K-theory of the Chair Tiling via AF-algebras

TL;DR

This paper calculates the K-theory groups of the chair tiling's groupoid C*-algebra using a novel approach based on exact sequences and AF-algebras, advancing understanding of tiling C*-algebras.

Contribution

It introduces a new method employing Putnam's exact sequences to compute K-theory from AF-algebras associated with substitution tilings.

Findings

K-theory groups of the chair tiling computed

Method applies to substitution tilings with AF-algebras

Enhances computational techniques for tiling C*-algebras

Abstract

We compute the $K$-theory groups of the groupoid C$^\ast$-algebra of the chair tiling, using a new method. We use exact sequences of Putnam to compute these groups from the $K$-theory groups of the $AF$-algebras of the substitution and the induced lower dimensional substitutions on edges and vertices.