Communications in Analysis and Geometry

Volume 31 (2023)

Number 2

Small knots of large Heegaard genus

Pages: 381 – 406



William Worden (Department of Mathematics, Rice University, Houston, Texas, U.S.A.)


Building off ideas developed by Agol, we construct a family of hyperbolic knots $K_n$ whose complements contain no closed incompressible surfaces and have Heegaard genus exactly $n$. These are the first known examples of such knots. Using work of Futer and Purcell, we are able to bound the crossing number for each $K_n$ in terms of $n$.

Received 7 January 2020

Accepted 6 October 2020

Published 6 December 2023