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