A ribbon knot group which has no free base

Yoshikawa, Katsuyuki

doi:10.1090/S0002-9939-1988-0934891-7

A ribbon knot group which has no free base
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by Katsuyuki Yoshikawa

Proc. Amer. Math. Soc. 102 (1988), 1065-1070

DOI: https://doi.org/10.1090/S0002-9939-1988-0934891-7

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Abstract:

We consider the following problem: If a group $G$ satisfies the conditions (1) $G$ has a finite presentation with $r + 1$ generators and $r$ relators, and (2) there exists an element $x$ of $G$ such that $G = {\left \langle {\left \langle x \right \rangle } \right \rangle ^G}$ where ${\left \langle {\left \langle x \right \rangle } \right \rangle ^G}$ is the normal closure of $x$ in $G$, then is $G$ an HNN (Higman-Neumann-Neumann) extension of a free group of finite rank? In this paper, we give a negative answer to the problem. Thus it follows that there exists a ribbon $n$-knot group $(n \geq 2)$ which has no free base.

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On

-knot groups which have abelian bases