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C*-algebras of Penrose's hyperbolic tilings
Penrose hyperbolic tilings are tilings of the hyperbolic plane which admit,
up to affine transformations a finite number of prototiles. In this paper, we
give a complete description of the C*-algebras and of the K-theory for such
tilings. Since the continuous hull of these tilings have no transversally
invariant measure, these C*-algebras are traceless. Nevertheless, harmonic
currents give rise to 3-cyclic cocycles and we discuss in this setting a
higher-order version of the gap-labelling.Comment: 36 pages. v2: some mistakes corrected, a section on topological
invariants of the continuous hull of the Penrose hyperbolic tilings adde
MLD Relations of Pisot Substitution Tilings
We consider 1-dimensional, unimodular Pisot substitution tilings with three
intervals, and discuss conditions under which pairs of such tilings are locally
isomorhphic (LI), or mutually locally derivable (MDL). For this purpose, we
regard the substitutions as homomorphisms of the underlying free group with
three generators. Then, if two substitutions are conjugated by an inner
automorphism of the free group, the two tilings are LI, and a conjugating outer
automorphism between two substitutions can often be used to prove that the two
tilings are MLD. We present several examples illustrating the different
phenomena that can occur in this context. In particular, we show how two
substitution tilings can be MLD even if their substitution matrices are not
equal, but only conjugate in . We also illustrate how the (in
our case fractal) windows of MLD tilings can be reconstructed from each other,
and discuss how the conjugating group automorphism affects the substitution
generating the window boundaries.Comment: Presented at Aperiodic'09 (Liverpool
Self-dual tilings with respect to star-duality
The concept of star-duality is described for self-similar cut-and-project
tilings in arbitrary dimensions. This generalises Thurston's concept of a
Galois-dual tiling. The dual tilings of the Penrose tilings as well as the
Ammann-Beenker tilings are calculated. Conditions for a tiling to be self-dual
are obtained.Comment: 15 pages, 6 figure
On the Classification of Brane Tilings
We present a computationally efficient algorithm that can be used to generate
all possible brane tilings. Brane tilings represent the largest class of
superconformal theories with known AdS duals in 3+1 and also 2+1 dimensions and
have proved useful for describing the physics of both D3 branes and also M2
branes probing Calabi-Yau singularities. This algorithm has been implemented
and is used to generate all possible brane tilings with at most 6
superpotential terms, including consistent and inconsistent brane tilings. The
collection of inconsistent tilings found in this work form the most
comprehensive study of such objects to date.Comment: 33 pages, 12 figures, 15 table
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