Twist-spun torus knots

Gordon, C. McA.

doi:10.1090/S0002-9939-1972-0288752-8

Twist-spun torus knots
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by C. McA. Gordon

Proc. Amer. Math. Soc. 32 (1972), 319-322

DOI: https://doi.org/10.1090/S0002-9939-1972-0288752-8

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Abstract:

Zeeman has shown that the complement of a twist-spun knot fibres over the circle. He also proves that the group of the 5-twist-spun trefoil is just the direct product of the fundamental group of the fibre with the integers. We generalise this by showing that, for torus knots, the group of the twist-spun knot is such a direct product whenever the fibre is a homology sphere. This then suggests the question (asked by Zeeman for the case of the 5-twist-spun trefoil) as to whether there is a corresponding product structure in the geometry. We answer in the negative.

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