# Bulletin of theKorean Mathematical SocietyBKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc.

Published online March 16, 2022

## Knots in homology lens spaces determined by their complements

Kazuhiro Ichihara and Toshio Saito

Nihon University, College of Humanities and Sciences, Joetsu University of Education

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

Keywords: knot complement, homology lens space

MSC numbers: 57N10; 57M25; 57M27