Title: Alexander invariants of periodic virtual knots,
Abstract: We show that every periodic virtual knot can be realized as the closure of a
periodic virtual braid and use this to study the Alexander invariants of
periodic virtual knots. If $K$ is a $q$-periodic and almost classical knot, we
show that its quotient knot $K_*$ is also almost classical, and in the case
$q=p^r$ is a prime power, we establish an analogue of Murasugi's congruence
relating the Alexander polynomials of $K$ and $K_*$ over the integers modulo
$p$. This result is applied to the problem of determining the possible periods
of a virtual knot $K$. One consequence is that if $K$ is an almost classical
knot with a nontrivial Alexander polynomial, then it is $p$-periodic for only
finitely many primes $p$. Combined with parity and Manturov projection, our
methods provide conditions that a general virtual knot must satisfy in order to
be $q$-periodic.