Differential expansion for link polynomials

The differential expansion is one of the key structures reflecting group theory properties of colored knot polynomials, which also becomes an important tool for evaluation of non-trivial Racah matrices. This makes highly desirable its extension from knots to links, which, however, requires knowledge of the $6j$-symbols, at least, for the simplest triples of non-coincident representations. Based on the recent achievements in this direction, we conjecture a shape of the differential expansion for symmetrically-colored links and provide a set of examples. Within this study, we use a special framing that is an unusual extension of the topological framing from knots to links. In the particular cases of Whitehead and Borromean rings links, the differential expansions are different from the previously discovered.Comment: 11 page

Correlation of Solid Dosage Porosity and Tensile Strength with Acoustically Extracted Mechanical Properties

Currently, the compressed tablet and its oral administration is the most popular drug delivery modality in medicine. The accurate porosity and tensile strength characterization of a tablet design is vital for predicting its performance such as disintegration, dissolution, and drug-release efficiency upon administration as well as ensuring its mechanical integrity. In current work, a non-destructive contact ultrasonic approach and an associated testing procedure are presented and employed to quantify and relate the acoustically extracted mechanical properties of pharmaceutical compacts to direct porosity and tensile strength measurements. Based on a comprehensive set of experimental data, it is demonstrated how strongly the acoustic wave propagation is modulated and correlated to the tablet porosity and tensile strength of a compact made using spray-dried lactose and microcrystalline cellulose with varying mixture ratios. The effect of mixing ratio on the porosity and tensile strength on the resulting compacts is quantified and, with the acoustic experimental data, mixing ratio is related to the compact ultrasonic characteristics. The ultrasonic techniques provide a rapid, non-destructive means for evaluating compacts in formulation development and manufacturing. The presented approach and data could find critical applications in continuous tablet manufacturing, its real-time quality monitoring, as well as minimizing batch-to-batch quality variations

Gaussian distribution of LMOV numbers

Recent advances in knot polynomial calculus allowed us to obtain a huge variety of LMOV integers counting degeneracy of the BPS spectrum of topological theories on the resolved conifold and appearing in the genus expansion of the plethystic logarithm of the Ooguri-Vafa partition functions. Already the very first look at this data reveals that the LMOV numbers are randomly distributed in genus (!) and are very well parameterized by just three parameters depending on the representation, an integer and the knot. We present an accurate formulation and evidence in support of this new puzzling observation about the old puzzling quantities. It probably implies that the BPS states, counted by the LMOV numbers can actually be composites made from some still more elementary objects.Comment: 23 page

Eigenvalue hypothesis for multi-strand braids

Computing polynomial form of the colored HOMFLY-PT for non-arborescent knots obtained from three or more strand braids is still an open problem. One of the efficient methods suggested for the three-strand braids relies on the eigenvalue hypothesis which uses the Yang-Baxter equation to express the answer through the eigenvalues of the ${\cal R}$-matrix. In this paper, we generalize the hypothesis to higher number of strands in the braid where commuting relations of non-neighbouring $\mathcal{R}$ matrices are also incorporated. By solving these equations, we determine the explicit form for $\mathcal{R}$-matrices and the inclusive Racah matrices in terms of braiding eigenvalues (for matrices of size up to 6 by 6). For comparison, we briefly discuss the highest weight method for four-strand braids carrying fundamental and symmetric rank two $SU_q(N)$ representation. Specifically, we present all the inclusive Racah matrices for representation $[2]$ and compare with the matrices obtained from eigenvalue hypothesis.Comment: 23 page