Cohomology Groups for Spaces of Twelve-Fold Tilings

Bédaride, N., Gähler, F. and Lecuona, A. G. (2022) Cohomology Groups for Spaces of Twelve-Fold Tilings. International Mathematics Research Notices, 2022(18), pp. 14181-14254. (doi: 10.1093/imrn/rnab117)

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Abstract

We consider tilings of the plane with twelve-fold symmetry obtained by the cutand-projection method. We compute their cohomology groups using the techniques introduced in [9]. To do this, we completely describe the window, the orbits of lines under the group action, and the orbits of 0-singularities. The complete family of generalized twelve-fold tilings can be described using two-parameters and it presents a surprisingly rich cohomological structure. To put this finding into perspective, one should compare our results with the cohomology of the generalized five-fold tilings (more commonly known as generalized Penrose tilings). In this case, the tilings form a one-parameter family, which fits in simply one of the two types of cohomology.

Item Type:Articles
Additional Information:Funding: This work was supported by the German Research Foundation [CRC 1283 to F.G.]; and the Engineering and Physical Sciences Research Council [EP/T028408/1 to A.G.L.].
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Garcia Lecuona, Professor Ana
Authors: Bédaride, N., Gähler, F., and Lecuona, A. G.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:International Mathematics Research Notices
Publisher:Oxford University Press
ISSN:1073-7928
ISSN (Online):1687-0247
Published Online:02 June 2021
Copyright Holders:Copyright © 2021 The Authors
First Published:First published in International Mathematics Research Notices 2022(18): 14181-14254
Publisher Policy:Reproduced under a Creative Commons licence

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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
309070Development and categorification of a new link invariantAna LecuonaEngineering and Physical Sciences Research Council (EPSRC)EP/T028408/1M&S - Mathematics