Alexander invariants of periodic virtual knots

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 $K_{\ast }$ 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 $K_{\ast }$ 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.