Small knots of large Heegaard genus (Q6185121)

\textit{W. P. Thurston} [Bull. Am. Math. Soc., New Ser. 6, 357--379 (1982; Zbl 0496.57005)] proved his celebrated geometrization conjecture for Haken \(3\)-manifolds. The theorem was proved in full generality by \textit{G. Perelman} [``The entropy formula for the Ricci flow and its geometric applications'', Preprint, \url{arXiv:math.DG/0211159}; ``Finite extinction time for the solutions to the Ricci flow on certain three-manifolds'', Preprint, \url{arXiv:math.DG/0307245}; ``Ricci flow with surgery on three-manifolds'', Preprint, \url{arXiv:math.DG/0303109}]. So, small \(3\)-manifolds became an important topic in low dimensional topology and play an important role in the study of \(3\)-manifolds. For example, \textit{I. Agol} [Geom. Dedicata 102, 53--64 (2003; Zbl 1039.57008)] constructed the first examples of small link complements having arbitrarily many components for answering a question of Reid. By Dehn filling such a link, Agol was able to give the first examples of small closed manifolds having large Heegaard genus.In the paper under review, the author proves the existence of small knots with large Heegaard genus. These are the first known examples of such knots. Also, the author bounds the crossing number for such knots.