# Double branched covers of tunnel number one knots

## TL;DR

This paper investigates conditions under which a tunnel number one knot's double branched cover does not uniquely identify the knot, showing that different knots can share the same double branched cover.

## Contribution

It introduces criteria to determine when a tunnel number one knot is not uniquely identified by its double branched cover, highlighting non-uniqueness in knot invariants.

## Key findings

- Identifies conditions for non-uniqueness of double branched covers
- Provides examples of distinct knots with the same double branched cover
- Enhances understanding of knot invariants and their limitations

## Abstract

We provide criteria ensuring that a tunnel number one knot $K$ is not determined by its double branched cover, in the sense that the double branched cover is also the double branched cover of a knot $K'$ not equivalent to $K$.