Construction of high-dimensional knot groups from classical knot groups

Lien, Magnhild

doi:10.1090/S0002-9947-1986-0860388-1

Construction of high-dimensional knot groups from classical knot groups
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by Magnhild Lien

Trans. Amer. Math. Soc. 298 (1986), 713-722

DOI: https://doi.org/10.1090/S0002-9947-1986-0860388-1

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Abstract:

In this paper we study constructions of high dimensional knot groups from classical knot groups. We study certain homomorphic images of classical knot groups. Specifically, let $K$ be a classical knot group and $w$ any element in $K$. We are interested in the quotient groups $G$ obtained by centralizing $w$, i.e. $G = K/[K,w]$, and ask whether $G$ is itself a knot group. For certain $K$ and $w$ we show that $G$ can be realized as the group of a knotted $3$-sphere in $5$-space, but $G$ is not realizable by a $2$-sphere in $4$-space. By varying $w$, we also obtain quotients that are groups of knotted $2$-spheres in $4$-space, but they cannot be realized as the groups of classical knots. We have examples of quotients $K/[K,w]$ that have nontrivial second homology. Hence these groups cannot be realized as knot groups of spheres in any dimension. However, we show that these groups are groups of knotted tori in ${S^4}$.

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