1)$ L-space knot in $S^3$ is a positive braid closure \cite{nie2021explicit}.

When a link is represented by a braid closure, it is not straightforward to determine whether it is a knot or a multi-component link. For this reason, we work in the more general setting of links rather than restricting ourselves to knots. Our main theorem constructs taut foliations on the complement of a non-split positive braid closure in $S^3$ with at least one nontrivial component\footnote{It is not sufficient to merely assume $L$ is nontrivial. The Hopf link provides a counterexample.}.

\begin{namedtheorem}[Main Theorem]
Suppose $L$ is the closure of a non-split positive braid in $S^3$ with components $L_1,\ldots,L_n$, and assume that at least one component is not the unknot. Then the Dehn surgery along a multislope $(s_1,\ldots,s_n)\in\mathbb{Q}^n$ satisfying $s_i<2g(L_i)-1$ for all $i=1,\ldots,n$ yields a $3$-manifold admitting a co-oriented taut foliation.
\end{namedtheorem}

As an immediate corollary, the resulting manifolds are non-L-spaces.

Lyu proved that $(1,1)$ non-L-space knots are persistently foliar \cite{lyu2025persistent}. It is also known that positive $(1,1)$ L-space knots in $S^3$ are positive braid closures, and that the L-spaces obtained by Dehn surgeries on $(1,1)$ knots have non-left-orderable fundamental groups \cite{nie2021explicit,Li2024TautFoliations}. Combining these results with our main theorem yields the following corollary.

A central theme in $3$-manifold topology is the relationship between taut foliations, left-orderability, and Heegaard Floer homology. The L-space conjecture \cite{boyer2013spaces,juhasz2015survey} predicts that the following are equivalent for an irreducible rational homology $3$-sphere $Y$:
\begin{enumerate}[label=(\alph*)]
 \item $Y$ is a non-L-space;
 \item $\pi_1(Y)$ is left-orderable;
 \item $Y$ admits a co-oriented taut foliation.
\end{enumerate}
In general, only the implication (c)$\Rightarrow$(a) \cite{ozsvath2004holomorphic,bowden2016approximating,kazez2017c0} is known.

Dehn surgery on knots in $S^3$ provides a natural source of both L-spaces and non-L-spaces. For a nontrivial knot $K\subset S^3$, the set of slopes yielding L-spaces \cite{kronheimer2007monopoles,rasmussen2017floer} is known to be one of the following three possibilities:
\begin{enumerate}
\item $[2g(K)-1,\infty)$, in which case $K$ is called a positive L-space knot;
\item $(-\infty,-2g(K)+1]$, in which case $K$ is called a negative L-space knot;
\item the empty set, in which case $K$ is called a non-L-space knot.
\end{enumerate}
Here $g(K)$ denotes the Seifert genus of $K$. Thus, the L-space interval is completely determined once we know which case $K$ falls into. In contrast, the corresponding interval for taut foliations or left-orderability is much less understood.

Most known positive L-space knots in $S^3$ arise as positive braid closures. For example, among the $632$ positive L-space knots in the SnapPy census with at most $9$ tetrahedra, $631$ are positive braid closures \cite{baker2024census}, and the remaining example $o9\_30634$ is the only known exception. Moreover, every positive $(1,1)$ L-space knot in $S^3$ is a positive braid closure \cite{nie2021explicit}.

\begin{namedtheorem}[Main Theorem]
Suppose $L$ is the closure of a non-split positive braid in $S^3$ with components $L_1,\ldots,L_n$, and assume that at least one component is not the unknot. Then the Dehn surgery along a multislope $(s_1,\ldots,s_n)\in\mathbb{Q}^n$ satisfying $s_i<2g(L_i)-1$ for all $i=1,\ldots,n$ yields a $3$-manifold admitting a co-oriented taut foliation.
\end{namedtheorem}

As an immediate corollary, the resulting manifolds are non-L-spaces.

Lyu proved that $(1,1)$ non-L-space knots are persistently foliar \cite{lyu2025persistent}. It is also known that positive $(1,1)$ L-space knots in $S^3$ are positive braid closures, and that the L-spaces obtained by Dehn surgeries on $(1,1)$ knots have non-left-orderable fundamental groups \cite{nie2021explicit,Li2024TautFoliations}. Combining these results with our main theorem yields the following corollary.

\begin{abstract}
We study taut foliations on the complements of non-split positive braid closures in $S^3$. If $L$ is such a link with components $L_1,\ldots,L_n$ and at least one component is not the unknot, then the Dehn surgery along a multislope $(s_1,\ldots,s_n)\in\mathbb{Q}^n$ satisfying $s_i<2g(L_i)-1$ for $i=1,2,\ldots, n$ yields a non-L-space that admits a co-oriented taut foliation.
\end{abstract}
\maketitle
\section{Introduction}
A central theme in $3$-manifold topology is the relationship between taut foliations, left-orderability, and Heegaard Floer homology. The L-space conjecture \cite{boyer2013spaces,juhasz2015survey} predicts that the following are equivalent for an irreducible rational homology $3$-sphere $Y$:
\begin{enumerate}[label=(\alph*)]
 \item $Y$ is a non-L-space;
 \item $\pi_1(Y)$ is left-orderable;
 \item $Y$ admits a co-oriented taut foliation.
\end{enumerate}
In general, only the implication (c)$\Rightarrow$(a) \cite{ozsvath2004holomorphic,bowden2016approximating,kazez2017c0} is known.

Dehn surgery on knots in $S^3$ provides a natural source of both L-spaces and non-L-spaces. For a nontrivial knot $K\subset S^3$, the set of slopes yielding L-spaces \cite{kronheimer2007monopoles,rasmussen2017floer} is known to be one of the following three possibilities:
\begin{enumerate}
\item $[2g(K)-1,\infty)$, in which case $K$ is called a positive L-space knot;
\item $(-\infty,-2g(K)+1]$, in which case $K$ is called a negative L-space knot;
\item the empty set, in which case $K$ is called a non-L-space knot.
\end{enumerate}
Here $g(K)$ denotes the Seifert genus of $K$. Thus, the L-space interval is completely determined once we know which case $K$ falls into. In contrast, the corresponding interval for taut foliations or left-orderability is much less understood.

In this paper, we study the existence of taut foliations on non-L-spaces obtained by Dehn surgeries on L-space knots in $S^3$.

\begin{namedtheorem}[Main Theorem]
Suppose $L$ is the closure of a non-split positive braid in $S^3$ with components $L_1,\ldots,L_n$, and assume that at least one component is not the unknot. Then the Dehn surgery along a multislope $(s_1,\ldots,s_n)\in\mathbb{Q}^n$ satisfying $s_i<2g(L_i)-1$ for all $i=1,\ldots,n$ yields a $3$-manifold admitting a co-oriented taut foliation.
\end{namedtheorem}

Lyu proved that $(1,1)$ non-L-space knots are persistently foliar \cite{lyu2025persistent}. It is also known that positive $(1,1)$ L-space knots in $S^3$ are positive braid closures, and that the L-spaces obtained by Dehn surgeries on $(1,1)$ knots have non-left-orderable fundamental groups \cite{nie2021explicit,Li2024TautFoliations}. Combining these results with our main theorem yields the following corollary.

The remaining implication $(a)\Rightarrow(b)$ is still open, even for the $(-2,3,7)$-pretzel knot \cite{varvarezos2021representations}.

Next, define a homotopy
\[
f \colon \tau \times [0,3] \times [0,1] \longrightarrow \tau \times [0,3]
\quad\text{by}\quad
f(p,t,x) := \bigl(p,\max(t,3x)\bigr).
\]
This homotopy gives a deformation retraction from
\[
\bigl\{ (p,t)\in \tau\times[0,3] : t \ge h(p) \bigr\}
\]
onto $\tau \times \{3\}$.

\begin{theorem}
Suppose $L$ is the closure of a non-split positive braid in $S^3$ with components $L_1,\ldots,L_n$, and assume that at least one component is not the unknot. Then the Dehn surgery along a multislope $(s_1,\ldots,s_n)\in\mathbb{Q}^n$ satisfying $s_i<2g(L_i)-1$ for all $i=1,\ldots,n$ yields a $3$-manifold admitting a co-oriented taut foliation.
\end{theorem}
\begin{proof}
 Since $L$ is non-split and non-trivial, $S^3\setminus \operatorname{int}(\nu(L))$ is an irreducible, orientable $3$-manifold whose boundary is a union of incompressible tori. By \Cref{laminar}, $B'$ is a laminar branched surface properly embedded in $S^3 \setminus \operatorname{int}(\nu(L))$. By \Cref{bigon}, $\partial \nu(L)\setminus B'$ is a union of bigons. By \Cref{realizable-slopes}, any multislope $(s_1,\ldots,s_n)\in\mathbb{Q}^n$ satisfying $s_i<2g(L_i)-1$ for all $i=1,\ldots,n$ is realized by the boundary train track $B'\cap \partial \nu(L)$.

The remaining implication $(a)\Rightarrow(b)$ is still open, even for the $(-2,3,7)$-pretzel knot \cite{varvarezos2021representations}.

Our construction differs from previous approaches. Most existing methods begin with a Seifert surface and then apply sutured manifold decompositions and splittings to produce a laminar branched surface. This has been the standard approach for over two decades \cite{roberts2000taut,roberts2001taut,li2003boundary,nakae20072,li2014taut,krishna2020taut,santoro2024spaces,krishna2025taut}.

Instead, we first pinch the Seifert surface to obtain a simpler branched surface, and then perform splittings. From the theory of laminar branched surfaces \cite{li2002laminar}, this intermediate step is not logically necessary. However, for explicit constructions and algorithmic implementation, simplifying the branched surface provides substantial advantages.

In Li's general framework, finding suitable splittings may require searching an exponentially large space of possibilities. In contrast, we design a greedy splitting algorithm of linear complexity (Algorithm~\ref{alg:brick-curve}) that produces a laminar branched surface from the simplified model.

In Krishna's work \cite{krishna2020taut,krishna2025taut} on taut foliations in braid closure complements, the branched surface is constructed by specifying coorientations for the product disks in Gabai's disk decomposition. This is equivalent to choosing curves $\gamma_1,\gamma_2,\ldots,\gamma_k$ in Section~\ref{sec:curve} with the requirement that each $\gamma_i$ lies in the closed region between the $i$-th row and the $(i+1)$-st row. Our construction removes this restriction, which allows greater flexibility and leads to a more general result.

The paper is organized as follows. Section~\ref{sec:2} constructs the branched surface $B$. Section~\ref{sec:3} splits $B$ into a laminar branched surface $B'$, assuming the existence of simple closed curves \(\gamma_1, \gamma_2, \ldots, \gamma_k\) satisfying Properties~\ref{G1}--\ref{G8}. Section~\ref{sec:curve} describes the algorithm producing the curves $\gamma_1,\dots,\gamma_k$ used in the splitting. Section~\ref{sec:5} analyzes the boundary train track. Section~\ref{sec:6} combines these ingredients to prove the main theorem.
\subsection*{Acknowledgements.}
The author is grateful to Qingfeng Lyu for many stimulating and ongoing discussions, and to Jacob Rasmussen for valuable advice and insightful discussions.

The post-theorem introduction does not give a proof sketch for the stated implications in the theorem. It only notes that the remaining implication $(a)\Rightarrow(b)$ is open, and then describes the authors' overall construction/organization: they "first pinch the Seifert surface to obtain a simpler branched surface, and then perform splittings," use a "greedy splitting algorithm of linear complexity (Algorithm~\ref{alg:brick-curve})" to produce a laminar branched surface, and outline sections: Section~\ref{sec:2} constructs $B$, Section~\ref{sec:3} splits $B$ into laminar $B'$ assuming curves $\gamma_1,\dots,\gamma_k$ satisfying Properties~\ref{G1}--\ref{G8}, Section~\ref{sec:curve} produces these curves, Section~\ref{sec:5} analyzes the boundary train track, and Section~\ref{sec:6} "combines these ingredients to prove the main theorem." No step-by-step argument establishing $(a)\Leftrightarrow(c)$ and $(a)\Leftarrow(b)$ is stated here.

The post-theorem introduction does not give a proof sketch for the stated implications in the theorem. It only notes that the remaining implication $(a)\Rightarrow(b)$ is open, and then describes the authors' overall construction/organization: they ``first pinch the Seifert surface to obtain a simpler branched surface, and then perform splittings,'' use a ``greedy splitting algorithm of linear complexity'' given by

\begin{algorithm}[!htbp]
\caption{Construction of the curves}
\label{alg:brick-curve}
\begin{algorithmic}[1]
\Require A positive braid diagram $D\subset S^1\times I$ and its associated brick diagram
\Ensure Simple closed curves satisfying Properties~\ref{G1}--\ref{G8}

\State Choose a non-vertex point $q$ in the $i$-th row lying on a component of $L$ which is not the unknot, with $i$ minimal.
\State $A\gets q$, $B\gets q$, $\gamma\gets\{q\}$, $\Gamma\gets\varnothing$
\While{$d_{\mathrm{up}}(B,A)=2$}
    \State \Call{StepBelow}{$A,B,\gamma$}
\EndWhile
\State \Call{CrossBrickStep}{$A,B,\gamma$} \label{line:CrossBrickStep}
\While{$A$ is not on the last row}
    \If{$d_{\mathrm{up}}(B,A)\ge2$}
        \State \Call{StepBelow}{$A,B,\gamma$}
    \Else
        \State \Call{CloseAndStore}{$\gamma,\Gamma,[B,A]$} \label{line:close-1}
        \State \Call{RestartBelow}{$A,B,\gamma$} \label{line:restart-below}
    \EndIf
\EndWhile
\State \Call{CloseAndStore}{$\gamma,\Gamma,[B,A]$}\label{line:close-2}
\If{$q$ is not on the first row}
    \State $A\gets q$, $B\gets q$
    \State \Call{RestartAbove}{$A,B,\gamma$}\label{line:restart-above1}
    \While{$A$ is not on the first row}
        \If{$d_{\mathrm{up}}(A,B)\ge2$}
            \State \Call{StepAbove}{$A,B,\gamma$}
        \Else
            \State \Call{CloseAndStore}{$\gamma,\Gamma,[A,B]$}\label{line:close-3}
            \State \Call{RestartAbove}{$A,B,\gamma$}\label{line:restart-above2}
        \EndIf
    \EndWhile
    \State \Call{CloseAndStore}{$\gamma,\Gamma,[A,B]$}\label{line:close-4}
\EndIf

\State \Return $\Gamma$
\end{algorithmic}

\end{algorithm}

to produce a laminar branched surface, and outlines the later sections: next they construct $B$; then they split $B$ into laminar $B'$ assuming curves $\gamma_1,\dots,\gamma_k$ satisfying the following properties:

\begin{enumerate}[label=(G\arabic*)]
    \item\label{G1} Each curve $\gamma_i$ is transverse to every vertical line and hence intersects each vertical line $\{x\}\times I$ once.
    \item\label{G2} Each curve $\gamma_i$ is composed of two arcs: one arc connects the two vertices of a bigon in the complement of $\tau$, while the other arc lies on the train track $\tau$.
    \item\label{G3} For each vertical line $\{x\}\times I$ with $x\in S^1$, the intersection point
    $\gamma_i \cap (\{x\}\times I)$ lies above or coincides with
    $\gamma_{i+1} \cap (\{x\}\times I)$ for all $i=1,\ldots,k-1$.
    \item\label{G4} The bigons associated to the curves $\gamma_1,\gamma_2, \ldots, \gamma_k$ are pairwise distinct.
    \item\label{G5} There exists a point $p\in \gamma_1$ such that $p$ lies on an under-type arc of $\tau$, and there is no bigon directly above $p$.
    \item\label{G6} There exists a point $p\in \gamma_k$ such that $p$ lies on an under-type arc of $\tau$, and there is no bigon directly below $p$.
    \item\label{G7} For each $1\le i\le k-1$, there exists a vertical segment connecting a point $p_1 \in \gamma_i$ to a point $p_2 \in \gamma_{i+1}$ whose interior is disjoint from $\tau$, such that one of the points $p_1,p_2$ lies on an under-type arc of $\tau$ and the other does not lie on $\tau$.
    \item\label{G8} For each under-arc of $D$, there exists a point $p\in \tau$ lying on an under-type arc of $\tau$ arising from that under-arc, such that $p\notin \gamma_i$ for all $1\le i\le k$.
\end{enumerate}

After that they produce these curves; then they analyze the boundary train track; and finally they ``combine these ingredients to prove the main theorem.'' No step-by-step argument establishing $(a)\Leftrightarrow(c)$ and $(a)\Leftarrow(b)$ is stated here.

Implication
Biconditional or Equivalence

Question:
Let \(Y\) be a 3-manifold obtained by Dehn surgery on a \((1,1)\) knot in \(S^3\), where a \((1,1)\) knot means a knot admitting a genus-one Heegaard splitting of \(S^3\) in which the knot meets each solid torus in a single boundary-parallel arc. In the notation of the L-space conjecture, let
\((a)\) mean that \(Y\) is a non-L-space,
\((b)\) mean that \(\pi_1(Y)\) is left-orderable, and
\((c)\) mean that \(Y\) admits a co-oriented taut foliation.
Which statement about these conditions holds for all such surgery manifolds \(Y\)?

For every 3-manifold \(Y\) obtained by Dehn surgery on a \((1,1)\) knot in \(S^3\), conditions \((a)\) and \((c)\) are equivalent, and condition \((b)\) implies \((a)\); that is, \(Y\) is a non-L-space if and only if \(Y\) admits a co-oriented taut foliation, and if \(\pi_1(Y)\) is left-orderable then \(Y\) is a non-L-space.

Choices:
B. For every 3-manifold \(Y\) obtained by Dehn surgery on a \((1,1)\) knot in \(S^3\), conditions \((a)\) and \((b)\) are equivalent, and condition \((c)\) implies \((a)\); that is, \(Y\) is a non-L-space if and only if \(\pi_1(Y)\) is left-orderable, and if \(Y\) admits a co-oriented taut foliation then \(Y\) is a non-L-space.
C. For every 3-manifold \(Y\) obtained by Dehn surgery on a \((1,1)\) knot in \(S^3\), if \(Y\) admits a co-oriented taut foliation, then \(Y\) is a non-L-space; that is, condition \((c)\) implies condition \((a)\).
D. For every 3-manifold \(Y\) obtained by Dehn surgery on a \((1,1)\) knot in \(S^3\), conditions \((a)\), \((b)\), and \((c)\) are all equivalent; that is, \(Y\) is a non-L-space if and only if \(\pi_1(Y)\) is left-orderable if and only if \(Y\) admits a co-oriented taut foliation.
E. For every 3-manifold \(Y\) obtained by Dehn surgery on a \((1,1)\) knot in \(S^3\), condition \((a)\) implies condition \((b)\), and condition \((a)\) is equivalent to condition \((c)\); that is, if \(Y\) is a non-L-space then \(\pi_1(Y)\) is left-orderable, and \(Y\) is a non-L-space if and only if \(Y\) admits a co-oriented taut foliation.

which implication involving left-orderability is established

drops the converse (a)⇒(c) and the separate implication (b)⇒(a)

reverses the one-way implication between non-L-space and left-orderability


Question:
Let \(Y\) be a \(3\)-manifold obtained by Dehn surgery on a \((1,1)\) knot in \(S^3\). In the notation of the L-space conjecture, let
\((a)\) mean that \(Y\) is a non-L-space,
\((b)\) mean that \(\pi_1(Y)\) is left-orderable, and
\((c)\) mean that \(Y\) admits a co-oriented taut foliation.
Which statement correctly describes the implications among \((a)\), \((b)\), and \((c)\) for such \(Y\)?

\((a)\) and \((c)\) are equivalent, and \((b)\) implies \((a)\); equivalently, \(Y\) is a non-L-space if and only if \(Y\) admits a co-oriented taut foliation, and if \(\pi_1(Y)\) is left-orderable, then \(Y\) is a non-L-space.

Choices:
B. \((a)\), \((b)\), and \((c)\) are all equivalent; equivalently, \(Y\) is a non-L-space if and only if \(\pi_1(Y)\) is left-orderable if and only if \(Y\) admits a co-oriented taut foliation.
C. \((c)\) implies \((a)\); equivalently, whenever \(Y\) admits a co-oriented taut foliation, \(Y\) is a non-L-space.
D. \((a)\) and \((b)\) are equivalent, and \((c)\) implies \((a)\); equivalently, \(Y\) is a non-L-space if and only if \(\pi_1(Y)\) is left-orderable, and every co-oriented taut foliation on \(Y\) forces \(Y\) to be a non-L-space.
E. \((a)\Rightarrow(c)\) and \((a)\Rightarrow(b)\); equivalently, every non-L-space obtained by Dehn surgery on a \((1,1)\) knot in \(S^3\) admits a co-oriented taut foliation and has left-orderable fundamental group.


[Song--Wang {\cite{wang2}}]\label{conj}
There exists, for each level $\ell$, a combinatorial bijection
\begin{equation}
\label{e:psi}
\psi_\ell : \mathrm{ParMat}_{\nu,\mu}^{\flat} \longrightarrow
\bigsqcup_{\lambda \vdash_\ell n}
\mathrm{SST}_\ell(\lambda,\mu)\times \mathrm{SST}_\ell(\lambda,\nu),
\end{equation}
which extends the level $\ell-1$ bijection and recovers the classical RSK correspondence when $\ell=1$.

The Robinson--Schensted--Knuth correspondence is a bijection between (bi)words or nonnegative integer matrices and pairs of Young tableaux of the same shape, with applications to symmetric functions, permutation statistics such as longest increasing subsequences, and the representation theory of the symmetric and general linear groups. With the development of web categories, representation theories traditionally governed by Young tableaux have been reformulated in diagrammatic terms that incorporate additional structure, such as multiple components, block decompositions, and restrictions on how labels may appear. In this setting, the relevant combinatorial data consist not of a single word or matrix but of several interrelated pieces, and the resulting tableau data record this multi-component structure. This viewpoint naturally raises the question of whether classical correspondences such as RSK admit higher--level analogues that organize such data in a way compatible with the additional structure.

In \cite{beeo}, the authors introduced the \emph{Schur category}, whose path algebra recovers certain subalgebras of Schur algebras, alongside a category \emph{Web}, a $\mathfrak{gl}_{\infty}$--analogue of the $\mathfrak{sl}_n$ web categories. They showed that these categories are isomorphic, and that both contain the (degenerate) Hecke category via fully faithful functors. Building on this framework, Song and Wang \cite{wang1,wang2} defined \emph{affine Schur} and \emph{affine web} categories. In contrast to the finite case, the affine web category embeds as a full subcategory of the affine Schur category rather than being equivalent to it. Their work further introduced \emph{cyclotomic} quotients of these categories, which retain embeddings of the cyclotomic Hecke category and exhibit rich combinatorial structure.

A key result of \cite{wang2} identifies the path algebra of the cyclotomic Schur category with the cyclotomic Schur algebra of Dipper--James--Mathas \cite{djm}. Song and Wang describe two natural bases for this algebra. The first is indexed by pairs of tableaux and corresponds, under this identification, to the cellular basis of the cyclotomic Schur algebra. The second is a family of basis elements indexed by enriched block matrices arising from elementary ``chicken--foot'' diagrams in the cyclotomic web category. The coexistence of these two parametrizations naturally suggests a bijection relating block-matrix data to pairs of tableaux in the cyclotomic setting.

A key result of \cite{wang2} identifies the path algebra of the cyclotomic Schur category with the cyclotomic Schur algebra of Dipper--James--Mathas \cite{djm}. Song and Wang describe two natural bases for this algebra. The first is indexed by pairs of tableaux and corresponds, under this identification, to the cellular basis of the cyclotomic Schur algebra. The second is a family of basis elements indexed by enriched block matrices arising from elementary ``chicken--foot'' diagrams in the cyclotomic web category. The coexistence of these two parametrizations naturally suggests a bijection relating block-matrix data to pairs of tableaux in the cyclotomic setting.

We confirm this conjecture by giving an explicit bijection at level~$\ell$ that uses only the classical RSK correspondence. The key step is to recast the combinatorial parameters in the set $\mathrm{ParMat}^{\flat}$ as \emph{flagged block composition matrices}. These matrices encode, in a transparent way, collections of biwords subject to the level-dependent labeling restrictions. With this reformulation in hand, the bijection is obtained by applying the classical RSK correspondence independently in each component and then assembling the outputs as semistandard multitableaux of common shape.

\begin{theorem}\label{classicalRSK}[Robinson--Schensted--Knuth \cite{RSK}]
There is a bijection
\[
\mathrm{BW}(\mathbb N)
\longleftrightarrow
\coprod_{\lambda\vdash n}
\mathrm{SST}(\mathbb N,\lambda)
\times
\mathrm{SST}(\mathbb N,\lambda).
\]
For biword $w$ corresponding to $(P,Q)$,
\(
\mathrm{wt}(P)=\text{content of } w_{\mathrm{bot}}\)
and
\(\mathrm{wt}(Q)=\text{content of } w_{\mathrm{top}}.
\)
\end{theorem}

Each biword $w\in \mathrm{BW}(\mathbb N)$ naturally determines an
$\mathbb N$-matrix $A$ by setting $a_{ij}$ equal to the multiplicity of the
biletter $\begin{pmatrix} i \\ j \end{pmatrix}$ in $w$.
Under this correspondence,
\[
\mathrm{row}(A)_i=\#\{\text{occurrences of } i \text{ in } w_{\mathrm{top}}\}
\quad\text{and}\quad
\mathrm{col}(A)_j=\#\{\text{occurrences of } j \text{ in } w_{\mathrm{bot}}\}.
\]
Thus the RSK correspondence may equivalently be written as a bijection
\begin{equation}
\label{e rskmatrix}
 \left\{\NN\mathrm{-matrices}\,\,A\setmid\begin{tabular}{c}$\mathrm{row}(A)=\nu$\\$\mathrm{col}(A)=\mu$\end{tabular}\right\}\longleftrightarrow\coprod_{\lambda\vdash n}\text{\rm SST}(\NN,\lambda,\mu)\times\text{\rm SST}(\NN,\lambda,\nu).
 \end{equation}

The cyclotomic web category is connected in \cite{wang2} to a combinatorial model called
(elementary) chicken foot diagrams.
The diagrammatic parameters indexing the cyclotomic web basis may be encoded by certain partition-enriched block matrices. These matrices are packaged into a set
\[
\mathrm{ParMat}^{\flat}_{\nu,\mu},
\]
indexed by multicompositions $\nu,\mu \vDash_\ell n$ which record the prescribed row and column data of the underlying block matrix.
Informally, an element of $\mathrm{ParMat}^{\flat}_{\nu,\mu}$ consists of:
\begin{itemize}
\item an $\ell\times\ell$ block matrix of nonnegative integers whose block row and column sums are fixed by $\nu$ and $\mu$, and
\item a choice of a partition in each matrix entry, subject to natural size and length constraints determined by the block position.
\end{itemize}

\begin{lemma}\label{lem:parmat-to-bcm}
Fix $\mu,\nu \vDash_\ell n$. There is a bijection
\begin{align*}
\mathrm{ParMat}^{\flat}_{\nu,\mu}&\longleftrightarrow\text{BCM}_\ell(\nu,\mu),\\
(A,P)&\mapsto B,
\end{align*}
satisfying $a_{ij}^{(pq)}=\left|b^{(pq)}_{ij}\right|$ and $\mathrm{length}\left(\eta_{ij}^{(pq)}\right)=\mathrm{length}\left(b_{ij}^{(pq)}\right)-1$.
\end{lemma}
\begin{proof}
There is a natural, length-respecting bijection between compositions of $k$ and partitions with parts at most $k$: if $\lambda$ is such a partition,
the associated composition $c$ is given by
\(
c_t=\lambda_{t-1}-\lambda_t,
\)
with the convention $\lambda_0=k$ and $\lambda_t=0$ for $t>\ell(\lambda)$.
The inverse map recovers $\lambda$ from $c$ by taking partial sums
$\lambda_t=\sum_{s>t} c_s$.

\begin{example}
For $\ell=3$, $\mu=(212,3,23)$ and $\nu=(13,3,222)$, the flagged compositional
block matrix $B$ from Example~\ref{ex:BCM-with-B}, written with
a letter-coordinate key, is sent to the flagged multi biword of
Example~\ref{uvword}\\
\begin{tabular}{c | c c | c | c c c}
\color{violet}\text{key:}& \color{violet}$1_1$ & \color{violet}$2_1$ & \color{violet}$1_2$ & \color{violet}$1_3$ & \color{violet}$2_3$ & \color{violet}$3_3$\\
\hline
{\color{violet}$1_1$}&$\varnothing$ & $2$ & $\varnothing$ & $\varnothing$ & $\varnothing$ & $\varnothing$\\
{\color{violet}$2_1$}&$\varnothing$ & $\varnothing$ & $\varnothing$ & $1$ & $\varnothing$ & $\varnothing$\\
{\color{violet}$3_1$}&$1$ & $\varnothing$ & $\varnothing$ & $\varnothing$ & $\varnothing$ & $1$\\
\hline
{\color{violet}$1_2$}&$\varnothing$ & $\varnothing$ & $03$ & $\varnothing$ & $\varnothing$ & $\varnothing$\\
\hline
{\color{violet}$1_3$}&$\varnothing$ & $\varnothing$ & $\varnothing$ & $001$ & $\varnothing$ & $01$\\
{\color{violet}$2_3$}&$\varnothing$ & $1$ & $\varnothing$ & $\varnothing$ & $011$ & $\varnothing$\\
\end{tabular}
$\longleftrightarrow$
$\left(\begin{pmatrix}1_1\ 1_1\ 2_1 \ 3_1 \ 3_1\ 2_3 \\ 2_1\ 2_1 \ 1_3 \ 1_1 \ 3_3\ 2_1\end{pmatrix},
\begin{pmatrix}1_2\ 1_2\ 1_2 \ 1_3 \ 2_3 \\ 1_2\ 1_2 \ 1_2 \ 3_3\ 2_3\end{pmatrix}\,,
\begin{pmatrix}1_3\ 2_3\\ 1_3\ 2_3\end{pmatrix}\right)$.
\end{example}
\begin{lemma}\label{lem:biwords-to-tableaux}
Fix $\mu,\nu \models_\ell n$. There is a bijection
\[
BW_\ell(\nu,\mu)\;
\longleftrightarrow\coprod_{\lambda\vdash_\ell n}\text{\rm SST}_\ell(\lambda,\mu)\times\text{\rm SST}_\ell(\lambda,\nu),
\]
obtained by applying the classical Robinson--Schensted--Knuth correspondence
componentwise, after an order-preserving relabeling of the $\mu$-- and
$\nu$--alphabets.
\end{lemma}
\begin{proof}
Let $w=(w^{(1)},\dots,w^{(\ell)})\in BW_\ell(\nu,\mu)$.
Construct the pair $(P,Q)$ of semistandard multitableaux componentwise using the usual RSK correspondence of Thm~\ref{classicalRSK}, where
we set $(P^{(i)},Q^{(i)})=\text{RSK}(w^{(i)})$, but with the modification that the (totally ordered) $\mu$- and $\nu$-alphabets serve in place of the usual letters $[n]$.

\section{The correspondence}
\begin{theorem}
Fix $\mu,\nu \models_\ell n$. Applying the classical RSK correspondence of Thm~\ref{classicalRSK}
componentwise defines a bijection
\begin{equation}
\text{ParMat}^\flat_{\nu,\mu}
\longleftrightarrow\coprod_{\lambda\vdash_\ell n}\text{\rm SST}_\ell(\lambda,\mu)\times\text{\rm SST}_\ell(\lambda,\nu).
\end{equation}
When restricted to multipartitions satisfying
$\mu^{(\ell)}=\nu^{(\ell)}=\lambda^{(\ell)}=\varnothing$, this bijection agrees with the
level--$(\ell-1)$ correspondence. For $\ell=1$, it reduces to the usual RSK
correspondence.
\end{theorem}
\begin{proof}
We first use Lemma~\ref{lem:parmat-to-bcm} to replace the domain with
$\text{\rm BCM}_\ell(\nu,\mu)$. From there,
Lemma~\ref{lem:BCM-to-biwords} takes flagged block composition matrices bijectively to
flagged biwords. Lemma~\ref{lem:biwords-to-tableaux} then applies
classical RSK componentwise to obtain the desired correspondence. The
specialization and restriction properties follow directly from the definitions.
\end{proof}

We confirm this conjecture by giving an explicit bijection at level~$\ell$ that uses only the classical RSK correspondence. The key step is to recast the combinatorial parameters in the set $\mathrm{ParMat}^{\flat}$  as \emph{flagged block composition matrices}. These matrices encode, in a transparent way, collections of biwords subject to the level-dependent labeling restrictions. With this reformulation in hand, the bijection is obtained by applying the classical RSK correspondence independently in each component and then assembling the outputs as semistandard multitableaux of common shape.

Our approach proceeds in three steps. First, we reinterpret the $\mathrm{ParMat}^{\flat}$-indexed data in terms of the {flagged block composition matrices}, which present the same information in a form amenable to combinatorial manipulation. Second, we establish a natural bijection between these matrices and collections of \emph{flagged biwords}, extending the classical correspondence between nonnegative integer matrices and biwords. Finally, we apply the classical RSK correspondence componentwise (after a simple relabeling of the alphabets) to obtain pairs of semistandard multitableaux of common shape.

The resulting correspondence is compatible with restriction to lower levels, specializes to the usual RSK correspondence when $\ell=1$, and reflects the multipartition data inherent in the cyclotomic setting. In particular, it provides a direct combinatorial explanation for the relationship between the tableau--indexed and diagrammatic bases of the cyclotomic Schur category.

\noindent {\bf Acknowledgement.} We thank Linliang Song and Weiqiang Wang for formulating the initiating conjecture. We are especially grateful to Weiqiang Wang for personally recommending this combinatorial problem.

We “confirm this conjecture by giving an explicit bijection at level~$\ell$ that uses only the classical RSK correspondence.” The “key step is to recast the combinatorial parameters in the set $\mathrm{ParMat}^{\flat}$ as \emph{flagged block composition matrices},” which “encode, in a transparent way, collections of biwords subject to the level-dependent labeling restrictions.”

The approach “proceeds in three steps.” (1) “Reinterpret the $\mathrm{ParMat}^{\flat}$-indexed data in terms of the flagged block composition matrices,” putting the data “in a form amenable to combinatorial manipulation.” (2) “Establish a natural bijection between these matrices and collections of \emph{flagged biwords}, extending the classical correspondence between nonnegative integer matrices and biwords.” (3) “Apply the classical RSK correspondence componentwise (after a simple relabeling of the alphabets) to obtain pairs of semistandard multitableaux of common shape,” i.e. “apply the classical RSK correspondence independently in each component and then assemble the outputs as semistandard multitableaux of common shape.”

The resulting correspondence is stated to be “compatible with restriction to lower levels” and to “specialize[] to the usual RSK correspondence when $\ell=1$,” matching the properties required for the bijection $\psi_\ell$ in Theorem~\ref{conj}.

We “confirm this conjecture by giving an explicit bijection at level~$\ell$ that uses only the classical RSK correspondence.” The “key step is to recast the combinatorial parameters in the set $\mathrm{ParMat}^{\flat}$ as \emph{flagged block composition matrices},” which “encode, in a transparent way, collections of biwords subject to the level-dependent labeling restrictions.”

The approach “proceeds in three steps.” (1) “Reinterpret the $\mathrm{ParMat}^{\flat}$-indexed data in terms of the flagged block composition matrices,” putting the data “in a form amenable to combinatorial manipulation.” (2) “Establish a natural bijection between these matrices and collections of \emph{flagged biwords}, extending the classical correspondence between nonnegative integer matrices and biwords.” (3) “Apply the classical RSK correspondence componentwise (after a simple relabeling of the alphabets) to obtain pairs of semistandard multitableaux of common shape,” i.e. “apply the classical RSK correspondence independently in each component and then assemble the outputs as semistandard multitableaux of common shape.”

The resulting correspondence is stated to be “compatible with restriction to lower levels” and to “specialize[] to the usual RSK correspondence when $\ell=1$,” matching the properties required for the bijection $\psi_\ell$ in establishing the main theorem.

Classification or Bijection
Universal–Existential

Question: Fix a level \(\ell\) and \(\ell\)-compositions \(\mu,\nu\models_\ell n\). Let \(\mathrm{ParMat}^{\flat}_{\nu,\mu}\) denote the paper’s flagged block-matrix/combinatorial matrix data with parameters \((\nu,\mu)\), and for an \(\ell\)-multipartition \(\lambda\vdash_\ell n\), let \(\mathrm{SST}_\ell(\lambda,\mu)\) and \(\mathrm{SST}_\ell(\lambda,\nu)\) be the sets of semistandard multitableaux of shape \(\lambda\) and weights \(\mu\) and \(\nu\), respectively. Which statement holds for these objects?

For each level \(\ell\), there exists a combinatorial bijection
\[
\psi_\ell:\mathrm{ParMat}^{\flat}_{\nu,\mu}\longrightarrow \bigsqcup_{\lambda\vdash_\ell n}\mathrm{SST}_\ell(\lambda,\mu)\times \mathrm{SST}_\ell(\lambda,\nu),
\]
which extends the level \(\ell-1\) bijection and recovers the classical Robinson--Schensted--Knuth correspondence when \(\ell=1\).

Choices:
B.
For each level \(\ell\), there exists a combinatorial bijection
\[
\psi_\ell:\mathrm{ParMat}^{\flat}_{\nu,\mu}\longrightarrow \bigsqcup_{\lambda\vdash_\ell n}\mathrm{SST}_\ell(\lambda,\mu)\times \mathrm{SST}_\ell(\lambda,\nu),
\]
which extends the level \(\ell-1\) bijection and, for \(\ell=1\), agrees with the classical Robinson--Schensted--Knuth correspondence only after restricting to ordinary partitions \(\lambda\) with a single nonempty component.
C.
For each level \(\ell\), there exists a combinatorial bijection
\[
\psi_\ell:\mathrm{ParMat}^{\flat}_{\nu,\mu}\longrightarrow \bigsqcup_{\lambda\vdash_\ell n}\mathrm{SST}_\ell(\lambda,\mu)\times \mathrm{SST}_\ell(\lambda,\nu).
\]
D.
For each level \(\ell\), there exists a combinatorial bijection
\[
\psi_\ell:\mathrm{ParMat}^{\flat}_{\nu,\mu}\longrightarrow \bigsqcup_{\lambda\vdash_\ell n}\mathrm{SST}_\ell(\lambda,\mu)\times \mathrm{SST}_\ell(\lambda,\nu),
\]
which recovers the classical Robinson--Schensted--Knuth correspondence when \(\ell=1\), and whose restriction to multipartitions satisfying \(\mu^{(\ell)}=\nu^{(\ell)}=\lambda^{(\ell)}=\varnothing\) is canonically the level \(\ell\) correspondence.
E.
For each level \(\ell\), there exists a combinatorial bijection
\[
\psi_\ell:\mathrm{ParMat}^{\flat}_{\nu,\mu}\longrightarrow \bigsqcup_{\lambda\vdash_\ell n}\mathrm{SST}_\ell(\lambda,\nu)\times \mathrm{SST}_\ell(\lambda,\mu),
\]
which extends the level \(\ell-1\) bijection and recovers the classical Robinson--Schensted--Knuth correspondence when \(\ell=1\).

exact \ell=1 specialization to classical RSK

dropped compatibility with lower levels and \ell=1 specialization

lower-level restriction identifies with level \ell-1, not level \ell

componentwise RSK preserves ordered pairing of \mu- and \nu-weight tableau factors