{"paper":{"title":"Immersions of the circle into a surface","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Sergey A. Melikhov","submitted_at":"2016-12-01T20:48:10Z","abstract_excerpt":"We classify immersions $f$ of $S^1$ in a $2$-manifold $M$ in terms of elementary invariants: the parity $S(f)$ of the number of double points of a self-transverse $C^1$-approximation of $f$, and the turning number $T(e\\bar f)$ of the immersion $e\\bar f:S^1\\to M_f\\subset\\Bbb R^2$, where $\\bar f$ is a lift of $f$ to the cover $M_f$ of $M$ corresponding to the subgroup $\\left<[f]\\right>\\subset\\pi_1(M)$. Namely, immersions $f,g:S^1\\to M$ are regular homotopic if and only if they are homotopic, and if $M=S^2$ or $\\Bbb R P^2$ or the normal bundle $\\nu(f)$ is non-orientable, then $S(f)=S(g)$, whereas"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.00428","kind":"arxiv","version":4},"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"}