Extensions of augmented racks and surface ribbon cocycle invariants

A rack is a set with binary operation that right-invertible and self-distributive, properties diagrammatically corresponding to Reidemeister moves II III, respectively. said be an augmented if the written by group action. Racks their cohomology theories have been extensively used for knot knotted surface invariants. Similarly cohomology, 2-cocycles relate extensions, natural question arises characterize extensions of racks are themselves racks. In this paper, we such in terms what call fibrant additive Simultaneous groups considered, where respective related through certain formula. Furthermore, construct coloring cocycle invariants compact orientable surfaces boundary ribbon forms embedded 3-space.