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Abstract. We prove that the near Frattini subgroup of every knot group is trivial. The argument first identifies the lower and upper near Frattini subgroups for finitely generated groups, and then shows that every nontrivial element of a finitely generated torsion-free virtually RFRS group is a near generator. Virtual specialness of knot groups then gives the result. This answers both parts of Problem 19.1 of the Kourovka Notebook.
Abstract. For every prime $p>3$ we construct an explicit family of finite two-generator $p$-groups of deficiency zero whose nilpotency classes tend to infinity. The construction gives, for every $n\geq1$, a group of order $p^{2n+1}$ and class $n+1$. This gives an affirmative answer to Problem 17.113 of the Kourovka Notebook.