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
DOI: https://doi.org/10.1090/S0002-9947-1972-0295327-8
MathSciNet review: 0295327
Full-text PDF Free Access

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