Geometric indices and the Alexander polynomial of a knot

Fujii, Hirozumi

doi:10.1090/S0002-9939-96-03489-2

Geometric indices and the Alexander polynomial of a knot
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by Hirozumi Fujii

Proc. Amer. Math. Soc. 124 (1996), 2923-2933

DOI: https://doi.org/10.1090/S0002-9939-96-03489-2

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Abstract:

It is well-known that any Laurent polynomial $\Delta (t)$ satisfying $\Delta (t) \doteq \Delta (t^{-1})$ and $\Delta (1) = \pm 1$ is the Alexander polynomial of a knot in $S^3$. We show that $\Delta (t)$ can be realized by a knot which has the following properties simultaneously: (i) tunnel number 1; (ii) bridge index 3; and (iii) unknotting number 1.

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