Journal of Knot Theory and Its Ramifications

In this paper, we introduce the tied links, i.e. ordinary links provided with some “ties” between strands. The motivation for introducing such objects originates from a diagrammatical interpretation of the defining generators of the so-called algebra of braids and ties; indeed, one half of such generators can be interpreted as the usual generators of the braid algebra, and the other half can be interpreted as ties between consecutive strands; this interpretation leads to the definition of tied braids. We define an invariant polynomial for the tied links via a skein relation. Furthermore, we introduce the monoid of tied braids and we prove the corresponding theorems of Alexander and Markov for tied links. Finally, we prove that the invariant of tied links we defined can be obtained also by using the Jones recipe.

In this paper, we introduce pseudo braid monoids and show that, such a monoid is isomorphic to a singular braid monoid. We also prove an analogue of Markov’s Theorem for pseudo braids.

There are three constructions of which I know that yield higher dimensional analogues of Sierpinski’s triangle. The most obvious is to remove the open convex hull of the midpoints of the edges of the $n$-simplex. The complement is a union of simplices. Continue the removal recursively in each of the remaining sub-simplices. The result is an uncountably infinite figure in $n$-dimensional space that is Cantor-like in a manner analogous to the Sierpinski triangle. A countable analogue is obtained by means of playing the chaos game in the $n$-simplex. In this “game” one chooses a random $(n+1)$-ary sequence; starting from the initial point (that is identified with a vertex of the simplex), one continues to plot points by moving half-again as much towards the next point in the sequence. The resulting plot converges to the figure described above. Similarly, coloring the multinomial coefficients black or white according to their parity results in a similar figure, when the $n$-dimensional analogue of the Pascal triangle is rescaled and embedded in space.

In this paper, we study properties of the Markov trace tr${}_d$ and the specialized trace ${\mathrm{\text{tr}}}_{d,D}$ on the Yokonuma–Hecke algebras, such as behavior under inversion of a word, connected sums and mirror imaging. We then define invariants for framed, classical and singular links through the trace ${\mathrm{\text{tr}}}_{d,D}$ and also invariants for transverse links through the trace tr${}_d$. In order to compare the invariants for classical links with the Homflypt polynomial, we develop computer programs and we evaluate them on several Homflypt-equivalent pairs of knots and links. Our computations lead to the result that these invariants are topologically equivalent to the Homflypt polynomial on knots. However, they do not demonstrate the same behavior on links.

In this note, we first give an alternative elementary proof of the relation between the determinant of a link and the spanning trees of the corresponding Tait graph. Then, we use this relation to give an extremely short, knot theoretical proof of a theorem due to Shank stating that a link has component number one if and only if the number of spanning trees of its Tait graph is odd.

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.

When the signed weighted resolution set (or were-set) was defined as an invariant of pseudoknots, it was unknown whether this invariant was complete. Using the Gauss-diagrammatic invariants of pseudoknots introduced by Dorais et al., we show that the signed were-set cannot distinguish all non-equivalent pseudoknots. This goal is achieved through studying the effects of a flype-like local move on a pseudodiagram.

We call a delta diagram any diagram of a knot or link whose regions (including the unbounded one) have 3, 4, or 5 sides. We prove that any knot or link admits a delta diagram. We define and estimate combinatorial link invariants stemming from this definition.

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.

We introduce concepts of the maximum unknotting number and the mixed unknotting number, taking into consideration the Bernhard–Jablan Conjecture about computing the unknotting number based only on minimal knot diagrams. The existence of Kauffman knots (alternating knots, such that a crossing change does not change their minimal crossing number) was first suggested by Kauffman. We extend the concept and offer three related classes of knots named: Kauffman knots, Zeković knots and Taniyama knots.

Twisted links are a generalization of virtual links. As virtual links correspond to abstract links on orientable surfaces, twisted links correspond to abstract links on (possibly non-orientable) surfaces. In this paper, we introduce the notion of the double covering of a twisted link. It is defined by considering the orientation double covering of an abstract link or alternatively by constructing a diagram called a double covering diagram. We also discuss links in thickened surfaces, their diagrams and their stable equivalence classes. Bourgoin’s twisted knot group is understood as the virtual knot group of the double covering.

We describe an algorithm for computing the invariants of classical links arising from the Yokonuma–Hecke algebras. A detailed overview of the algorithm is given, following closely its implementation, a program used to calculate the invariants on several Homflypt-equivalent pairs of links.

Looking at the structure of minimal prime knot presentations, one can notice that there are often, perhaps always, segments that present either the trefoil or the figure-eight knot. This note explores the question as to whether this is always the case, reporting on conversations with Jablan Slavik that began at a $2009$ conference in Trieste, Italy and which never reached a conclusion. Evidence supporting this conjectured presence is reported and potential consequences are described.

We show that every alternating link of two components and $12$ crossings can be reduced by $4$-moves to the trivial link or the Hopf link. It answers the question asked in one of the last papers by Slavik Jablan.

In this paper we will discuss knots in material culture, giving an overview of their importance and range of usage. We will then discuss a method of characterization for these knots and a method to study these knots through cultural evolution. We then give some preliminary results and ideas for further study.

This note is dedicated to the memory of Slavik Jablan, and the last work we had together on the computation and properties of Cherns–Simons invariant of knots and links.