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

Abstract

We characterize sequences of positive integers $(a_1,a_2,\ldots,a_n)$ for which 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 the identity matrix $\mathrm Id$, its negative $-\mathrm Id$, or square root of $-\mathrm Id$. This extends a theorem of Conway and Coxeter that classifies such solutions subject to a total positivity restriction.