Bulletin of the
Korean Mathematical Society BKMS

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 $H_1(L(p,q))$.

Keywords: knot complement, homology lens space

MSC numbers: 57N10; 57M25; 57M27