Gauss studied knots and links, and introduced a word theory of plane curves. Tait gave a table of knots, and obtained a famous conjecture of knots. After Jones polynomial was appeared and this Tait conjecture was solved, we noticed an importance of studies of knot projections (plane curves). We introduce recent studies. From 1984, Jones polynomial, its generalization (Quantum invariants), and their categorifications have been hot topics in Geometry and Physics. From 2005, by Turaev’s nanoword theory, we can describe which information of a knot produces the Jones polynomial and the Khovanov homology. In this report, we introduce recent works of mathematicians and the author with respect to these topics.

Coulombic interaction is inversely proportional to the distance between two charged particles and converges to zero very slowly; a quite large cutoff-radius distance is required to treat molecular dynamics in solution. One of theoretical approaches to solve the problem is the Ewald summation. A further development, the particle-mesh Ewald (PME) summation, accelerates the original approach and is nowadays available from variety of MD program packages. In the present work, the PME summation was applied to the ONIOM scheme (PME-ONIOM) for validating Coulombic interaction in solution. The two different photochemical reactions were actually performed as nonadiabatic ab initio MD with the Zhu-Nakamura trajectory surface hopping, (i) the photoisomerization of a (Z)-penta-2,4-dieniminium cation (PSB3) in methanol solution, and (ii) the photochromic reaction between 1,3-Cyclohexadiene (CHD) and 1,3,5-cis-Hexatriene (HT) in both hexane and ethanol solvents. The two different S₁-state lifetimes of PSB3 were found to be 3.483 ps and 82 fs, which were consistent with the experimental trend, moreover the mystery of the product branching ratio could be solved. On the other hand, the high nonpolar-nonpolar affinity between CHD and the solvent was observed in hexane solvent, which definitely affected the S₁ lifetimes, the product branching ratio of CHD : HT, and solute (CHD) dynamics. In ethanol solvent, however, the CHD solute was isomerized in the solvent cage caused by the first solvation shell. The photochemical dynamics in ethanol solvent results in the similar property to the process appeared in vacuo (isolated CHD dynamics).

Proton plays important roles in protein function, stability and folding. Change in pH induces folding and acid-denaturation via (de) protonation of the ionizable groups. We introduce a statistical mechanical model of proton-induced folding that incorporates the protonation mechanisms of conventional models. The model considers both specific affinity of proton to the protein molecules, and Coulombic interactions among the net charges on them as the major determinants of pH-induced folding events. The model reproduces the pH-dependence of folding of apomyoglobin and chymotrypsin inhibitor 2. For both proteins, Coulombic interactions induce the conformational changes in early stages of folding, and specific affinities are enhanced only in late stages of folding. This model is potentially applicable to other ligand-binding systems coupled with conformational changes.

We report an experimental study of ultracold collisions between atomic ions and neutral atoms. The atom-ion interactions in a hybrid trap, where a RF Paul trap and an optical dipole trap are combined, are investigated. We characterized the collisional properties of elastic and inelastic collisions between Ca⁺ ions and Li atoms by determining the cross sections in a milli kelvin temperature.

Recent nuclear magnetic resonance and specific-heat experiments have revealed that Cu_xBi₂Se₃ exhibits spin-triplet nematic superconductivity, in which the rotational symmetry is spontaneously broken both in the Cooper-pair spins and the superconducting gap amplitude. This review describes this superconductivity with novel symmetry breakings.