Twisted Links Research Articles

From mind‐deadness to mindedness, from collaboration to cooperation

AbstractLooking at the October 7, 2023, events in Israel, this paper suggests a unique configuration of Bion's minus links, called "broken links." The broken links not only represent a negation of a connection—but also a negation of the object. When the link of Love is broken, it is replaced by a unique formation of Love without an object, in which the lack‐of‐object installs itself as the sole object, replacing the work of mourning with pseudo, blank mourning. When the link of Hate is broken, the result is a murderous version of hatred, one that exceeds any proportion or context. When the link of Knowledge is broken, what ensues is a language that functions as a sealing material: instead of signifying and making differences, it drives to overgeneralize, erase differences, and remove distinctions, losing its metaphorical qualities and reducing to its literal meaning. These broken links lead to another twisted link: the link of solidarity, turning positive solidarity, which is based on cooperation—into negative solidarity, based on collaboration. The discussion suggests the shift from the state of mind‐deadness, characterizing broken links on both sides of the conflict, to the state of mindedness, in which the links are reclaimed and restored.

Geometric triangulations and highly twisted links

Geometric triangulations and highly twisted links

Twisted intersection colorings, invariants and double coverings of twisted links

Twisted links are a generalization of classical links and correspond to stably equivalence classes of links in thickened surfaces. In this paper, we introduce twisted intersection colorings of a diagram and construct two invariants of a twisted link using such colorings. As an application, we show that there exist infinitely many pairs of twisted links such that for each pair the two twisted links are not equivalent but their double coverings are equivalent. We also introduce a method of constructing a pair of twisted links whose double coverings are equivalent.

One conjecture on cut points of virtual links and the arrow polynomial of twisted links

Checkerboard framings are an extension of checkerboard colorings for virtual links. According to checkerboard framings, in 2017, Dye obtained 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. We prove that one of the conjectures in Dye’s paper is correct. Moreover, we analyze the connection and difference between checkerboard framing obtained from virtual link diagram by adapting cut points and twisted link diagram obtained from virtual link diagram by introducing bars. By adjusting the normalized arrow polynomial of virtual links, we generalize it to twisted links. We show that it is an invariant for twisted link. Finally, we figure out three characteristics of the normalized arrow polynomial of a checkerboard colorable twisted link, which is a tool of detecting checkerboard colorability of a twisted link. The latter two characteristics are the same as in the case of checkerboard colorable virtual link diagram.

Virtual links which are equivalent as twisted links

A virtual link is a generalization of a classical link that is defined as an equivalence class of certain diagrams, called virtual link diagrams. It is further generalized to a twisted link. Twisted links are in one-to-one correspondence with stable equivalence classes of links in oriented thickenings of (possibly nonorientable) closed surfaces. By definition, equivalent virtual links are also equivalent as twisted links. In this paper, we discuss when two virtual links are equivalent as twisted links, and give a necessary and sufficient condition for this to be the case.

Dihedral semigroups, their defining relations and an application to describing knot semigroups of rational links

It is known that a knot (link) is rational if and only if its $$\pi $$ -orbifold group is dihedral. A semigroup version of this result has been formulated as a conjecture. Working towards proving the conjecture, we describe certain semigroups associated with twist links, clarify how these semigroups are related to dihedral groups and find defining relations of these semigroups.

A twisted link invariant derived from a virtual link invariant

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.

Link invariant and $G_2$ web space

In this paper, we reconstruct Kuperberg’s $G_2$ web space [5, 6]. We introduce a new web diagram (a trivalent graph with only double edges) and new relations between Kuperberg’s web diagrams and the new web diagram. Using the web diagrams, we give crossing formulas for the $R$-matrices associated to some irreducible representations of $U_q(G_2)$ and calculate $G_2$ quantum link invariants for generalized twist links.

Double coverings of twisted links

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.

Robustness of discrete semifluxons in closed Bose–Hubbard chains

We present the properties of the ground state and low-energy excitations of Bose–Hubbard chains with a geometry that varies from open to closed and with a tunable twisted link. In the vicinity of the symmetric π-flux case the system behaves as an interacting gas of discrete semifluxons for finite chains and interactions in the Josephson regime. The energy spectrum of the system is studied by direct diagonalization and by solving the corresponding Bogoliubov–de Gennes equations. The atom–atom interactions are found to enhance the presence of strongly correlated macroscopic superpositions of semifluxons.

Character varieties of double twist links

We compute both natural and smooth models for the $SL_2(\mathbb C)$ character varieties of the two component double twist links, an infinite family of two-bridge links indexed as $J(k,l)$. For each $J(k,l)$, the component(s) of the character variety containing characters of irreducible representations are birational to a surface of the form $C\times \mathbb C$ where $C$ is a curve. The same is true of the canonical component. We compute the genus of this curve, and the degree of irrationality of the canonical component. We realize the natural model of the canonical component of the $SL_2(\mathbb C)$ character variety of the $J(3,2m+1)$ links as the surface obtained from $\mathbb{P}^1\times \mathbb{P}^1$ as a series of blow-ups.

Essential surfaces in highly twisted link complements

We prove that in the complement of a highly twisted link, all closed, essential, meridionally incompressible surfaces must have high genus. The genus bound is proportional to the number of crossings per twist region. A similar result holds for surfaces with meridional boundary: such a surface either has large negative Euler characteristic, or is an n-punctured sphere visible in the diagram.

Surface pole bracket polynomials of virtual knots and twisted knots

Dye and Kauffman defined surface bracket polynomials for virtual links by use of surface states, and found a relationship between the surface states and the minimal genus of a surface in which a virtual link diagram is realized. They and Miyazawa independently defined a multivariable polynomial invariant of virtual links. This invariant is deeply related to the surface states. In this paper, we introduce the notion of surface pole bracket polynomials for link diagrams in closed surfaces, as a generalization of surface bracket polynomials by Dye and Kauffman. The polynomials induce the invariant of twisted links defined by the author before as a generalization of Dye, Kauffman and Miyazawa's polynomial invariant. Furthermore we discuss a relationship between curves in surface pole states and variables of the polynomial invariant.

INDEX POLYNOMIAL INVARIANTS OF TWISTED LINKS

Bourgoin defined the notion of a twisted link. In a sense, it is a non-orientable version of a virtual link. Im, Lee and Lee defined a polynomial invariant of a virtual link by using the virtual intersection index. In this paper, we give an alternative definition of index polynomial by using indices of real crossings and extend it to a twisted links.

Polynomial invariants and quandles of twisted links

Bourgoin defined the notion of a twisted link which corresponds to a stable equivalence class of links in oriented thickenings. It is a generalization of a virtual link. Some invariants of virtual links are extended for twisted links including the knot group and the Jones polynomial. In this paper, we generalize a multivariable polynomial invariant of a virtual link to a twisted link. We also introduce a quandle of a twisted link.

Generating family invariants for Legendrian links of unknots

Theory is developed for linear-quadratic at infinity generating families for Legendrian knots in R^3. It is shown that the unknot with maximal Thurston--Bennequin invariant of -1 has a unique linear-quadratic at infinity generating family, up to fiber-preserving diffeomorphism and stabilization. From this, invariant generating family polynomials are constructed for 2-component Legendrian links where each component is a maximal unknot. Techniques are developed to compute these polynomials, and computations are done for two families of Legendrian links: rational links and twist links. The polynomials allow one to show that some topologically equivalent links with the same classical invariants are not Legendrian equivalent. It is also shown that for these families of links the generating family polynomials agree with the polynomials arising from a linearization of the differential graded algebra associated to the links.