# Bulletin of theKorean Mathematical SocietyBKMS

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

## Article

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Bull. Korean Math. Soc. 2022; 59(4): 869-877

Published online July 31, 2022 https://doi.org/10.4134/BKMS.b210503

## Knots in homology lens spaces determined by their complements

Kazuhiro Ichihara, Toshio Saito

3-25-40 Sakurajosui, Setagaya-ku; 1 Yamayashiki

### 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 greater 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: Primary 57K10; Secondary 57K31

Supported by: This work is partially supported by JSPS KAKENHI Grant Number 18K03287.