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Non-uniqueness of optimal $1$-embeddings on surfaces up to weak equivalence

  • Kengo Enami
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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.

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Cite

BibTeXRIS
BibTeX
@misc{enami2026nonuniqueness,
  title = {{Non-uniqueness of optimal $1$-embeddings on surfaces up to weak equivalence}},
  author = {Kengo Enami},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.09386},
  eprint = {2610.09386},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.09386},
}
RIS
TY  - UNPB
TI  - Non-uniqueness of optimal $1$-embeddings on surfaces up to weak equivalence
AU  - Kengo Enami
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.09386
UR  - https://arxiv.org/abs/2610.09386
AB  - 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.
ER  -