Ljiljana Radovic

THE THEORY OF PSEUDOKNOTS

Classical knots in ℝ3 can be represented by diagrams in the plane. These diagrams are formed by curves with a finite number of transverse crossings, where each crossing is decorated to indicate which strand of the knot passes over at that point. A pseudodiagram is a knot diagram that may be missing crossing information at some of its crossings. At these crossings, it is undetermined which strand passes over. Pseudodiagrams were first introduced by Ryo Hanaki in 2010. Here, we introduce the notion of a pseudoknot, i.e. an equivalence class of pseudodiagrams under an appropriate choice of Reidemeister moves. In order to begin a classification of pseudoknots, we introduce the concept of a weighted resolution set, or WeRe-set, an invariant of pseudoknots. We compute the WeRe-set for several pseudoknot families and discuss extensions of crossing number, homotopy, and chirality for pseudoknots.

Do you like paleolithic op‐art?

Purpose – The purpose of this paper is to consider the history of certain modular elements: Truchet tiles, Op‐tiles, Kufic tiles, and key‐patterns, which occur as ornamental archetypes from Paleolithic times until the present. The appearance of the same ornamental archetypes at the same level of the development in different cultures, distant in space and time can be described from the cybernetics point of view as a specific kind of self‐referential systems or cellular automata present in the intellectual and cultural development of mankind. The aim of this research is to show a continuity of the development of ornamental structures based on modular elements used as ornamental archetypes.Design/methodology/approach – Research of the material from archaeological findings, history of art, painting, architecture, and applied arts.Findings – Existence of universal geometrical construction principles based on modularity.Practical implications – Creation of new patterns or designs (e.g. TeX‐fonts, tiles, etc.) b...

REDUCED RELATIVE TUTTE, KAUFFMAN BRACKET AND JONES POLYNOMIALS OF VIRTUAL LINK FAMILIES

This paper contains general formulae for the reduced relative Tutte, Kauffman bracket and Jones polynomials of families of virtual knots and links given in Conway notation and discussion of a counterexample to the Z-move conjecture of Fenn, Kauffman and Manturov.

Knots in Art

We analyze applications of knots and links in the Ancient art, beginning from Babylonian, Egyptian, Greek, Byzantine and Celtic art. Construction methods used in art are analyzed on the examples of Celtic art and ethnical art of Tchokwe people from Angola or Tamil art, where knots are constructed as mirror-curves. We propose different methods for generating knots and links based on geometric polyhedra, suitable for applications in architecture and sculpture.

Symmetry of “Twins”

The idea of construction of twin buildings is as old as architecture itself, and yet there is hardly any study emphasizing their specificity. Most frequently there are two objects or elements in an architectural composition of “twins” in which there may be various symmetry relations, mostly bilateral symmetries. The classification of “twins” symmetry in this paper is based on the existence of bilateral symmetry, in terms of the perception of an observer. The classification includes both, 2D and 3D perception analyses. We start analyzing a pair of twin buildings with projection of the architectural composition elements in 2D picture plane (plane of the composition) and we distinguish four 2D keyframe cases based on the relation between the bilateral symmetry of the twin composition and the bilateral symmetry of each element. In 3D perception for each 2D keyframe case there are two sub-variants, with and without a symmetry plane parallel to the picture plane. The bilateral symmetry is dominant if the corresponding symmetry plane is orthogonal to the picture plane. The essence of the complete classification is relation between the bilateral (dominant) symmetry of the architectural composition and the bilateral symmetry of each element of that composition.

In Memoriam: Slavik Jablan 1952–2015

After a long and brave battle with a serious illness, our dear friend and colleague Slavik Jablan passed away on 26 February 2015. [...]

The Vasarely Playhouse: Invitation to a Mathematical and Combinatorial Visual Game

The authors analyze Victor Vasarely’s works from the viewpoints of the theory of visual perception, mathematics and modularity. The chapter concludes that almost all construction methods, modular elements, optical effects and visual illusions belonging to these fields were (re)discovered by Vasarely, mostly by intuition, creative visual thinking and experimenting, and then used in his artworks.

Meanders, knots, labyrinths and mazes

There are strong indications that the history of design may have begun with the concept of a meander. This paper explores the application of meanders to new classes of meander and semi-meander knots, meander friezes, labyrinths and mazes. A combinatorial system is introduced to classify meander knots and labyrinths. Mazes are analyzed with the use of graphs. Meanders are also created with the use of simple proto-tiles upon which a series of lines are etched.

Plaited polyhedra: A knot theory point of view

Plaited polyhedra, discovered by P. Gerdes in African dance rattle capsules are analyzed from the knot-theory point of view. Every plaited polyhedron can be derived from a knot or link diagram as its dual. In the attempt to classify obtained plaited polyhedra, we propose different methods based on families of knots and links in Conway notation or their corresponding braid families, both leading to the notion of families of plaited polyhedra.