Small knots of large Heegaard genus

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 $n$.

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Received 7 January 2020

Accepted 6 October 2020

Published 6 December 2023