{"paper":{"title":"Mappings from R^3 to R^3 and signs of swallowtails","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Justyna Bobowik, Zbigniew Szafraniec","submitted_at":"2014-03-21T07:06:53Z","abstract_excerpt":"Let M be an oriented 3-manifold. For a generic f \\in C^ \\infty(M,R^3), there is a discrete set of swallowtail critical points. In that case, at any swallowtail point p there exists a well-oriented coordinate system centered at p, and a coordinate system centered at f(p), such that locally f has the form f_\\pm(x,y,z)=(\\pm xy+x^2 z+x^4,y,z), so one may associate with p a sign I(f,p)\\in \\{\\pm 1\\}. A geometric definition of the sign associated with a swallowtail was recently introduced by Goryunov. We shall show how to compute the number of swallowtail points having the positive/negative sign, in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.5379","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"}