Ribbon 2-knot groups of Coxeter type

TL;DR

This paper introduces Coxeter-type labeled oriented trees (LOTs) for ribbon 2-knot groups, demonstrating their existence for any rank and establishing asphericity for label separated Coxeter LOTs, thus contributing to knot group theory.

Contribution

It defines Coxeter-type LOTs, proves their existence for any rank, and shows that label separated Coxeter LOTs are aspherical, advancing understanding of knot group presentations.

Findings

Existence of Coxeter-type LOTs for all ranks

Label separated Coxeter LOTs are aspherical

Connections to Whitehead's asphericity conjecture

Abstract

Wirtinger presentations of deficiency 1 appear in the context of knots, long virtual knots, and ribbon 2-knots. They are encoded by (word) labeled oriented trees and, for that reason, are also called LOT presentations. These presentations are a well known and important testing ground for the validity (or failure) of Whitehead's asphericity conjecture. In this paper we define LOTs of Coxeter type and show that for every given $n$ there exists a (prime) LOT of Coxeter type with group of rank $n$. We also show that label separated Coxeter LOTs are aspherical.