On pretzel knots and conjecture \(\mathbb{Z}\) (Q2796954)

In combinatorial group theory, the Kervaire conjecture claims that for any non-trivial group \(G\), \(\mathbb{Z}*G\) cannot be normally generated by a single element. This is still open, and there is also a knot theoretical equivalent form called Conjecture \(\mathbb{Z}\) introduced by \textit{F. González-Acuña} and \textit{A. Ramírez} [J. Knot Theory Ramifications 15, No. 4, 471--478 (2006; Zbl 1092.57006)]. It claims that if \(F\) is a compact orientable non-separating surface properly embedded in a knot exterior \(E\), then \(\pi_1(E/F)\cong \mathbb{Z}\). They verified this when the boundary of \(F\) is connected. Unfortunately, such a surface can have a disconnected boundary.NEWLINENEWLINEThe main result of the paper under review is to verify Conjecture \(\mathbb{Z}\) for all pretzel knots of the form \(P(p,q,-r)\) where \(p,q,r\) are odd positive integers.NEWLINENEWLINEAs the first reduction, the authors show that it is sufficient to examine incompressible surfaces. Then they use the classification of incompressible surfaces in the exterior of Montesinos knots by \textit{A. Hatcher} and \textit{U. Oertel} [Topology 28, No. 4, 453--480 (1989; Zbl 0686.57006)] to determine the zero boundary-slope surfaces which are connected with an odd number of boundary components. This part occupies most of the paper. The last part uses a technical argument which reduces the number of boundary components of the surface without changing the fundamental group of the complement.