Small knots of large Heegaard genus

Abstract

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 small knots having large Heegaard genus. Using work of Futer and Purcell, we are able to bound the crossing number for each $K_n$ in terms of $n$.

Final version to appear in Communications in Analysis and Geometry. Mostly minor changes/corrections, proof of Lemma 5.3 expanded for clarity and some errors corrected. 18 pages, 10 figures