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...

The transport of Balbiani ring (BR) premessenger RNP particles in the larval salivary gland cells of the dipteran Chironomus tentans can be followed using electron microscopy. A BR RNP particle consists of an RNP ribbon bent into a ringlike structure. Upon translocation through the nuclear pore complex (NPC), the ribbon is straightened and enters the central channel of the NPC with the 5' end o...

Over 100 synaptic ribbons were studied in the intact animal from birth to the 23rd day and over 500 were studied in the isolated organ up to 24 days in culture. Our findings suggest that synaptogenesis in the cochlea of the mouse occurs mainly postnatally and lasts at least 14 days. Afferent synapses of young cochleas are characterized by round ribbons which are attached to the presynaptic memb...

We introduce a new operation on skew diagrams called composition of transpositions, and use it and a Jacobi-Trudi style formula to derive equalities on skew Schur Q-functions whose indexing shifted skew diagram is an ordinary skew diagram. When this skew diagram is a ribbon, we conjecture necessary and sufficient conditions for equality of ribbon Schur Q-functions. Moreover, we determine all re...

A bottom tangle is a tangle in a cube consisting of arc components whose boundary points are on a line in the bottom square of the cube. A ribbon bottom tangle is a bottom tangle whose closure is a ribbon link. For every n-component ribbon bottom tangle T , we prove that the universal invariant JT of T associated to the quantized enveloping algebra Uh(sl2) of the Lie algebra sl2 is contained in...

In mammalian cells, the Golgi reassembly stacking protein 65 (GRASP65) has been implicated in both Golgi stacking and ribbon linking by forming trans-oligomers through the N-terminal GRASP domain. Because the GRASP domain is globular and relatively small, but the gaps between stacks are large and heterogeneous, it remains puzzling how GRASP65 physically links Golgi stacks into a ribbon. To expl...

A bottom tangle is a tangle in a cube consisting of arc components whose boundary points are on a line in the bottom square of the cube. A ribbon bottom tangle is a bottom tangle whose closure is a ribbon link. For every n-component ribbon bottom tangle T , we prove that the universal invariant JT of T associated to the quantized enveloping algebra Uh(sl2) of the Lie algebra sl2 is contained in...

We show that a bilayer graphene ribbon deposited above a ferromagnetic insulator can behave as a spin-filtering device. The ferromagnetic material induces exchange splitting in the graphene ribbon, and due to the Fano antiresonances occurring in the transmission of the graphene ribbon as a function of ribbon length and energy, it is possible to obtain a net spin current. This happens when an an...

The influence of winding-induced mechanical stresses on the magnetic anisotropy and core loss in toroidal cores made of Fe73Cu1Nb3Si16B7 ribbon is studied. The ribbon for the cores was rapidly pre-heated under tensile stress up to 120 MPa. It was found that magnetic characteristics of the material (magnetic anisotropy energy and the core loss) can be controlled by varying the tensile stress dur...