Baker matrices for general parameters

Establish whether the matrices $A_k=\begin{pmatrix}1&k+1\\ k&1\end{pmatrix}$ and $B_k=\begin{pmatrix}1&k(k+1)\\ 1&1\end{pmatrix}$ are strong shift equivalent for every integer $k\geq 3$.

Background

Baker’s matrices form a concrete family of similar two-by-two nonnegative integer matrices whose strong shift equivalence was not previously known in general. The paper proves strong shift equivalence for several additional values, namely k≤10k\leq 10 and k=12k=12, but leaves the general family unresolved.

The unresolved cases include all sufficiently large parameters and, according to the paper’s earlier formulation, the result remains open for k≥3k\geq 3 before the new computational results are taken into account.

References

The first open question is from Baker: Are the matrices $\begin{pmatrix} 1 & k+1 \ k & 1 \end{pmatrix}$ and $\begin{pmatrix} 1 & k(k+1) \ 1 & 1 \end{pmatrix}$ strong shift equivalent? The result is known for $k = 2$ but remains open for $k \geq 3$.

— Combinatorial Search for Strong Shift Equivalence  (2609.03567 - Jeandel, 3 Sep 2026) in Introduction; Problem environment “Baker’s matrices,” Section 2, “Open problems”

Is it true that $A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$ and $B = \begin{pmatrix} a & bc \ 1 & d \end{pmatrix}$ are SSE ?

— Combinatorial Search for Strong Shift Equivalence  (2609.03567 - Jeandel, 3 Sep 2026) in Section “A generalization of Baker,” Problem \ref{pb:gen}

Are $A = \begin{pmatrix} 1 & t+1 \ t & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & t2 + t \ 1 & 1 \end{pmatrix}$ SSE in $\mathbb{Z}_+[t]$ ?

— Combinatorial Search for Strong Shift Equivalence  (2609.03567 - Jeandel, 3 Sep 2026) in Section “A generalization of Baker,” final problem environment