The optimal twisted paper cylinder
-
Noah Montgomery
Abstract
An embedded twisted paper cylinder of aspect ratio λ is a smooth isometric embedding of a flat λ × 1 cylinder into ℝ3 such that the images of the boundary components are linked. We prove that for such an object to exist we must have λ > 2 and that this bound is sharp. We also show that any sequence of examples having aspect ratio converging to 2 must converge to a (non-smooth) 4-fold wrapping of a right-angled isosceles triangle.
Funding statement: R. E. S. is supported by N. S. F. Grant DMS-2102802 and a Simons Sabbatical Fellowship.
Acknowledgements
R. E. S. would like to thank Brienne Brown, Elizabeth Denne, Eliot Fried, Jeremy Kahn, Curtis McMullen and Sergei Tabachnikov for conversations related to this paper. R. E. S. also thanks the National Science Foundation and the Simons Foundation for their support.
Communicated by: R. Löwen
References
[1] B. E. Brown, R. E. Schwartz, The crisscross and the cup: Two short 3-twist paper Moebius bands. Preprint 2023, arXiv:2310.10000Suche in Google Scholar
[2] E. Denne, Folded ribbon knots in the plane. In: Encyclopedia of knot theory, ed. Colin Adams et al., Chapter 88, pp. 885–897. CRC Press, Boca Raton, FL 2021. MR4439733 Zbl 1468.57001Suche in Google Scholar
[3] E. Denne, T. Larsen, Linking number and folded ribbon unknots. J. Knot Theory Ramifications 32 (2023), Paper No. 2350003, 42 pages. MR4557484 Zbl 1516.5700510.1142/S0218216523500037Suche in Google Scholar
[4] B. Halpern, C. Weaver, Inverting a cylinder through isometric immersions and isometric embeddings. Trans. Amer. Math. Soc. 230 (1977), 41–70. MR474388 Zbl 0305.5300310.2307/1997711Suche in Google Scholar
[5] A. Hennessey, Constructing many-twist Möbius bands with small aspect ratios. C. R. Math. Acad. Sci. Paris 362 (2024), 1837–1845. MR4834586 Zbl 794999110.5802/crmath.690Suche in Google Scholar
[6] R. E. Schwartz, The optimal paper Moebius band. Ann. of Math. (2) 201 (2025), 291–305. MR4848672 Zbl 799569010.4007/annals.2025.201.1.5Suche in Google Scholar
[7] R. E. Schwartz, On nearly optimal paper Moebius bands. Adv. Geom. 25 (2025), 207–214. MR4908002 Zbl 805163110.1515/advgeom-2025-0006Suche in Google Scholar
[8] M. Spivak, A comprehensive introduction to differential geometry. Vol. III. Third Edition. Publish or Perish, Inc., Wilmington, DE 1979. MR532832 Zbl 3686285Suche in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Eigenforms of hyperelliptic curves with many automorphisms
- The optimal twisted paper cylinder
- Birational rigidity of quartic three-folds with a double point of rank 3
- Special divisors on real trigonal curves
- The Hoffman–Singleton manifold
- Classification of trees that quasi-inscribe rectangles in the hyperbolic plane
- Ramification points of homotopies: Enumeration and general theory
- On partially ample Ulrich bundles
- Rotational cmc surfaces in terms of Jacobi elliptic functions
- Locally classical stable planes
- Flocks in topological circle planes and their representation in generalised quadrangles
Artikel in diesem Heft
- Frontmatter
- Eigenforms of hyperelliptic curves with many automorphisms
- The optimal twisted paper cylinder
- Birational rigidity of quartic three-folds with a double point of rank 3
- Special divisors on real trigonal curves
- The Hoffman–Singleton manifold
- Classification of trees that quasi-inscribe rectangles in the hyperbolic plane
- Ramification points of homotopies: Enumeration and general theory
- On partially ample Ulrich bundles
- Rotational cmc surfaces in terms of Jacobi elliptic functions
- Locally classical stable planes
- Flocks in topological circle planes and their representation in generalised quadrangles
