# Small knots of large Heegaard genus

## TL;DR

This paper constructs a new family of hyperbolic knots with small complements that have arbitrarily large Heegaard genus, providing the first known examples of such knots and bounding their crossing numbers.

## Contribution

It introduces the first known small knots with large Heegaard genus and establishes bounds on their crossing numbers based on genus.

## Key findings

- Constructed hyperbolic knots with no closed incompressible surfaces.
- Proved these knots have Heegaard genus exactly n.
- Bounded crossing number in terms of 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 $n$.