{"paper":{"title":"Topology of a class of $p2$-crystallographic replication tiles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Beno\\^it Loridant, Shu-Qin Zhang","submitted_at":"2016-11-15T15:58:24Z","abstract_excerpt":"We study the topological properties of a class of planar crystallographic replication tiles. Let $M\\in\\mathbb{Z}^{2\\times2}$ be an expanding matrix with characteristic polynomial $x^2+Ax+B$ ($A,B\\in\\mathbb{Z}$, $B\\geq 2$) and ${\\bf v}\\in\\mathbb{Z}^2$ such that $({\\bf v},M{\\bf v})$ are linearly independent. Then the equation $$MT+\\frac{B-1}{2}{\\bf v} =T\\cup(T+{\\bf v})\\cup (T+2{\\bf v})\\cup \\cdots\\cup(T+(B-2){\\bf v})\\cup(-T) $$ defines a unique nonempty compact set $T$ satisfying $\\overline{T^o}=T$. Moreover, $T$ tiles the plane by the crystallographic group $p2$ generated by the $\\pi$-rotation a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.04903","kind":"arxiv","version":1},"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"}