A central result in the space geometry of closed twisted ribbons is Călugăreanu’s theorem (also known as White’s formula, or the Călugăreanu-White-Fuller theorem). This enables the integer linking number of the two edges of the ribbon to be written as the sum of the ribbon twist (the rate of rotation of the ribbon about its axis) and its writhe. We show that twice the twist is the average, over...

We introduce the notion of a half-ribbon Hopf algebra, which is a Hopf algebra H along with a distinguished element t ∈ H such that (H, R,C) is a ribbon Hopf algebra, where R = (t ⊗ t)∆(t) and C = t. The element t is closely related to the topological ‘half-twist’, which twists a ribbon by 180 degrees. We construct a functor from a topological category of ribbons with half-twists to the categor...

A central result in the space geometry of closed twisted ribbons is Călugăreanu’s theorem (also known as White’s formula, or the Călugăreanu-White-Fuller theorem). This enables the integer linking number of the two edges of the ribbon to be written as the sum of the ribbon twist (the rate of rotation of the ribbon about its axis) and its writhe. We show that twice the twist is the average, over...

We describe a phenomenon of ribbon-to-fiber transformation observed in the process of spinning of single wall carbon nanotubes dispersed in polymer solutions. In the process of spinning, a gel-like ribbon comprised of nanotube bundles bound by polymer is withdrawn from a solvent bath. We show that upon crossing the liquid-air interface, the ribbon may either retain its flat shape or fold into a...

A close to native structure of bulk biological specimens can be imaged by cryo-electron microscopy of vitreous sections (CEMOVIS). In some cases structural information can be combined with X-ray data leading to atomic resolution in situ. However, CEMOVIS is not routinely used. The two critical steps consist of producing a frozen section ribbon of a few millimeters in length and transferring the...

The sensory epithelium lining the ampulla of Lorenzini in the skate was examined by the freeze-fracture technique. Anastomosing tight junctions (zonula occludens) completely encircle the apex of each receptor cell joining it to neighbouring support cells. The tight junctions separate two distinctly different regions of the receptor-cell surface. The apical P-face has numerous large particles wh...

We extend the notion of k-ribbon tableaux to the Fibonacci lattice, a differential poset defined by R. Stanley in 1975. Using this notion, we describe an insertion algorithm that takes k-colored permutations to pairs of k-ribbon Fibonacci tableaux of the same shape, and we demonstrate a colorto-spin property, similar to that described by Shimozono and White for ribbon tableaux. We give an evacu...

he ribbon complex of retinal photoreceptor synapses represents a specialization of the cytomatrix at the active zone (CAZ) present at conventional synapses. In mice deficient for the CAZ protein Bassoon, ribbons are not anchored to the presynaptic membrane but float freely in the cytoplasm. Exploiting this phenotype, we dissected the molecular structure of the photoreceptor ribbon complex. Iden...

Visual and auditory information is encoded by sensory neurons that tonically release neurotransmitter at high rates. The synaptic ribbon is an essential organelle in nerve terminals of these neurons. Its precise function is unknown, but if the ribbon could be visualized in a living terminal, both its own dynamics and its relation to calcium and vesicle dynamics could be studied. We designed a s...