We define a group-valued invariant of virtual knots and relate it to various other group-valued invariants of virtual knots, including the extended group of Silver-Williams and the quandle group of Manturov and Bardakov-Bellingeri. A virtual knot is called almost classical if it admits a diagram with an Alexander numbering, and in that case we show that the group factors as a free product of the usual knot group and Z. We establish a similar formula for mod p almost classical knots, and we use these results to derive obstructions to a virtual knot K being mod p almost classical. Viewed as knots in thickened surfaces, almost classical knots correspond to those that are homologically trivial. We show they admit Seifert surfaces and relate their Alexander invariants to the homology of the associated infinite cyclic cover. We prove the first Alexander ideal is principal, recovering a result first proved by Nakamura et al. using different methods. The resulting Alexander polynomial is shown to satisfy a skein relation, and its degree gives a lower bound for the Seifert genus. We tabulate almost classical knots up to 6 crossings and determine their Alexander polynomials and virtual genus.Comment: 44 page

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.