Bulletin of the
Korean Mathematical Society
BKMS

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

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Bull. Korean Math. Soc.

Published online March 16, 2022

Copyright © The Korean Mathematical Society.

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

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