{
  "abstract": "We study the uniqueness of optimal $1$-embeddings on closed surfaces up to weak equivalence. Answering a question of Suzuki in the negative, we show that for every closed surface other than the sphere, the projective plane and the Klein bottle, there exists an optimal $1$-embedded graph admitting two weakly inequivalent optimal $1$-embeddings. Our construction is given explicitly on the torus and is then extended to other surfaces through a suitable connected-sum construction that preserves weak inequivalence. As a consequence, we also obtain optimal $1$-embedded graphs with exponentially many pairwise weakly inequivalent optimal $1$-embeddings as the genus increases.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09386",
  "author_total": 1,
  "authors": [
    {
      "name": "Kengo Enami"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5dx43y54w9120.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09386",
  "fulltext_url": null,
  "id": "5dx43y54w9120",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09386",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7220894359",
  "pdf_url": "https://arxiv.org/pdf/2610.09386",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5dx43y54w9120.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.09386",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09386"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Non-uniqueness of optimal $1$-embeddings on surfaces up to weak equivalence",
  "title_text": "Non-uniqueness of optimal $1$-embeddings on surfaces up to weak equivalence",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/5dx43y54w9120",
  "venue": "arXiv",
  "versions": []
}