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...

Andruskiewitsch and Schneider classify a large class of pointed Hopf algebras with abelian coradical. The quantum double of each such Hopf algebra is investigated. The quantum doubles of a family of Hopf algebras from the above classification are ribbon Hopf algebras. Introduction Quasitriangular Hopf algebras have an universal R-matrix which is a solution of the Yang-Baxter equation and their ...

BACKGROUND AND PURPOSE Large admission DWI infarct volume (>70 mL) is an established marker for poor clinical outcome in acute stroke. Outcome is more variable in patients with small infarcts (<70 mL). Percentage insula ribbon infarct correlates with infarct growth. We hypothesized that percentage insula ribbon infarct can help identify patients with stroke likely to have poor clinical outcome,...

This week in class we treat quantum traces and dimensions in general ribbon categories (from the syllabus of week 46, see also [2, Chapter 6]) and we define the general notion of ribbon algebras and discuss how these give rise relation to ribbon categories. This material will be used in later to give some important explicit constructions of ribbon categories. The syllabus of this week 47 contai...

Oriented ribbon graphs (dessins d’enfant) are graphs embedded in oriented surfaces. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. We show that for any link diagram L , there is an associated ribbon graph whose quasi-trees correspond bijectively to spanning trees of the graph obtained by checkerboard coloring L . This corresp...