# Simple 4-knots

## Author

J. A. Hillman and C. Kearton

## Status

Research Report 97-26
Date: 26 September 1997

## Abstract

We show that the isotopy type of a 1-simple $n$-knot $K$ is determined by the Postnikov $(n-1)$-stage of its exterior $X(K)$, together with the homotopy class of the longitude $\lambda_K$ in $\pi_n(X(K))$. Moreover any pair $(X,j)$ where $X$ is a 4-dimensional homology circle with $\pi_1(X)\cong Z$ and $j:S^4\times S^1\to X$ is a map such that $(X,j)=(MCyl(j),S^4\times S^1)$ is an orientable $PD_6$-pair is realizable by some simple 4-knot. As a consequence we are able to characterize completely the Artin spins of fibred simple 3-knots.

## Key phrases

fibred knot. 4-knot. Poincare duality. Postnikov. simple knot.

Primary: 57Q45
Secondary:

## Content

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Sydney Mathematics and Statistics