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We characterize sequences of positive integers $(a_1,a_2,\ldots,a_n)$ forwhich the $2\times2$ matrix $\left( \begin{array}{cc} a_n&-1 1&0 \end{array}\right) \left( \begin{array}{cc} a_{n-1}&-1 1&0 \end{array} \right) \cdots\left( \begin{array}{cc} a_1&-1 1&0 \end{array} \right) $ is either theidentity matrix $\mathrm Id$, its negative $-\mathrm Id$, or square root of$-\mathrm Id$. This extends a theorem of Conway and Coxeter that classifiessuch solutions subject to a total positivity restriction.

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