Arithmetic infinite friezes from punctured discs
arXiv:1503.04352
Abstract
We define the notion of infinite friezes of positive integers as a variation of Conway-Coxeter frieze patterns and study their properties. We introduce useful gluing and cutting operations on infinite friezes. It turns out that triangulations of once-punctured discs give rise to periodic infinite friezes having special properties, a notable example being that each diagonal consists of a collection of arithmetic progressions. Furthermore, we work out a combinatorial interpretation of the entries of infinite friezes associated to triangulations of once-punctured discs via matching numbers for certain combinatorial objects, namely periodic triangulations of strips. Alternatively, we consider a known algorithm that as we show computes as well these entries.
26 pages, 22 figures. Changes in notation throughout the whole article. New Proposition 3.13 providing a link between the occurrence of the entry 1 and peripheral arcs in the triangulations. Additional Proposition 5.4 showing that the common differences can be calculated by using the labeling algorithm introduced in Section 5