Some Problems and Pictures Associated with the Kauffman Bracket

This talk is a summary and continuation of conversations with Vassily Manturov and Alyona Akimova. Akimova and Manturov have recently formulated a generalized diagrammatic or picture-based bracket whose formalism encompasses both the usual Kauffman bracket and also the Kuperberg brackets and their generalizations for virtual knots. We are discussing structure and further generalizations. This talk will discuss relationships with the Arrow polynomial for virtuals and its possible generalizations and also possible generalizations related to the Homflypt polynomial. More ideas may occur as we continue this exploration.