# Every lens space contains a genus one homologically fibered knot

## TL;DR

This paper proves that all lens spaces contain a genus one homologically fibered knot, using number theory tools like the Chebotarev density theorem and quadratic forms, and discusses their Alexander polynomials.

## Contribution

It establishes the existence of genus one homologically fibered knots in every lens space, contrasting with the absence of such knots in some lens spaces.

## Key findings

- Every lens space contains a genus one homologically fibered knot.
- Number theory methods are crucial in the proof.
- Discussion of Alexander polynomials of these knots.

## Abstract

We prove that every lens space contains a genus one homologically fibered knot, which is contrast to the fact that some lens spaces contain no genus one fibered knot. In the proof, the Chebotarev density theorem and binary quadratic forms in number theory play a key role. We also discuss the Alexander polynomial of homologically fibered knots.