We consider codimension-2 ribbon knottings of circles and 2-spheres. We find that if a given ribbon knot has two ribbon disks, those disks are related by ambient isotopy together with a finite number of local modifications to be described. This allows a complete set of moves to be developed for the representation of ribbon 2-knots by abstract or planar graphs. Similar results hold for classical...

An open ribbon is a square with one side called the seam. A closed ribbon is a cylinder with one boundary component called the seam. We sew an open (resp. closed) ribbon onto a graph by identifying the seam with an open (resp. closed) walk in the graph. A ribbon complex is a graph with a nite number of ribbons sewn on. We investigate when a ribbon complex embeds in 3-dimensional Euclidean space...

the miocene pyroxene andesite lava flows are exposed in southeastern edge of urumieh dokhtar magmatic assemblage in iran. the hypocrystalline andesite in parts contain conspicuous co-magmatic igneous enclaves which are dark grey and occur mostly as spherical and occasionally as ribbon shapes with some showing chilled margins. petrographic study shows that the ribbon type enclaves have been form...