Partitions of unity in $\mathrm{SL}(2,\mathbb{Z})$, negative continued fractions, and dissections of polygons

Abstract

We characterize sequences of positive integers $(a_1,a_2,\dots ,a_n)$ for which the $2\times 2$ matrix $\left(\begin{array}{ccc}{a}_n& -11& 0\end{array}\right)\left(\begin{array}{ccc}{a}_{n-1}& -11& 0\end{array}\right)\cdots \left(\begin{array}{ccc}{a}_1& -11& 0\end{array}\right)$ is either the identity matrix $\mathrm{I}d$, its negative $-\mathrm{I}d$, or square root of $-\mathrm{I}d$. This extends a theorem of Conway and Coxeter that classifies such solutions subject to a total positivity restriction.

final version

References in corpus (1)

• Rotundus: triangulations, Chebyshev polynomials, and Pfaffians

Cited by in corpus (1)

• Symplectic Frieze Patterns