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.