Relative helicity and tiling twist

Khesin, Boris; Saldanha, Nicolau

doi:10.1090/tran/9456

Relative helicity and tiling twist
HTML articles powered by AMS MathViewer

by Boris Khesin and Nicolau C. Saldanha;

Trans. Amer. Math. Soc.

DOI: https://doi.org/10.1090/tran/9456

Published electronically: May 8, 2025

HTML | PDF | Request permission

Abstract:

We consider domino tilings of 3D cubiculated regions. The tilings have two invariants, flux and twist, often integer-valued, which are given in purely combinatorial terms. These invariants allow one to classify the tilings with respect to certain elementary moves, flips and trits. In this paper we present a construction associating a divergence-free vector field $\xi _{\mathbf t}$ to any domino tiling ${\mathbf t}$, such that the flux of the tiling ${\mathbf t}$ can be interpreted as the (relative) rotation class of the field $\xi _{\mathbf t}$, while the twist of ${\mathbf t}$ is proved to be the relative helicity of the field $\xi _{\mathbf t}$.

References

V. I. Arnol′d, The one-dimensional cohomologies of the Lie algebra of divergence-free vector fields, and the winding numbers of dynamical systems, Funkcional. Anal. i Priložen. 3 (1969), no. 4, 77–78 (Russian). MR 263101
V. I. Arnol′d, The asymptotic Hopf invariant and its applications, Selecta Math. Soviet. 5 (1986), no. 4, 327–345. Selected translations. MR 891881
Vladimir I. Arnold and Boris A. Khesin, Topological methods in hydrodynamics, Applied Mathematical Sciences, vol. 125, Springer-Verlag, New York, 1998. MR 1612569
G. Bednik, Hopfions in lattice dimer model, arXiv:1901.04527, 2019.
Mitchell A. Berger and George B. Field, The topological properties of magnetic helicity, J. Fluid Mech. 147 (1984), 133–148. MR 770136, DOI 10.1017/S0022112084002019
A. Fathi, Transformations et homéomorphismes préservant la mesure, Systèmes dynamiques minimaux, Thèse Orsay, 1980.
M. Freedman, M. B. Hastings, C. Nayak, and X. L. Qi, Weakly coupled non-Abelian anyons in three dimensions, arXiv:1107.2731, 2011.
Juliana Freire, Caroline J. Klivans, Pedro H. Milet, and Nicolau C. Saldanha, On the connectivity of spaces of three-dimensional domino tilings, Trans. Amer. Math. Soc. 375 (2022), no. 3, 1579–1605. MR 4378071, DOI 10.1090/tran/8532
Jean-Marc Gambaudo and Étienne Ghys, Enlacements asymptotiques, Topology 36 (1997), no. 6, 1355–1379 (French). MR 1452855, DOI 10.1016/S0040-9383(97)00001-3
Pedro H. Milet and Nicolau C. Saldanha, Flip invariance for domino tilings of three-dimensional regions with two floors, Discrete Comput. Geom. 53 (2015), no. 4, 914–940. MR 3341585, DOI 10.1007/s00454-015-9685-y
Pedro H Milet and Nicolau C Saldanha. Domino tilings of three-dimensional regions: flips and twists. arXiv:1410.7693.
H. K. Moffatt, The degree of knottedness of tangled vortex lines, J. Fluid Mech. 35 (1969), no.1, 117–129.
H. K. Moffatt and Renzo L. Ricca, Helicity and the Călugăreanu invariant, Proc. Roy. Soc. London Ser. A 439 (1992), no. 1906, 411–429. MR 1193010, DOI 10.1098/rspa.1992.0159
Nicolau C. Saldanha, Domino tilings of cylinders: the domino group and connected components under flips, Indiana Univ. Math. J. 71 (2022), no. 3, 965–1002. MR 4448575
N. C. Saldanha, C. Tomei, M. A. Casarin Jr., and D. Romualdo, Spaces of domino tilings, Discrete Comput. Geom. 14 (1995), no. 2, 207–233. MR 1331927, DOI 10.1007/BF02570703
Martin W. Scheeler, Wim M. van Rees, Hridesh Kedia, Dustin Kleckner, and William T. M. Irvine, Complete measurement of helicity and its dynamics in vortex tubes, Science 357 (2017), no. 6350, 487–491. MR 3700475, DOI 10.1126/science.aam6897
Egor Shelukhin, “Enlacements asymptotiques” revisited, Ann. Math. Qué. 39 (2015), no. 2, 205–208 (English, with English and French summaries). MR 3418805, DOI 10.1007/s40316-015-0035-5
William P. Thurston, Conway’s tiling groups, Amer. Math. Monthly 97 (1990), no. 8, 757–773. MR 1072815, DOI 10.2307/2324578

AMS Website Down

Transactions of the American Mathematical Society

Relative helicity and tiling twist
HTML articles powered by AMS MathViewer

Abstract: