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

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## Abstract

We characterize sequences of positive integers \((a_1,a_2,\ldots ,a_n)\) for which the \(2\times 2\) matrix \(\left( \begin{array}{ll} a_n&{} \quad -\,1\\ 1&{}\quad 0 \end{array} \right) \left( \begin{array}{ll} a_{n-1}&{}\quad -\,1\\ 1&{}\quad 0 \end{array} \right) \cdots \left( \begin{array}{ll} a_1&{}\quad -\,1\\ 1&{}\quad 0 \end{array} \right) \) 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.

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