Journal of Knot Theory and Its Ramifications

In this paper, we study unitary braid group representations associated with Majorana fermions. Majorana fermions are represented by Majorana operators, elements of a Clifford algebra. The paper proves a general result about braid group representations associated with Clifford algebras and compares this result with the Ivanov braiding associated with Majorana operators and with other braiding representations associated with Majorana fermions such as the Fibonacci model for universal topological quantum computing.

In this paper, we observe that two-dimensional 0-surgery occurs in natural processes, such as tornado formation and other phenomena reminiscent of hole drilling. Inspired by such phenomena, we introduce new theoretical concepts which enhance the formal definition of two-dimensional 0-surgery with the observed dynamics. To do this, we first present a schematic model which extends the formal definition to a continuous process caused by local forces. Next, for modeling phenomena which do not happen on surfaces but are three-dimensional, we fill the interior space by defining the notion of solid two-dimensional 0-surgery. Finally, we connect these new theoretical concepts with a dynamical system and present it as a model for both two-dimensional 0-surgery and natural phenomena exhibiting it. We hope that through this study, topology and dynamics of many natural phenomena will be better understood.

The main result of this paper is the construction of two Hyperbolic manifolds, $M_1$ and $M_2$, with several remarkable properties:

• ⁽¹⁾Every closed orientable $3$-manifold is homeomorphic to the quotient space of the action of a group of order $16$ on some covering space of $M_1$ or $M_2$.
• ⁽²⁾$M_1$ and $M_2$ are tesselated by 16 dodecahedra such that the pentagonal faces of the dodecahedra fit together in a certain way.
• ⁽³⁾There are 12 closed non-orientable hyperbolic surfaces of Euler characteristic $-2$ each of which is tesselated by regular right angled pentagons and embedded in $M_1$ or $M_2$. The union of the pentagonal faces of the tesselating dodecahedra equals the union of the 12 images of the embedded surfaces of Euler characteristic $-2$.

Based on symbolic dynamics of Lorenz maps, we prove that, provided one conjecture due to Morton is true, then Lorenz knots associated to orbits of points in the renormalization intervals of Lorenz maps with reducible kneading invariant of type $(X,Y)\ast S$, where the sequences $X$ and $Y$ are Farey neighbors verifying some conditions, are hyperbolic.

In this paper, we first present the construction of the new 2-variable classical link invariants arising from the Yokonuma–Hecke algebras ${\mathrm{\text{Y}}}_{d,n}(q)$, which are not topologically equivalent to the Homflypt polynomial. We then present the algebra ${\mathrm{\text{FTL}}}_{d,n}(q)$ which is the appropriate Temperley–Lieb analogue of ${\mathrm{\text{Y}}}_{d,n}(q)$, as well as the related 1-variable classical link invariants, which in turn are not topologically equivalent to the Jones polynomial. Finally, we present the algebra of braids and ties which is related to the Yokonuma–Hecke algebra, and also its quotient, the partition Temperley–Lieb algebra ${\mathrm{\text{PTL}}}_n(q)$ and we prove an isomorphism of this algebra with a subalgebra of ${\mathrm{\text{FTL}}}_{d,n}(q)$.

Checkerboard framings are an extension of checkerboard colorings for virtual links. From checkerboard framings, we obtain an independent invariant of virtual links: the cut point number. Checkerboard framings and cut points can be used as a tool to extend other classical invariants to virtual links.

Virtual knot theory is a generalization of knot theory which is based on Gauss chord diagrams and link diagrams on closed oriented surfaces. A twisted knot is a generalization of a virtual knot, which corresponds to a link diagram on a possibly non-orientable surface. In this paper, we discuss an invariant of twisted links which is obtained from the JKSS invariant of virtual links by use of double coverings. We also discuss some properties of double covering diagrams.

In this paper, we first investigate the minimal sufficient sets of colors for $p=11$ and $13$. For odd prime $p$ and any $p$-colorable link $L$ with $detL\ne 0$, we give alternative proofs of ${\mathrm{\text{mincol}}}_{p}L\ge 5$ for $p\ge 11$ and ${\mathrm{\text{mincol}}}_{p}L\ge 6$ for $p\ge 17$. We also elaborate on equivalence classes of sets of distinct colors (on a given modulus) and prove that there are two such classes of five colors modulo $11$, and only one such class of five colors modulo 13.

We revisit the issue of the existence of infinitely many distinct prime knots with the same Alexander invariant. We present infinitely many distinct families, each family made up of infinitely many distinct knots. Within each family, the Alexander invariant is the same. Unlike, other examples in the literature, ours are elementary and based on a sub-collection of pretzel knots with three tassels.