2019/12/27 by Nicolau C. Saldanha, Saldanha, Nicolau C. · 1 citation
Materials Science · Mathematics · #05B45 #Combinatorics (math.CO) #FOS: Mathematics #Liquid Crystal Research Advancements #Mathematical Dynamics and Fractals #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.1912.12102
openalex publication_date 2019/12/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider domino tilings of three-dimensional cubiculated regions. A flip is a local move: two neighboring parallel dominoes are removed and placed back in a different position. The twist is an integer associated to each tiling, which is invariant under flips. A balanced quadriculated disk D is regular if whenever two tilings t0 and t1 of D × [0,N] have the same twist then t0 and t1 can be joined by a sequence of flips provided some extra vertical space is allowed. We define the domino group of a quadriculated disk and prove that D is regular if and only if its domino group is isomorphic to Z ⊕ Z/(2). We prove that a rectangle D = [0,L] × [0,M] with LM even is regular if and only if min\L,M\ ≥ 3 and conjecture that in general "large" disks are regular. In the cases where D is not regular we prove partial results concerning the structure of the domino group: the group is not abelian and has exponential growth. We also prove that if D is regular then the extra vertical space necessary to join by flips two tilings of D × [0,N] with the same twist depends only on D, not on the height N.