# Remarks on SU(2)-simple knots and SU(2)-cyclic 3-manifolds

## TL;DR

This paper explores properties of SU(2)-simple knots and SU(2)-cyclic 3-manifolds, establishing a characterization of SU(2)-simple Montesinos knots and proposing a conjecture relating SU(2)-cyclic rational homology 3-spheres to L-spaces.

## Contribution

It proves that a Montesinos knot is SU(2)-simple if and only if it is a 2-bridge knot, extending previous results, and conjectures a link between SU(2)-cyclic rational homology 3-spheres and L-spaces.

## Key findings

- Montesinos knots are SU(2)-simple iff they are 2-bridge knots.
- Conjecture: SU(2)-cyclic rational homology 3-spheres are L-spaces.
- Extension of Zentner's result on pretzel knots.

## Abstract

We give some remarks on two closely related issues as stated in the title. In particular we show that a Montesinos knot is SU(2)-simple if and only if it is a 2-bridge knot, extending a result of Zentner for 3-tangle summand pretzel knots. We conjecture with some evidence that an SU(2)-cyclic rational homology 3-sphere is an L-space.