Filtrations on the knot contact homology of transverse knots

We construct a new invariant of transverse links in the standard contact structure on R^3. This invariant is a doubly filtered version of the knot contact homology differential graded algebra (DGA) of the link. Here the knot contact homology of a link in R^3 is the Legendrian contact homology DGA of its conormal lift into the unit cotangent bundle S^*R^3 of R^3, and the filtrations are constructed by counting intersections of the holomorphic disks of the DGA differential with two conormal lifts of the contact structure. We also present a combinatorial formula for the filtered DGA in terms of braid representatives of transverse links and apply it to show that the new invariant is independent of previously known invariants of transverse links.Comment: 23 pages, v2: minor corrections suggested by refere

Selenium supplemented fertilization - effects on the selenium content of foods and the selenium intake in Finland

The effect of Se fertilization was distinct. Winter cereals were not affected as much as spring cereals due to the different cultivation and fertilization practice. The variation between the farms was large. In organic cultivation Se content of cereals was low

The Minimal Length of a Lagrangian Cobordism between Legendrians

To investigate the rigidity and flexibility of Lagrangian cobordisms between Legendrian submanifolds, we investigate the minimal length of such a cobordism, which is a $1$-dimensional measurement of the non-cylindrical portion of the cobordism. Our primary tool is a set of real-valued capacities for a Legendrian submanifold, which are derived from a filtered version of Legendrian Contact Homology. Relationships between capacities of Legendrians at the ends of a Lagrangian cobordism yield lower bounds on the length of the cobordism. We apply the capacities to Lagrangian cobordisms realizing vertical dilations (which may be arbitrarily short) and contractions (whose lengths are bounded below). We also study the interaction between length and the linking of multiple cobordisms as well as the lengths of cobordisms derived from non-trivial loops of Legendrian isotopies.Comment: 33 pages, 9 figures. v2: Minor corrections in response to referee comments. More general statement in Proposition 3.3 and some reorganization at the end of Section

A Hardy inequality in twisted waveguides

We show that twisting of an infinite straight three-dimensional tube with non-circular cross-section gives rise to a Hardy-type inequality for the associated Dirichlet Laplacian. As an application we prove certain stability of the spectrum of the Dirichlet Laplacian in locally and mildly bent tubes. Namely, it is known that any local bending, no matter how small, generates eigenvalues below the essential spectrum of the Laplacian in the tubes with arbitrary cross-sections rotated along a reference curve in an appropriate way. In the present paper we show that for any other rotation some critical strength of the bending is needed in order to induce a non-empty discrete spectrum.Comment: LaTeX, 20 page

New insight on the structural features of the cytotoxic auristatins MMAE and MMAF revealed by combined NMR spectroscopy and quantum chemical modelling

Antibody-drug conjugates (ADCs) are emerging as a promising class of selective drug delivery systems in the battle against cancer and other diseases. The auristatins monomethyl auristatin E (MMAE) and monomethyl auristatin F (MMAF) appear as the cytotoxic drug in almost half of the state-of-the-art ADCs on the market or in late stage clinical trials. Here, we present the first complete NMR spectroscopic characterisation of these challenging molecules, and investigate their structural properties by a combined NMR and quantum chemical modelling approach. We find that in solution, half of the drug molecules are locked in an inactive conformation, severely decreasing their efficiency, and potentially increasing the risk of side-effects. Furthermore, we identify sites susceptible to future modification, in order to potentially improve the performance of these drugs.Peer reviewe