The three loop isotopy and framed isotopy invariants of virtual knots

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.