Aspherical Word Labeled Oriented Graphs and Cyclically Presented Groups

TL;DR

This paper introduces a class of aspherical word labeled oriented graphs (WLOGs) that generalize knot presentations and can generate aspherical cyclically presented groups, advancing understanding of Whitehead's Asphericity Conjecture.

Contribution

The paper presents a new class of aspherical WLOGs, expanding the toolkit for constructing aspherical groups and exploring their properties.

Findings

Constructed a class of aspherical WLOGs.

Produced highly non-injective aspherical labeled oriented trees.

Generated aspherical cyclically presented groups.

Abstract

A {\em word labeled oriented graph} (WLOG) is an oriented graph $\cal G$ on vertices $X=\{ x_1,\ldots ,x_k\}$, where each oriented edge is labeled by a word in $X^{\pm1}$. 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.