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

TL;DR

This paper characterizes integer sequences related to specific matrix products in SL(2,Z), extending Conway and Coxeter's classification of solutions with total positivity, and explores their connections to negative continued fractions and polygon dissections.

Contribution

It extends the classification of sequences associated with SL(2,Z) matrices to include cases involving the identity, negative identity, and square root of negative identity, broadening Conway and Coxeter's theorem.

Findings

Characterization of sequences producing specific matrix products

Extension of Conway and Coxeter's classification

Connections to polygon dissections and negative continued fractions

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.