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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Knots whose branched cyclic coverings have periodic homology
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by C. McA. Gordon PDF
Trans. Amer. Math. Soc. 168 (1972), 357-370 Request permission

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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Additional Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 168 (1972), 357-370
  • MSC: Primary 55A25
  • DOI: https://doi.org/10.1090/S0002-9947-1972-0295327-8
  • MathSciNet review: 0295327