{"paper":{"title":"Geometric complexity of embeddings in ${\\mathbb R}^d$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG","math.CO","math.GT"],"primary_cat":"math.MG","authors_text":"Michael Freedman, Vyacheslav Krushkal","submitted_at":"2013-11-12T03:50:41Z","abstract_excerpt":"Given a simplicial complex $K$, we consider several notions of geometric complexity of embeddings of $K$ in a Euclidean space ${\\mathbb R}^d$: thickness, distortion, and refinement complexity (the minimal number of simplices needed for a PL embedding). We show that any $n$-complex with $N$ simplices which topologically embeds in ${\\mathbb R}^{2n}$, $n>2$, can be PL embedded in ${\\mathbb R}^{2n}$ with refinement complexity $O(e^{N^{4+{\\epsilon}}})$. Families of simplicial $n$-complexes $K$ are constructed such that any embedding of $K$ into ${\\mathbb R}^{2n}$ has an exponential lower bound on t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.2667","kind":"arxiv","version":2},"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"}