Author ORCID Identifier
Date Available
5-15-2024
Year of Publication
2024
Degree Name
Doctor of Philosophy (PhD)
Document Type
Doctoral Dissertation
College
Arts and Sciences
Department/School/Program
Mathematics
First Advisor
Dr. Khrystyna Serhiyenko
Abstract
An SLₖ-frieze is a bi-infinite array of integers where adjacent entries satisfy a certain diamond rule. SL₂-friezes were introduced and studied by Conway and Coxeter. Later, these were generalized to infinite matrix-like structures called tilings as well as higher values of k. A recent paper by Short showed a bijection between bi-infinite paths of reduced rationals in the Farey graph and SL₂-tilings. We extend this result to higher k by constructing a bijection between SLₖ-tilings and certain pairs of bi-infinite strips of vectors in ℤᵏ called paths. The key ingredient in the proof is the relation to Plucker friezes and Grassmannian cluster algebras. As an application, we obtain results about periodicity, duality, and positivity for tilings.
Digital Object Identifier (DOI)
https://doi.org/10.13023/etd.2024.192
Recommended Citation
Peterson, Zachery T., "SLₖ-Tilings and Paths in ℤᵏ" (2024). Theses and Dissertations--Mathematics. 115.
https://uknowledge.uky.edu/math_etds/115
