{"paper":{"title":"Partitions of unity in $\\mathrm{SL}(2,\\mathbb Z)$, negative continued fractions, and dissections of polygons","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Valentin Ovsienko","submitted_at":"2017-10-09T08:59:11Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.02996","kind":"arxiv","version":5},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}