International Journal of Mathematics

In this paper, we investigate the monotonicity and one-dimensional symmetry of solutions for parabolic equations related to the weighted fractional Laplacian on the whole space. We first establish a generalized weighted average inequality and the maximum principle in unbounded domains, and then carry out the sliding method to obtain monotonicity and one-dimensional symmetry of solutions.

We introduce a quandle version of the normalized (twisted) Alexander polynomial, which is an invariant of a pair of an oriented link and a quandle representation. The invariant can be constructed by fixing each Alexander pair, and we find various invariants in our framework, which include the quandle cocycle invariant and the normalized (twisted) Alexander polynomial of a knot. In this paper, we develop the theory of normalization with row and column relations of matrices. The theory works for several row and column relations, although the twisted Alexander polynomial is defined with one column relation. We give a formula of our invariant for the mirror image of an oriented link, which explains why the Alexander polynomial fails to detect the chirality of knots and why the quandle cocycle invariant effectively detects it from a unified point of view. We also show that cohomologous Alexander pairs yield the same invariant.

The LMO invariant is a universal quantum invariant of closed oriented 3-manifolds. In this paper, we present the degree 2 part of the LMO invariant of cyclic branched covers of knots by using the 3-loop invariant of knots, and we calculate it concretely for knots obtained by plumbing the doubles of two knots.

A locally conformally product (LCP) structure on compact manifold $M$ is a conformal structure $c$ together with a closed, non-exact and non-flat Weyl connection $D$ with reducible holonomy. Equivalently, an LCP structure on $M$ is defined by a reducible, non-flat, incomplete Riemannian metric $h_D$ on the universal cover $\tilde{M}$ of $M$, with respect to which the fundamental group ${\pi }_1(M)$ acts by similarities. It was recently proved by Kourganoff that in this case $(\tilde{M},h_D)$ is isometric to the Riemannian product of the flat space ${ℝ}^q$ and an incomplete irreducible Riemannian manifold $(N,g_N)$. In this paper, we show that for every LCP manifold $(M,c,D)$, there exists a metric $g\in c$ such that the Lee form of $D$ with respect to $g$ vanishes on vectors tangent to the distribution on $M$ defined by the flat factor ${ℝ}^q$, and use this fact in order to construct new LCP structures from a given one by taking products. We also establish links between LCP manifolds and number field theory, and use them in order to construct large classes of examples, containing all previously known examples of LCP manifolds constructed by Matveev–Nikolayevsky, Kourganoff and Oeljeklaus–Toma (OT-manifolds).

Let $\phi$ be a function in the complex Sobolev space $W^{\ast }(U)$, where $U$ is an open subset in ${ℂ}^k$. We show that the complement of the set of Lebesgue points of $\phi$ is pluripolar. The key ingredient in our approach is to show that $|\phi {|}^{\alpha }$ for $\alpha \in [1,2)$ is locally bounded from above by a plurisubharmonic function.

We prove a simple necessary and sufficient condition for a two-bridge knot $K(p,q)$ to be quasipositive, based on the continued fraction expansion of $p/q$. As an application, coupled with some classification results in contact and symplectic topology, we give a new proof of the fact that smoothly slice two-bridge knots are non-quasipositive. Another proof of this fact using methods within the scope of knot theory is presented in Appendix A, by Stepan Orevkov.