{"paper":{"title":"Linkless embeddings of graphs in $3$-space","license":"","headline":"","cross_cats":["math.GT"],"primary_cat":"math.CO","authors_text":"Neil Robertson, Paul Seymour, Robin Thomas","submitted_at":"1993-01-01T00:00:00Z","abstract_excerpt":"We announce results about flat (linkless) embeddings of graphs in 3-space. A piecewise-linear embedding of a graph in 3-space is called {\\it flat} if every circuit of the graph bounds a disk disjoint from the rest of the graph. We have shown:\n  (i) An embedding is flat if and only if the fundamental group of the complement in 3-space of the embedding of every subgraph is free.\n  (ii) If two flat embeddings of the same graph are not ambient isotopic, then they differ on a subdivision of $K_5$ or $K_{3,3}$.\n  (iii) Any flat embedding of a graph can be transformed to any other flat embedding of t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9301216","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"}