Knots in homology lens spaces determined by their complements

TL;DR

This paper investigates when knots in homology lens spaces are uniquely determined by their complements, focusing on non-hyperbolic and hyperbolic cases with specific homological conditions.

Contribution

It establishes conditions under which knots in homology lens spaces are uniquely identified by their complements, extending previous results to new classes of knots and spaces.

Findings

Knots in non-hyperbolic homology lens spaces are determined by their complements.

Hyperbolic knots in these spaces are also uniquely determined if the space's first homology is a prime greater than 7.

In lens spaces, knots representing generators of the first homology are uniquely determined by their complements.

Abstract

In this paper, we consider the knot complement problem for not null-homologous knots in homology lens spaces. Let $M$ be a homology lens space with $H_1(M; \mathbb{Z}) \cong \mathbb{Z}_p$ and $K$ a not null-homologous knot in $M$. We show that $K$ is determined by its complement if $M$ is non-hyperbolic, $K$ is hyperbolic, and $p$ is a prime more than 7, or, if $M$ is actually a lens space $L(p,q)$ and $K$ represents a generator of $H_1(L(p,q))$.