Knots and k-width

We investigate several integer invariants of curves in 3-space. We demonstrate relationships of these invariants to crossing number and to total curvature

Magnon topology and thermal Hall effect in trimerized triangular lattice antiferromagnet

The non-trivial magnon band topology and its consequent responses have been extensively studied in two-dimensional magnetisms. However, the triangular lattice antiferromagnet (TLAF), the best-known frustrated two-dimensional magnet, has received less attention than the closely related Kagome system, because of the spin-chirality cancellation in the umbrella ground state of the undistorted TLAF. In this work, we study the band topology and the thermal Hall effect (THE) of the TLAF with (anti-)trimerization distortion under the external perpendicular magnetic field using the linearized spin wave theory. We show that the spin-chirality cancellation is removed in such case, giving rise to the non-trivial magnon band topology and the finite THE. Moreover, the magnon bands exhibit band topology transitions tuned by the magnetic field. We demonstrate that such transitions are accompanied by the logarithmic divergence of the first derivative of the thermal Hall conductivity. Finally, we examine the above consequences by calculating the THE in the hexagonal manganite YMnO$_3$, well known to have anti-trimerization.Comment: 6 + 7 pages, 3 + 5 figures, 0 + 1 table; Journal reference adde

Subdivision surface fitting to a dense mesh using ridges and umbilics

Fitting a sparse surface to approximate vast dense data is of interest for many applications: reverse engineering, recognition and compression, etc. The present work provides an approach to fit a Loop subdivision surface to a dense triangular mesh of arbitrary topology, whilst preserving and aligning the original features. The natural ridge-joined connectivity of umbilics and ridge-crossings is used as the connectivity of the control mesh for subdivision, so that the edges follow salient features on the surface. Furthermore, the chosen features and connectivity characterise the overall shape of the original mesh, since ridges capture extreme principal curvatures and ridges start and end at umbilics. A metric of Hausdorff distance including curvature vectors is proposed and implemented in a distance transform algorithm to construct the connectivity. Ridge-colour matching is introduced as a criterion for edge flipping to improve feature alignment. Several examples are provided to demonstrate the feature-preserving capability of the proposed approach

Can a charged dust ball be sent through the Reissner--Nordstr\"{o}m wormhole?

In a previous paper we formulated a set of necessary conditions for the spherically symmetric weakly charged dust to avoid Big Bang/Big Crunch, shell crossing and permanent central singularities. However, we did not discuss the properties of the energy density, some of which are surprising and seem not to have been known up to now. A singularity of infinite energy density does exist -- it is a point singularity situated on the world line of the center of symmetry. The condition that no mass shell collapses to $R = 0$ if it had $R > 0$ initially thus turns out to be still insufficient for avoiding a singularity. Moreover, at the singularity the energy density $\epsilon$ is direction-dependent: $\epsilon \to - \infty$ when we approach the singular point along a $t =$ const hypersurface and $\epsilon \to + \infty$ when we approach that point along the center of symmetry. The appearance of negative-energy-density regions turns out to be inevitable. We discuss various aspects of this property of our configuration. We also show that a permanently pulsating configuration, with the period of pulsation independent of mass, is possible only if there exists a permanent central singularity.Comment: 30 pages, 21 figures; several corrections after referee's comments, 4 figures modifie

Mixed topological semimetals driven by orbital complexity in two-dimensional ferromagnets

The concepts of Weyl fermions and topological semimetals emerging in three-dimensional momentum space are extensively explored owing to the vast variety of exotic properties that they give rise to. On the other hand, very little is known about semimetallic states emerging in two-dimensional magnetic materials, which present the foundation for both present and future information technology. Here, we demonstrate that including the magnetization direction into the topological analysis allows for a natural classification of topological semimetallic states that manifest in two-dimensional ferromagnets as a result of the interplay between spin-orbit and exchange interactions. We explore the emergence and stability of such mixed topological semimetals in realistic materials, and point out the perspectives of mixed topological states for current-induced orbital magnetism and current-induced domain wall motion. Our findings pave the way to understanding, engineering and utilizing topological semimetallic states in two-dimensional spin-orbit ferromagnets