The character variety of some classes of rational knots

We propose a method to determine the character variety of a class [Formula: see text] of rational knots, which includes the twist knots. The defining polynomials depend only on the variables [Formula: see text] and [Formula: see text]. This answers for these classes of knots a question posed in [H. M. Hilden, M. T. Lozano and J. M. Montesinos–Amilibia, On the character variety of group representations of a 2-bridge link [Formula: see text] into [Formula: see text], Bol. Soc. Mat. Mexicana 37(2) (1992) 241–262], and allows us to give an easy geometrical description of the considered character variety. Our results are obtained by using special presentations of the knot groups whose relators are palindromes.

Pattern generation for direct-write three-dimensional nanoscale structures via focused electron beam induced deposition

Fabrication of three-dimensional (3D) nanoarchitectures by focused electron beam induced deposition (FEBID) has matured to a level that highly complex and functional deposits are becoming available for nanomagnetics and plasmonics. However, the generation of suitable pattern files that control the electron beam’s movement, and thereby reliably map the desired target 3D structure from a purely geometrical description to a shape-conforming 3D deposit, is nontrivial. To address this issue we developed several writing strategies and associated algorithms implemented in C++. Our pattern file generator handles different proximity effects and corrects for height-dependent precursor coverage. Several examples of successful 3D nanoarchitectures using different precursors are presented that validate the effectiveness of the implementation.

Adaptive analysis of structures of complex geometrical and mechanical description with use of the transition finite elements

This work concerns inclusion of the transition finite elements in adaptive analysis of structures of complex mechanical and geometrical description. The paper addresses the influence of modifications of the transition model on the numerical solution. The ability to remove the internal boundary layer and the effectivity of error estimation and adaptation are analyzed.