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\newcommand{\0}{\mathtt{0}}
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\title{Coincidences of $q$-analogues of Markoff numbers\\ outside the known families:
an answer to a question of\\ Labb\'e, Lapointe and Steiner}
\author{David Niedbala Giraudin\\ \small ORCID 0009-0009-1526-1178 --- Independent researcher, Meaux, France}
\date{31 October 2026}

\begin{document}
\maketitle

\begin{abstract}
Labb\'e, Lapointe and Steiner proved that $w\mapsto\mu_q(w)_{12}$ is injective on Christoffel words,
gave two families of identities showing that it is not injective on $\{\0,\1\}^*$, and asked whether
every coincidence $\mu_q(x)_{12}=\mu_q(y)_{12}$ comes from these two families. We show that it does
not. The smallest counterexamples have length $24$; by an exhaustive computation, among words of
length at most $32$ there are exactly $50$ classes of coincidences not explained by the two families, $10$ of length $24$ and $40$
of length $30$. Each of them contains an instance of one of a few new identities, which we prove for a
generic $2\times2$ matrix in place of the matrix of a factor of the word. One of these identities
belongs to an infinite family, indexed by $k\equiv1\pmod 4$, which we prove for every such $k$.
\end{abstract}

\section{Introduction}

Let $L_q=\begin{pmatrix}q&0\\q&1\end{pmatrix}$, $R_q=\begin{pmatrix}q&1\\0&1\end{pmatrix}$, let
$M_q:\{\0,\1\}^*\to\mathrm{GL}_2(\Z[q^{\pm1}])$ be the morphism with $M_q(\0)=L_q$,
$M_q(\1)=R_q$, and let $\mu_q$ be the morphism with
\[
\mu_q(\0)=R_qL_q=M_q(\1\0),\qquad \mu_q(\1)=R_qR_qL_qL_q=M_q(\1\1\0\0),
\]
the $q$-analogue of the Markoff matrices introduced by Leclere and Morier-Genoud~\cite{LMG}. For a
Christoffel word $w$, $\mu_q(w)_{12}$ is a $q$-analogue of the Markoff number of $w$. Labb\'e,
Lapointe and Steiner proved that $w\mapsto\mu_q(w)_{12}$ is injective on Christoffel
words~\cite{LLS}, and showed that it is not injective on $\{\0,\1\}^*$ by two families of
identities, their equations (4) and (7):
\begin{align*}
\mu_q(\0\,w\,\1)_{12}&=\mu_q(\0\,\widetilde w\,\1)_{12}, \tag{4}\\
\mu_q(\0\,\psi_w(v)\,w\,\1)_{12}&=\mu_q(\0\,\psi_w(\widehat v)\,w\,\1)_{12}, \tag{7}
\end{align*}
for $w\in\{\0,\1\}^*$ and $v\in\{\0,\1,\mathtt2,\mathtt3\}^*$. Here $\widetilde w$ is the mirror image
of $w$, $\psi_w$ is the morphism $\0\mapsto w\,\0\1\,\widetilde w\,\0\1$,
$\1\mapsto w\,\1\0\,\widetilde w\,\1\0$, $\mathtt2\mapsto w\,\0\1\,\widetilde w\,\1\0$,
$\mathtt3\mapsto w\,\1\0\,\widetilde w\,\0\1$, and $\widehat v$ is the mirror image of $v$ with
$\0\leftrightarrow\1$ exchanged and $\mathtt2,\mathtt3$ fixed. (The published version of~\cite{LLS} has
$\overline v$ in place of $\widehat v$ in both identities of its Theorem~3.2; the correction is made in
the latest arXiv version, and we use the corrected form.) They asked~\cite[Question~3.5]{LLS}:

\smallskip
\emph{Do there exist $x,y\in\{\0,\1\}^*$ satisfying $\mu_q(x)_{12}=\mu_q(y)_{12}$ which are not
given by Equation (4) or (7)?}
\smallskip

\noindent They state that they believe (4) and (7) account for all coincidences. The answer to the
question is yes.

\begin{theorem}\label{thm:main}
\begin{enumerate}
\item[(a)] The words
\begin{align*}
A&=\0\0\1\0\1\1\0\0\1\0\1\1\0\1\0\0\1\1\0\0\1\0\1\1,\\
B&=\0\0\1\1\0\1\0\0\1\0\1\0\1\1\0\0\1\1\0\1\0\0\1\1
\end{align*}
satisfy $\mu_q(A)_{12}=\mu_q(B)_{12}$, a polynomial of degree $69$ whose value at $q=1$ is
$43\,633\,212\,873\,750$, and $B$ cannot be obtained from $A$ by any finite chain of identities (4)
and~(7).
\item[(b)] Among the words of length at most $32$, the coincidences of $\mu_q(\cdot)_{12}$ that are not
explained by (4) and (7) form exactly $50$ classes: $10$ of length $24$ and $40$ of length $30$
(Table~\ref{tab:census}). In particular $24$ is the smallest length at which a counterexample occurs.
\item[(c)] Each of these $50$ classes contains an instance of Theorem~\ref{thm:k5} ($30$ classes) or of
Theorem~\ref{thm:self} ($20$ classes).
\item[(d)] Theorem~\ref{thm:k5}(i) is the first member of an infinite family of identities
(Theorem~\ref{thm:family}). Its second member gives, for instance, a pair of words of length $50$,
listed in Section~\ref{sec:data}, which is again not explained by (4) and~(7).
\end{enumerate}
\end{theorem}

Parts (a) and (c), the closing assertion of (d), and the identities of Theorems~\ref{thm:k5}
and~\ref{thm:self} are checked in exact arithmetic by the script attached to this note
(Section~\ref{sec:data}). Part (b) rests on an exhaustive computation which is not part of the
deposit; see Section~\ref{sec:census}.

\section{Four facts}

Write $\overline z$ for the mirror image of $z\in\{\0,\1\}^*$ with $\0\leftrightarrow\1$ exchanged.
Let $Q=\mathrm{diag}(q,1)$, $S=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$, $J=SQ=\begin{pmatrix}0&-1\\q&0\end{pmatrix}$,
and for a $2\times2$ matrix $X$ put $\varphi(X)=(QXQ^{-1})^{\mathsf T}=Q^{-1}X^{\mathsf T}Q$. Let
$r=(q,1)$ and $s=(1,1)^{\mathsf T}$.

\begin{lemma}\label{lem:facts}
\begin{enumerate}
\item[(F1)] $M_q(\Pw)=M_q(\Nw)+(q^3+1)J$.
\item[(F2)] $XJ\varphi(X)=\det(X)\,J$ for every $2\times2$ matrix $X$.
\item[(F3)] $\varphi$ is an involutive anti-automorphism, and $\varphi(M_q(z))=M_q(\overline z)$.
\item[(F4)] $M_q(\1 z\1)_{12}=r\,M_q(z)\,s$; and $\mu_q(w)_{12}=M_q(\sigma(w))_{12}$, where $\sigma$ is the
morphism $\0\mapsto\1\0$, $\1\mapsto\1\1\0\0$, and this entry does not change if the final letters
$\0$ of $\sigma(w)$ are deleted.
\end{enumerate}
\end{lemma}

\begin{proof}
(F1) is a direct computation. (F2): $XJ\varphi(X)=XSQQ^{-1}X^{\mathsf T}Q=XSX^{\mathsf T}Q=\det(X)SQ$.
(F3): $\varphi$ reverses products, and $\varphi(L_q)=R_q$, $\varphi(R_q)=L_q$. (F4): the first row of
$R_q$ is $r$ and its second column is $s$; the second column of $L_q$ is $(0,1)^{\mathsf T}$, so
$(XL_q)_{12}=X_{12}$.
\end{proof}

Fact (F1), and (F2) for $X=M_q(z)$ in the form $M_q(z)\,SQ\,M_q(\overline z)=q^{|z|}SQ$, appear in the
proof of~\cite[Lemma~3.3]{LLS}; (F3) is equivalent to their equation~(5). By (F4), for $w\in\{\0,\1\}^*$ the word $\sigma(w)$ with its final $\0$'s deleted
begins and ends with $\1$, and coincidences of $\mu_q(\cdot)_{12}$ are coincidences of
$z\mapsto r\,M_q(z)\,s$ on the words obtained by deleting these two letters.

\section{Identities with five windows}

The two words $\Pw$ and $\Nw$ are called \emph{windows}; note $\overline{\Pw}=\Nw$. For a word $T$, put
$T^{+}=T$ and $T^{-}=\overline T$. For $\varepsilon=(\varepsilon_0,\dots,\varepsilon_k)\in\{\pm\}^{k+1}$
and windows $a_1,\dots,a_k$, put
\[
Z_\varepsilon(T;a_1,\dots,a_k)=\1\,T^{\varepsilon_0}a_1T^{\varepsilon_1}a_2\cdots a_kT^{\varepsilon_k}\,\1 .
\]
For a set $F$ of positions, $a^F$ denotes the sequence obtained from $a$ by exchanging $\Pw$ and
$\Nw$ at the positions of $F$.

\begin{theorem}\label{thm:k5}
Let $T\in\{\0,\1\}^*$ and let $a_1,\dots,a_5$ be windows. Then
$M_q(Z_\varepsilon(T;a))_{12}=M_q(Z_\varepsilon(T;a^F))_{12}$ in each of the following cases:
\begin{enumerate}
\item[(i)] $\varepsilon=(+,-,+,-,-,+)$, $F=\{1,3,5\}$, and $a_1=a_5\neq a_3$;
\item[(ii)] $\varepsilon=(+,-,-,+,-,+)$, $F=\{1,3,5\}$, and $a_1=a_5\neq a_3$;
\item[(iii)] $\varepsilon=(+,-,-,+,+,-)$, $F=\{2,4\}$, $a_1=a_5\neq a_3$ and $a_2=a_4$.
\end{enumerate}
\end{theorem}

\begin{proof}
By (F3) and (F4), $M_q(Z_\varepsilon(T;a))_{12}=r\,G^{\varepsilon_0}A_1G^{\varepsilon_1}\cdots A_5G^{\varepsilon_5}\,s$
with $G=M_q(T)$, $G^{+}=G$, $G^{-}=\varphi(G)$ and $A_i=M_q(a_i)$. It therefore suffices to prove
the corresponding identities with $G=\begin{pmatrix}\alpha&\beta\\\gamma&\delta\end{pmatrix}$ generic, that is, in
$\Z[\alpha,\beta,\gamma,\delta,q]$ after clearing the common power of $q$ coming from $\varphi(G)$. These are finitely
many polynomial identities ($4$, $4$ and $2$ in the three cases, up to exchanging the two sides), and they are verified
by exact expansion; see Section~\ref{sec:data}.
\end{proof}

\begin{remark}
For the generic matrix the conditions on $a$ are exact: for each of the three pairs $(\varepsilon,F)$,
the identity fails for every other choice of the five windows. In case (i) the factor $\Pw$ or
$\Nw$ that is exchanged produces, by (F1), a term in $J$, which (F2) absorbs when the two
neighbouring factors are $G$ and $\varphi(G)$ in this order or the reverse; this is the mechanism
behind the infinite family of Section~\ref{sec:family}.
\end{remark}

\section{Self-conjugate words}

If $\overline T=T$ (for instance $T=\0\1$), then all exponents $\varepsilon_i$ are irrelevant, and we
write $Z(T;a)$. Table~\ref{tab:self} lists $20$ pairs $(a,F)$ with $k=14$ windows, written with
$+$ for $\Pw$ and $-$ for $\Nw$.

\begin{theorem}\label{thm:self}
Let $T\in\{\0,\1\}^*$ with $\overline T=T$. For each of the $20$ pairs $(a,F)$ of Table~\ref{tab:self},
$M_q(Z(T;a))_{12}=M_q(Z(T;a^F))_{12}$.
\end{theorem}

\begin{proof}
By (F3), $G=M_q(T)$ satisfies $\varphi(G)=G$, that is, $G=\begin{pmatrix}\alpha&\beta\\q\beta&\delta\end{pmatrix}$. As in
the proof of Theorem~\ref{thm:k5}, it suffices to verify $20$ identities in $\Z[\alpha,\beta,\delta,q]$ for this
generic $G$, which is done by exact expansion.
\end{proof}

\begin{table}[ht]
\centering\small
\begin{tabular}{ll@{\qquad}ll}
\toprule
$a$ & $F$ & $a$ & $F$\\
\midrule
\texttt{+++-++---+-+++} & $\{2,3,7,8,12,13\}$ & \texttt{+++-+---+++++-} & $\{5,10\}$\\
\texttt{+++-+---++-+++} & $\{2,3,7,8,12,13\}$ & \texttt{++-++--+-+++-+} & $\{5,10\}$\\
\texttt{+++--+-----+++} & $\{2,3,7,8,12,13\}$ & \texttt{+-+++-+--++-++} & $\{5,10\}$\\
\texttt{+++-----+--+++} & $\{2,3,7,8,12,13\}$ & \texttt{-+++++---+-+++} & $\{5,10\}$\\
\texttt{+++-++-----+++} & $\{1,4,11,14\}$ & \texttt{++-++-+--++-++} & $\{2,3,5,10,12,13\}$\\
\texttt{+++-+---+--+++} & $\{1,4,11,14\}$ & \texttt{++-++--+-++-++} & $\{2,3,5,10,12,13\}$\\
\texttt{+++--+---+-+++} & $\{1,4,11,14\}$ & \texttt{++-+--+---+-++} & $\{2,3,5,10,12,13\}$\\
\texttt{+++-----++-+++} & $\{1,4,11,14\}$ & \texttt{++-+---+--+-++} & $\{2,3,5,10,12,13\}$\\
\texttt{++-++-+---+-++} & $\{2,3,12,13\}$ & \texttt{++-+--+--++-++} & $\{2,3,12,13\}$\\
\texttt{++-++--+--+-++} & $\{2,3,12,13\}$ & \texttt{++-+---+-++-++} & $\{2,3,12,13\}$\\
\bottomrule
\end{tabular}
\caption{The $20$ identities of Theorem~\ref{thm:self} ($k=14$; $+$ stands for $\Pw$, $-$ for $\Nw$).
Each set $F$ is invariant under $i\mapsto15-i$.}
\label{tab:self}
\end{table}

\section{An infinite family}\label{sec:family}

For $m\geq1$ put $k=4m+1$ and $\varepsilon=(+,-)^{2m}(-,+)\in\{\pm\}^{k+1}$. Call a sequence of windows
$a_1,\dots,a_k$ \emph{admissible} if its odd-indexed terms alternate, starting with $\Pw$:
$a_{2t+1}=\Pw$ for $t$ even and $a_{2t+1}=\Nw$ for $t$ odd; the even-indexed terms are arbitrary. Let
$F_{\mathrm{odd}}=\{1,3,\dots,k\}$.

\begin{theorem}\label{thm:family}
For every $m\geq1$, every word $T$ and every admissible $a$,
\[
M_q\bigl(Z_\varepsilon(T;a)\bigr)_{12}=M_q\bigl(Z_\varepsilon(T;a^{F_{\mathrm{odd}}})\bigr)_{12}.
\]
For $m=1$ this is Theorem~\ref{thm:k5}(i) with $a_1=\Pw$.
\end{theorem}

\begin{proof}
As before it suffices to treat a generic matrix $G=\begin{pmatrix}\alpha&\beta\\\gamma&\delta\end{pmatrix}$ over the field
$K=\mathbb{Q}(\alpha,\beta,\gamma,\delta,q)$, with $G'=\varphi(G)$, $P=M_q(\Pw)$, $N=M_q(\Nw)$; the identity to be proved
is a polynomial identity, so it holds after any specialisation of $G$. Put
\[
U=GNG',\quad V=G'NG,\quad K_0=(q^3+1)\det(G)\,J .
\]
By (F1) and (F2), $GPG'=U+K_0$ and $G'PG=V+K_0$ (for the second, apply (F2) to $X=G'$, noting
$\varphi(G')=G$ and $\det G'=\det G$). Writing $a_i$ also for $M_q(a_i)$, group the factors of
the product attached to $Z_\varepsilon(T;a)$ as
\[
(G\,a_1\,G')\,a_2\,(G\,a_3\,G')\,a_4\cdots(G\,a_{4m-1}\,G')\,a_{4m}\,(G'\,a_{4m+1}\,G).
\]
For admissible $a$
the bracketed factors are, in order, $U+K_0,U,U+K_0,\dots,U$ ($2m$ factors) followed by $V+K_0$;
after the exchange they are $U,U+K_0,\dots,U+K_0$ followed by $V$. With
\[
C_{XY}=(U+K_0)\,X\,U\,Y,\qquad C'_{XY}=U\,X\,(U+K_0)\,Y\qquad(X,Y\in\{P,N\}),
\]
four matrices each, the statement becomes: for all $X_1,\dots,X_{2m}\in\{P,N\}$,
\begin{equation}\label{eq:fam}
r\,C_{X_1X_2}\cdots C_{X_{2m-1}X_{2m}}\,(V+K_0)\,s=r\,C'_{X_1X_2}\cdots C'_{X_{2m-1}X_{2m}}\,V\,s .
\end{equation}
Write $C_1,\dots,C_4$ and $C'_1,\dots,C'_4$ for the two quadruples. The following are verified by
exact computation in $\Z[\alpha,\beta,\gamma,\delta,q]$ (Section~\ref{sec:data}):
(a) $\tr C_i=\tr C'_i$, $\tr(C_iC_j)=\tr(C'_iC'_j)$ for $i\leq j$, and
$\tr(C_iC_jC_l)=\tr(C'_iC'_jC'_l)$ for $i<j<l$;
(b) $C_1,\dots,C_4$ are linearly independent over $K$;
(c) $(V+K_0)s\neq0$;
(d) \eqref{eq:fam} for $m=1$, which is Theorem~\ref{thm:k5}(i).

\emph{Step 1: equal traces on all words.} For $2\times2$ matrices over a field of characteristic $0$,
the trace of any product of $X_1,\dots,X_r$ is a polynomial in $\tr X_i$, $\tr(X_iX_j)$ ($i\le j$)
and $\tr(X_iX_jX_l)$ ($i<j<l$): this follows from the Cayley--Hamilton identity
$XY+YX=\tr(X)Y+\tr(Y)X+(\tr(XY)-\tr X\tr Y)I$, see~\cite{P,P2,S}. Hence by (a),
$\tr w(C)=\tr w(C')$ for every word $w$ in four letters, and by linearity
$\tr f(C)=\tr f(C')$ for every $f$ in the free algebra $K\langle x_1,\dots,x_4\rangle$.

\emph{Step 2: conjugacy.} By (b) the $C_i$ span $M_2(K)$, and the trace form $(X,Y)\mapsto\tr(XY)$
is nondegenerate on $M_2(K)$, so the Gram matrix $(\tr C_iC_j)$ is invertible; it equals
$(\tr C'_iC'_j)$, hence the $C'_i$ span $M_2(K)$ too. If $f(C)=0$ then
$\tr(f(C')\,g(C'))=\tr(f(C)\,g(C))=0$ for all $g$, so $f(C')=0$. Thus $C_i\mapsto C'_i$ defines a
$K$-algebra endomorphism of $M_2(K)$, injective since $M_2(K)$ is simple, hence an automorphism, and
by the Skolem--Noether theorem there is $W\in\mathrm{GL}_2(K)$ with $C'_i=W^{-1}C_iW$ for all $i$.

\emph{Step 3: the two ends.} By (d), $r\,C_i\,(V+K_0)s=r\,C'_i\,Vs=(rW^{-1})\,C_i\,(WVs)$ for
$i=1,\dots,4$. The linear forms $X\mapsto r\,X\,(V+K_0)s$ and $X\mapsto (rW^{-1})\,X\,(WVs)$ on
$M_2(K)$ agree on the basis $C_1,\dots,C_4$, hence everywhere; that is, the rank-one matrices
$(V+K_0)s\,r$ and $WVs\,rW^{-1}$ are equal. By (c) and $r\neq0$ they are nonzero, so there is
$\lambda\in K^{\times}$ with $rW^{-1}=\lambda^{-1}r$ and $WVs=\lambda(V+K_0)s$.

\emph{Conclusion.} For any $i_1,\dots,i_m$,
$r\,C'_{i_1}\cdots C'_{i_m}Vs=rW^{-1}C_{i_1}\cdots C_{i_m}WVs=r\,C_{i_1}\cdots C_{i_m}(V+K_0)s$.
\end{proof}

\begin{remark}
The congruence $k\equiv1\pmod 4$ is where the last odd window falls on the factor $G'\,a_k\,G$, as
the grouping requires. The analogous statement with $k=3$, $\varepsilon=(+,-,-,+)$ and $a_1=\Pw$,
$a_3=\Nw$ is false (checked on several words $T$ with $\overline T\neq T$).
\end{remark}

\section{The census}\label{sec:census}

For each $n\leq32$, all $2^n$ words of length $n$ were sorted by the value of $\mu_q(\cdot)_{12}$, and
the classes of equal values were compared with the classes generated by (4) and (7) on words of
length $n$; both identities preserve length. Since a coincidence preserves the numbers of letters
$\0$ and $\1$ (\cite[Proposition~2.3]{LLS}, numbering of the arXiv version), the computation can be split by content. Every class found was
confirmed in exact polynomial arithmetic. Table~\ref{tab:census} gives the number of classes of
coincidence that are not reduced to a pair $\{\0w\1,\0\widetilde w\1\}$ given by (4), and among them
the number not explained by (4) and (7) together. No other length $n\leq32$ carries any class of
either kind. The computation was carried out with three independent implementations, and the counts
up to length $24$ were reproduced by a fourth, written separately.

\begin{table}[ht]
\centering\small
\begin{tabular}{l*{11}{r}}
\toprule
$n$ & 12 & 15 & 18 & 20 & 21 & 24 & 25 & 27 & 28 & 30 & total\\
\midrule
classes beyond (4) & 1 & 2 & 6 & 6 & 12 & 38 & 12 & 56 & 28 & 196 & 357\\
not explained by (4) and (7) & 0 & 0 & 0 & 0 & 0 & 10 & 0 & 0 & 0 & 40 & 50\\
\bottomrule
\end{tabular}
\caption{Classes of coincidences of $\mu_q(\cdot)_{12}$ among words of length $n\leq32$.}
\label{tab:census}
\end{table}

Each of the $50$ classes consists of exactly four words $\{A,A',B,B'\}$, where $A'$ and $B'$ are the
partners of $A$ and $B$ under~(4); identities (4) and (7) relate $A$ to $A'$ and $B$ to $B'$, and
nothing else. The census answers Question~3.5 only for $n\le32$; the pair of length $50$ of
Section~\ref{sec:data}, an instance of Theorem~\ref{thm:family}, shows that the phenomenon is not
confined to these lengths.

\section{Data and verification}\label{sec:data}

The $50$ pairs $(A,B)$ of Theorem~\ref{thm:main}(b), the pair of length $50$ of
Theorem~\ref{thm:main}(d), namely
\begin{align*}
&\0\0\0\1\0\1\1\0\0\0\1\0\0\1\1\0\1\0\0\0\1\0\0\1\0\1\1\0\0\0\1\0\0\1\1\0\1\0\0\0\1\1\0\0\0\1\0\0\1\1,\\
&\0\0\0\1\1\0\1\0\0\0\1\0\0\1\0\1\1\0\0\0\1\0\0\1\1\0\1\0\0\0\1\0\0\1\0\1\1\0\0\0\1\1\0\0\1\0\0\0\1\1,
\end{align*}
and all the identities of this note are checked by the script \texttt{anc/verify\_markoff\_q35.py} (Python~3,
no package, no argument, about one minute). It checks, in exact arithmetic over $\Z$ and
$\Z[\alpha,\beta,\gamma,\delta,q]$:
the values of~\cite{LLS} used for calibration;
$\mu_q(A)_{12}=\mu_q(B)_{12}$ for the $51$ pairs;
that for each of them $B$ is not reached from $A$ by any chain of instances of (4) and (7), the
set reached from $A$ being $\{A,A'\}$ and the set reached from $B$ being $\{B,B'\}$;
facts (F1) and (F2), and (F3)--(F4) on all words of length at most $8$;
the identities of Theorems~\ref{thm:k5} and~\ref{thm:self}, and the exactness of the conditions
in Theorem~\ref{thm:k5};
that each of the $50$ classes contains an instance of one of them;
the inputs (a)--(c) of the proof of Theorem~\ref{thm:family} (input (d) is Theorem~\ref{thm:k5}(i)), direct instances of it for
$m=1,2,3$, and the fact that the pair of length $50$ is an instance with $m=2$ and $T=\0\1\0\1\0\1\1\0\0\1$.

\section*{Acknowledgements}

The computations and the writing were carried out with the assistance of Claude (Anthropic).

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\end{document}