×

From dominoes to hexagons. (English) Zbl 1407.52021

Morrison, Scott (ed.) et al., Proceedings of the 2014 Maui and 2015 Qinhuangdao conferences in honour of Vaughan F. R. Jones’ 60th birthday. Canberra: Australian National University, Centre for Mathematics and its Applications. Proc. Cent. Math. Appl. Aust. Natl. Univ. 46, 399-414 (2017).
It is known that the space of domino tilings is connected under the domino flips. Instead of domino tilings, author considers its dual structures: triple diagrams. Each domino tiling produces a triple diagram, but there exist triple diagrams that can not be obtained from the domino tilings.
The author introduces a special type of triple diagrams: minimal triple diagrams and shows that two minimal triple point diagrams with the same matching on the endpoints can be related by a sequence of \(2\leftrightarrow 2\) moves (these moves are generalization of the domino flips). Also it is proved that any triple point diagram can be reduced to any minimal triple point diagram (with the same matching) by a sequence of three type of moves: \(2\leftrightarrow 2\) move, \(1\leftrightarrow 0\) move and dropping a simple loop with no crossings and an empty interior.
For the entire collection see [Zbl 1386.46002].

MSC:

52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
05B45 Combinatorial aspects of tessellation and tiling problems
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
PDFBibTeX XMLCite
Full Text: arXiv Euclid