Aspherical Word Labeled Oriented Graphs and Cyclically Presented Groups

Abstract

A {\em word labeled oriented graph} (WLOG) is an oriented graph $\mathcal{G}$ on vertices $X=\{x_1,\dots ,x_k\}$, where each oriented edge is labeled by a word in $X^{\pm 1}$. WLOGs give rise to presentations which generalize Wirtinger presentations of knots. WLOG presentations, where the underlying graph is a tree are of central importance in view of Whitehead's Asphericity Conjecture. We present a class of aspherical world labeled oriented graphs. This class can be used to produce highly non-injective aspherical labeled oriented trees and also aspherical cyclically presented groups.

7 pages, 3 figures