International Journal of Mathematics

For a virtual knot $K$ and an integer $r$ with $r\ge 2$, we introduce a method of constructing an $r$-component virtual link $L(K;r)$, which we call the $r$-multiplexing of $K$. Every invariant of $L(K;r)$ is an invariant of $K$. We give a way of calculating three kinds of invariants of $L(K;r)$ using invariants of $K$. As an application of our method, we also show that Manturov’s virtual $n$-colorings for $K$ can be interpreted as certain classical $n$-colorings for $L(K;2)$.

We prove the existence of a universal braided compact quantum group acting on a graph ${\mathrm{\text{C}}}^{\ast }$-algebra in the category of $𝕋$-${\mathrm{\text{C}}}^{\ast }$-algebras with a twisted monoidal structure, in the spirit of the seminal work of Wang. To achieve this, we construct a braided analogue of the free unitary quantum group and study its bosonization. As a concrete example, we compute this universal braided compact quantum group for the Cuntz algebra.

A lower bound of the Gordian distance is presented in terms of the Blanchfield pairing. Our approach, in particular, allows us to show at least for $195$ pairs of unoriented nontrivial prime knots with up to $10$ crossings that their Gordian distance is equal to $3$, most of which are difficult to treat otherwise.

Petty proved that a convex body in ${ℝ}^n$ has the minimal surface area among its $\mathrm{\text{SL}}(n)$ images if and only if its surface area measure is isotropic. In this paper, we seek the general minimal Orlicz surface area of a convex body containing the origin in its interior among its $\mathrm{\text{SL}}(n)$ images for a measure $\mu$ on ${ℝ}^n$ with density that has a positive degree of homogeneous, demonstrate the existence of the extreme body, and establish a necessary condition for the extreme body. Moreover, we also provide a counterexample to negate the uniqueness of the extreme body attaining the general minimal Orlicz surface area for $\mu$.

Let $X$ be a normal projective variety with klt singularities and $L_X$ a strictly nef $\mathbb{Q}$-divisor on $X$. In this paper, we study the singular version of Serrano’s conjecture, i.e. the ampleness of $K_X+t{L}_X$ for $t\gg 1$. We show that, if $X$ is a $\mathbb{Q}$-factorial Gorenstein terminal threefold, then $K_X+t{L}_X$ is ample for $t\gg 1$ unless $X$ is a weak Calabi–Yau variety (i.e. the canonical divisor $K_X$ is a $\mathbb{Q}$-torsion and the augmented irregularity $q^{\circ }(X)$ vanishes) with $L_X\cdot c_2(X)=0$.

In this paper, we first show that the monoid of separable surjective self-morphisms of a variety of Ueno type coincides with the group of automorphisms. We also give an explicit description of the automorphism group. As applications, we confirm Kawaguchi–Silverman conjecture for automorphisms of a variety of Ueno type and some Calabi–Yau three-fold, defined over $\overline{\mathbb{Q}}$.