The canonical genus for Whitehead doubles of a family of alternating knots

For any given integer $r \geq 1$ and a quasitoric braid $\beta_r=(\sigma_r^{-\epsilon} \sigma_{r-1}^{\epsilon}...$ $\sigma_{1}^{(-1)^{r}\epsilon})^3$ with $\epsilon=\pm 1$, we prove that the maximum degree in $z$ of the HOMFLYPT polynomial $P_{W_2(\hat\beta_r)}(v,z)$ of the doubled link $W_2(\hat\beta_r)$ of the closure $\hat\beta_r$ is equal to $6r-1$. As an application, we give a family $\mathcal K^3$ of alternating knots, including $(2,n)$ torus knots, 2-bridge knots and alternating pretzel knots as its subfamilies, such that the minimal crossing number of any alternating knot in $\mathcal K^3$ coincides with the canonical genus of its Whitehead double. Consequently, we give a new family $\mathcal K^3$ of alternating knots for which Tripp's conjecture holds.Comment: 33 pages, 27 figure

Influence of mixed alkali oxides on some melt properties of TV screen glass

The influence of K2O/R2O (R2O = Na2O+K2O) on some melt properties was studied in alkali-alkaline earth-silicate TV screen glasses. Viscosity, surface tension, electrical resistivity, volatilization and devitrification of glass melts were determined. No mixed alkali effect was observed in viscosity, surface tension and liquidus temperature. They showed a linear behavior with increase of K2O/R2O. On the other hand, electrical resistivity and weight loss by volatilization showed a strong mixed alkali effect against relative alkali concentration. According to the dependence of viscosity, electrical resistivity and volatilization on K2O/R2O, the slope change of those properties took place at K2O/R2O = 0.4 to ≈ 0.5. The compositional dependence of viscosity, surface tension and liquidus temperature was discussed in terms of field strength, polarizability and material diffusion, respectively. Α correlation was also discussed between the dependence of properties on K2O/R2O and the production process of TV screen glass. In conclusion, from the viewpoint of both production and application of TV glasses it was suggested that the mole fraction of K2O/R2O should lie between 0.2 and 0.5

Roles of Primary Cilia in the Developing Brain

Essential to development, primary cilia are microtubule-based cellular organelles that protrude from the surface of cells. Acting as cellular antenna, primary cilia play central roles in transducing or regulating several signaling pathways, including Sonic hedgehog (Shh) and Wnt signaling. Defects in primary cilia contribute to a group of syndromic disorders known as “ciliopathies” and can adversely affect development of the brain and other essential organs, including the kidneys, eyes, and liver. The molecular mechanisms of how defective primary cilia contribute to neurological defects, however, remain poorly understood. In this mini review, we summarize recent advances in understanding of the interactions between primary cilia and signaling pathways essential to cellular homeostasis and brain development