A knot group has weight one, so is normally generated by a single element called a weight element of the knot group. A meridian is a typical weight element, but some knot groups admit other weight elements. We show that for some infinite classes of three-strand pretzel knots and all prime knots with up to eight crossings, the knot groups admit weight elements that are not automorphic images of meridians.

In this paper we use the Alexander ideals of groups to solve the isomorphism problem for the Baumslag–Solitar groups and a family of parafree groups introduced by Baumslag and Cleary.

The geography and botany problems of irreducible non-spin symplectic 4-manifolds with a choice of fundamental group from $\{{\mathbb{Z}}_p, {\mathbb{Z}}_p\oplus {\mathbb{Z}}_q, {\mathbb{Z}}, {\mathbb{Z}}\oplus {\mathbb{Z}}_p, {\mathbb{Z}}\oplus {\mathbb{Z}}\}$ are studied by building upon the recent progress obtained on the simply connected realm. Results on the botany of simply connected 4-manifolds not available in the literature are extended.

An interesting result of Ivanov implies that a non-aspherical relative presentation that defines a torsion-free group would provide a potential counterexample to the Kaplansky zero-divisor conjecture. In this point of view, we prove the asphericity of the length-6 relative presentation $\langle H,x: xh_1xh_2xh_3xh_4xh_5xh_6\rangle$, provided that each coefficient is torsion free.

In previous work [2] calculations of subquadratic second order Dehn functions for various groups were carried out. In this paper we obtain estimates for upper and lower bounds of second order Dehn functions of HNN-extensions, and use these to exhibit an infinite number of different superquadratic second order Dehn functions. At the end of the paper several open questions concerning second order Dehn functions of groups are discussed.