Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 36 min ago
PreprintarXiv licencearXiv

ℓ1-Embeddability of Regular Quadrilateral Maps on the Torus and the Klein Bottle

  • Guangfu Wang
  • Wenchao Cong
  • Chunxiao Xu
Open at sourcePDFDOICite

Abstract

A connected graph is called ℓ1-embeddable if its shortest-path metric admits an isometric embedding into the ℓ1-space; equivalently, in the finite case, it admits a scale embedding into a hypercube. This paper gives a complete parameter-level classification of ℓ1-embeddability for regular quadrilateral maps on the torus and the Klein bottle. We treat the two toroidal families Qk,m,r and Qk,m,e, obtained from suitable identifications of finite square grids, and, using the known isomorphisms among Klein-bottle quadrilateral maps, the three essential Klein-bottle families Qk,m,a, Qk,m,b, and Qk,m,h. The ℓ1-embeddable toroidal maps are exactly Qk,m,0, Qk,2,1, Qk,1,e, Q5,2,e, Q2i,2,e (i≥ 2), and the ℓ1-embeddable Klein-bottle maps are exactly $ Q_{2,2,a}, Q_{4,2,a}, Q_{2,m,b}, Q_{1,2,h}.$ All other members of these five families are not ℓ1-embeddable.

Related

  1. Klein bottle and optimal systolic inequality for nonpositively curved surfaces
    Mikhail G. Katz, Stephane Sabourau
    arXivPreprintCC BY 4.0Full textGeometry & topologyarXiv
  2. Nearby Lagrangians in the cotangent bundle of the Klein bottle
    Yuan Gao, Weiwei Wu, Tianyu Yuan
    arXivPreprintCC BY 4.0Full textGeometry & topologyarXiv
  3. Sector-Switching Curl on the Klein Bottle — numerical verification report
    László Márk
    Zenodo (CERN European Organization for Nuclear Research)PreprintCC BY12 citedGeneral astronomyGeometry & topologyOpenAlex
  4. Symmetry breaking in the polygonal Szegö-Weinberger inequality as p→1+: the longest shortest-fence quadrilateral
    Beniamin Bogosel, Dorin Bucur, Ilaria Fragalà
    arXivPreprintCC BY-NC-ND 4.0Full textOptimisation & controlarXiv
  5. Connes-embeddability and twisted group von Neumann algebras
    Soham Chakraborty, Felipe Flores
    arXivPreprintarXiv licenceFull textAlgebra & number theoryAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@misc{wang2026ℓ1embeddability,
  title = {{$\ell_1$-Embeddability of Regular Quadrilateral Maps on the Torus and the Klein Bottle}},
  author = {Guangfu Wang and Wenchao Cong and Chunxiao Xu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.03770},
  eprint = {2610.03770},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.03770},
}
RIS
TY  - UNPB
TI  - ℓ_1-Embeddability of Regular Quadrilateral Maps on the Torus and the Klein Bottle
AU  - Guangfu Wang
AU  - Wenchao Cong
AU  - Chunxiao Xu
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.03770
UR  - https://arxiv.org/abs/2610.03770
AB  - A connected graph is called ℓ_1-embeddable if its shortest-path metric admits an isometric embedding into the ℓ_1-space; equivalently, in the finite case, it admits a scale embedding into a hypercube. This paper gives a complete parameter-level classification of ℓ_1-embeddability for regular quadrilateral maps on the torus and the Klein bottle. We treat the two toroidal families Q_k,m,r and Q_k,m,e, obtained from suitable identifications of finite square grids, and, using the known isomorphisms among Klein-bottle quadrilateral maps, the three essential Klein-bottle families Q_k,m,a, Q_k,m,b, and Q_k,m,h. The ℓ_1-embeddable toroidal maps are exactly Q_k,m,0, Q_k,2,1, Q_k,1,e, Q_5,2,e, Q_2i,2,e (i≥ 2), and the ℓ_1-embeddable Klein-bottle maps are exactly $ Q_{2,2,a}, Q_{4,2,a}, Q_{2,m,b}, Q_{1,2,h}.$ All other members of these five families are not ℓ_1-embeddable.
ER  -