{"paper":{"title":"Types of perfect matchings in toroidal square grids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MP"],"primary_cat":"math-ph","authors_text":"Marcos Kiwi, Martin Loebl","submitted_at":"2014-06-12T18:53:15Z","abstract_excerpt":"Let $T_{m,n}$ be toroidal square grid of size $m\\times n$ and let both $m$ and $n$ be even. Let $P$ be a perfect matching of $T_{m,n}$ and let $D(P)$ be the cycle-rooted spanning forest of $P$ obtained by the generalized Temperley's construction. The types of $P$ and $D(P)$ in the first homology group $H_1(\\mathbb{T},\\mathbb{Z})$ of torus $\\mathbb{T}$ with coefficients in $\\mathbb{Z}$ has been extensively studied. In this paper we study the types of $P$ and $D(P)$ in the first homology group $H_1(\\mathbb{T},\\mathbb{F}_2)$ with the coefficients in $\\mathbb{F}_2$. Our considerations connect two "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.3317","kind":"arxiv","version":3},"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"}