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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Knots whose branched cyclic coverings have periodic homology

Author: C. McA. Gordon
Journal: Trans. Amer. Math. Soc. 168 (1972), 357-370
MSC: Primary 55A25
MathSciNet review: 0295327
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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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PII: S 0002-9947(1972)0295327-8
Keywords: Classical knots, branched cyclic coverings, homology, periodicity
Article copyright: © Copyright 1972 American Mathematical Society