The AMS website will be down for maintenance on May 23 between 6:00am - 8:00am EDT. For questions please contact AMS Customer Service at or (800) 321-4267 (U.S. & Canada), (401) 455-4000 (Worldwide).


Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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$.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 55A25

Retrieve articles in all journals with MSC: 55A25

Additional Information

Keywords: Classical knots, branched cyclic coverings, homology, periodicity
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society