Virtual knot theory is a generalization of knot theory which is based on Gauss chord diagrams and link diagrams on closed orientable surfaces. Twisted knots are a generalization of virtual knots, which correspond to link diagrams in possibly non-orientable surfaces. In this paper, we construct a table of twisted knots with crossing numbers $3$. We use the multivariable polynomial invariants, JKSS invariants of double covering of twisted knots, and the twisted knot quandles for classification of twisted knots. We determine invertibility, chirality and checkerboard colorability for some twisted knots in our table.

A recent result of Funayoshi-Koda shows that a handlebody-knot of genus two has a finite symmetry group if and only if it is hyperbolic—the exterior admits a hyperbolic structure with totally geodesic boundary—or irreducible, atoroidal, cylindrical—the exterior contains no essential disks or tori but contains an essential annulus. Based on the Koda-Ozawa classification theorem, essential annuli in an irreducible, atoroidal handlebody-knots of genus two are classified into four classes: type $2$, type $3$-$2$, type $3$-$3$ and type $4$-$1$. We show that under mild conditions most genus two cylindrical handlebody-knot exteriors contain no essential disks or tori, and when a type $3$-$3$ annulus exists, it is often unique up to isotopy; a classification result for symmetry groups of such cylindrical handlebody-knots is also obtained.

We study topological zeta functions of complex plane curve singularities using toric modifications and further developments. As applications of the research method, we prove that the topological zeta function is a topological invariant for complex plane curve singularities, we give a short and new proof of the monodromy conjecture for plane curves.

We introduce a six-variable polynomial invariant of Niebrzydowski tribrackets analogous to quandle, rack and biquandle polynomials. Using the subtribrackets of a tribracket, we additionally define subtribracket polynomials and establish a sufficient condition for isomorphic subtribrackets to have the same polynomial regardless of their embedding in the ambient tribracket. As an application, we enhance the tribracket counting invariant of knots and links using subtribracket polynomials and provide examples to demonstrate that this enhancement is proper.

We study the Baily-Borel compactification of a family of four-dimensional orthogonal modular varieties arising as period spaces of compact hyperkähler manifolds of deformation generalised Kummer type. Our main results concern the classification of boundary components, their incidence relations and combinatorics.

Our purpose of this paper is to derive Strichartz estimates for solutions of magnetic Schrödinger, wave and Klein-Gordon equations in exterior to the star-shaped obstacle. For its proof we need the smoothing estimates for solutions of perturbed equations and the Strichartz estimates for solutions of free equations. Moreover as an application of them, we shall investigate the scattering theory for these equations with a power type nonlinearity in suitable space.

The group ${\rm{TL}}(2, 5)$ is a nontrivial double covering of $S_5$ with the center of order $2$, where $S_5$ is the symmetric group on five letters. In addition, ${\rm{TL}}(2, 5)$ has a free irreducible real representation of dimension $8$ and contains ${\rm{SL}}(2, 5)$ as a normal subgroup up to isomorphisms, where ${\rm{SL}}(2, 5)$ is the special linear group of degree $2$ over the field consisting of five elements. ${\rm{TL}}(2, 5)$ has been denoted by ${\rm{SL}}(2, 5).C_2$ as well as ${\rm{TL}}_2 (5)$. In this paper, we show that there never exists an effective smooth action of ${\rm{TL}}(2, 5)$ on a homology sphere of dimension $21$, with exactly one fixed point.

We develop rigorously a geometric theory of third-order partial differential equations for a scalar function. Under our framework, we can define a notion of nilpotent graded Lie algebras as an invariant useful to study geometry of third-order equations. In terms of these graded Lie algebras, we provide a classification for some classes of third-order equations under a contact equivalence. By this classification, together with model equations, we also clarify several aspects for each subcategory of equations.

For a non-isomorphic finite endomorphism of a germ of a complex analytic normal surface at a point, the pair of the surface and a completely invariant reduced divisor is shown to be log-canonical. It is also shown in many situations that the endomorphism or its square lifts to an endomorphism of another surface by an essential blowing up.