Ribbon \(2\)-knot groups of Coxeter type (Q6148485)

Wirtinger presentations of deficiency \(1\) appear in the context of knots, long virtual knots and ribbon \(2\)-knots (see [the authors, J. Knot Theory Ramifications 12, No. 7, 947--962 (2003; Zbl 1053.57005)]). They are encoded by labeled oriented trees and, for that reason, are also called \(\mathrm{LOT}\) presentations. Adding a generator to the set of relators in a Wirtinger presentation, \(P\) gives a balanced presentation of the trivial group. Thus, the associated \(2\)-complex \(K(P)\) is a subcomplex of an aspherical (contractible) \(2\)-complex. These presentations are a well known and important testing ground for the validity (or failure) of Whitehead's asphericity conjecture. In the paper under review, the authors define \(\mathrm{LOT}\)s of Coxeter type and show that for every given \(n\) there exists a prime \(\mathrm{LOT}\) of Coxeter type with group of rank \(n\). They also show that label separated Coxeter \(\mathrm{LOT}\)s are aspherical.