  This paper introduces two virtual knot theory ``analogues'' of a well-known
family of invariants for knots in thickened surfaces: the Grishanov-Vassiliev
finite-type invariants of order two. The first, called the three loop isotopy
invariant, is an invariant of virtual knots while the second, called the three
loop framed isotopy invariant, is a regular isotopy invariant of framed virtual
knots. The properties of these invariants are investigated at length. In
addition, we make precise the informal notion of ``analogue''. Using this
formal definition, it is proved that a generalized three loop invariant is a
virtual knot theory analogue of a generalization of the Grishanov-Vassiliev
invariants of order two.
