Knots whose branched cyclic coverings have periodic homology

Gordon, C. McA.

doi:10.1090/S0002-9947-1972-0295327-8

Knots whose branched cyclic coverings have periodic homology
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by C. McA. Gordon

Trans. Amer. Math. Soc. 168 (1972), 357-370

DOI: https://doi.org/10.1090/S0002-9947-1972-0295327-8

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Abstract:

Let ${M_k}$ be the $k$-fold branched cyclic covering of a (tame) knot of ${S^1}$ in ${S^3}$. Our main result is that the following statements are equivalent: (1) ${H_1}({M_k})$ is periodic with period $n$, i.e. ${H_1}({M_k}) \cong {H_1}({M_{k + n}})$ for all $k$, (2) ${H_1}({M_k}) \cong {H_1}({M_{(k,n)}})$ for all $k$, (3) the first Alexander invariant of the knot, ${\lambda _1}(t) = {\Delta _1}(t)/{\Delta _2}(t)$, divides ${t^n} - 1$.

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