A geometric classification of rod complements in the 3-torus

Hui, Connie On Yu

doi:10.1090/proc/16949

A geometric classification of rod complements in the $3$-torus
HTML articles powered by AMS MathViewer

by Connie On Yu Hui;

Proc. Amer. Math. Soc. 153 (2025), 381-394

DOI: https://doi.org/10.1090/proc/16949

Published electronically: October 29, 2024

HTML | PDF | Request permission

Abstract:

Rod packings are used in crystallography to describe crystal structures with linear or zigzag chains of particles, and each rod packing can be topologically viewed as a collection of disjoint geodesics in the $3$-torus. Hui and Purcell developed a method to study the complements of rods in the $3$-torus with the use of $3$-dimensional geometry and tools from the $3$-sphere, and they partially classified the geometry of some families of rod complements in the $3$-torus. In this paper, we provide a complete classification of the geometry of all rod complements in the $3$-torus using topological arguments.

References

C. Adams, C. Albors-Riera, B. Haddock, Z. Li, D. Nishida, B. Reinoso, and L. Wang, Hyperbolicity of links in thickened surfaces, Topology Appl. 256 (2019), 262–278. MR 3916014, DOI 10.1016/j.topol.2019.01.022
Colin Adams, Or Eisenberg, Jonah Greenberg, Kabir Kapoor, Zhen Liang, Kate O’Connor, Natalia Pacheco-Tallaj, and Yi Wang, tg-hyperbolicity of virtual links, J. Knot Theory Ramifications 28 (2019), no. 12, 1950080, 26. MR 4059928, DOI 10.1142/s0218216519500809
Alena Andreevna Akimova and Sergei Vladimirovich Matveev, Classification of genus 1 virtual knots having at most five classical crossings, J. Knot Theory Ramifications 23 (2014), no. 6, 1450031, 19. MR 3253963, DOI 10.1142/S021821651450031X
Abhijit Champanerkar, Ilya Kofman, and Jessica S. Purcell, Geometry of biperiodic alternating links, J. Lond. Math. Soc. (2) 99 (2019), no. 3, 807–830. MR 3977891, DOI 10.1112/jlms.12195
Tommaso Cremaschi and José A. Rodríguez-Migueles, Hyperbolicity of link complements in Seifert-fibered spaces, Algebr. Geom. Topol. 20 (2020), no. 7, 3561–3588. MR 4194288, DOI 10.2140/agt.2020.20.3561
Tommaso Cremaschi, José Andrés Rodriguŕz-Migueles, and Andrew Yarmola, On volumes and filling collections of multicurves, J. Topol. 15 (2022), no. 3, 1107–1153. MR 4442684, DOI 10.1112/topo.12246
Marc Culler, Nathan M. Dunfield, Matthias Goerner, and Jeffrey R. Weeks, SnapPy, a computer program for studying the geometry and topology of $3$-manifolds, http://snappy.computop.org, 2022.
Myfanwy E. Evans and Stephen T. Hyde, From three-dimensional weavings to swollen corneocytes, Journal of The Royal Society Interface 8 (2011), no. 62, 1274–1280.
Myfanwy E. Evans, Vanessa Robins, and Stephen T. Hyde, Periodic entanglement II: weavings from hyperbolic line patterns, Acta Crystallogr. Sect. A 69 (2013), no. 3, 262–275. MR 3047743, DOI 10.1107/S0108767313001682
Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
Allen Hatcher, Notes on basic 3-manifold topology, https://pi.math.cornell.edu/~hatcher/3M/3M.pdf, 2007.
Joshua A. Howie and Jessica S. Purcell, Geometry of alternating links on surfaces, Trans. Amer. Math. Soc. 373 (2020), no. 4, 2349–2397. MR 4069222, DOI 10.1090/tran/7929
Connie On Yu Hui and Jessica S. Purcell, On the geometry of rod packings in the 3-torus, Bull. Lond. Math. Soc. 56 (2024), no. 4, 1291–1309. MR 4735618, DOI 10.1112/blms.12993
William Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, RI, 1980. MR 565450, DOI 10.1090/cbms/043
M. O’Keeffe and Sten Andersson, Rod packings and crystal chemistry, Acta Crystallogr. Sect. A 33 (1977), no. 6, 914–923.
M. O’Keeffe, J. Plévert, Y. Teshima, Y. Watanabe, and T. Ogama, The invariant cubic rod (cylinder) packings: symmetries and coordinates, Acta Cryst. Sect. A 57 (2001), no. 1, 110–111. MR 1805573, DOI 10.1107/S010876730001151X
Dale Rolfsen, Knots and links, Mathematics Lecture Series, No. 7, Publish or Perish, Inc., Berkeley, CA, 1976. MR 515288
Nathaniel L. Rosi, Jaheon Kim, Mohamed Eddaoudi, Banglin Chen, Michael O’Keeffe, and Omar M. Yaghi, Rod Packings and Metal-Organic Frameworks Constructed from Rod-Shaped Secondary Building Units, J. Amer. Chem. Soc. 127 (2005), no. 5, 1504–1518 (eng).
William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. MR 648524, DOI 10.1090/S0273-0979-1982-15003-0