Virtual Crossings Research Articles

Virtual index cocycles and invariants of virtual links

Virtual index cocycle is the 1-cochain that counts virtual crossings in the arcs of a virtual link diagram. We show how this cocycle can be used to reformulate and unify some known invariants of virtual links.

Virtual mosaic knot theory

Mosaic diagrams for knots were first introduced in 2008 by Lomanoco and Kauffman for the purpose of building a quantum knot system. Since then, many others have explored the structure of these knot mosaic diagrams, as they are interesting objects of study in their own right. Knot mosaics have been generalized by Garduño to virtual knots, by including an additional tile type to represent virtual crossings. There is another interpretation of virtual knots, however, as knot diagrams on surfaces, which inspires this work. By viewing classical mosaic diagrams as [Formula: see text]-gons and gluing edges of these polygons, we obtain knots on surfaces that can be viewed as virtual knots. These virtual mosaics are our present objects of study. In this paper, we provide a set of moves that can be performed on virtual mosaics that preserve knot and link type, we show that any virtual knot or link can be represented as a virtual mosaic, and we provide several computational results related to virtual mosaic numbers for small classical and virtual knots.

Redrawing the Boundary: From Carlos Fuentes’s La frontera de cristal (1995) to Rosaura Sánchez and Beatrice Pita’s Lunar Braceros 2125-2148 (2009)*

This article traces the workings of the border in Carlos Fuentes’ La frontera de cristal , Alejandro Morales’ The Rag Doll Plagues , Alex Rivera’s film Sleep Dealer , and Rosaura Sanchez and Beatrice Pita’s Lunar Braceros 2125-2140 . All the narratives can be considered borderland texts that illustrate the dialogue between literature and issues such as globalization, technology and colonial relations of power. While the border is still a place of detention and interdiction in Fuentes’ novel, it mutates into virtual crossings in The Rag Doll Plagues and Sleep Dealer , to return to barbed wire in Sanchez and Pita’s novella. From the logic of the border as a mechanism that may open or close, the article moves on to address the liminal situation of those who, although situated within the new versions of the nation-state, are considered permanent outsiders. This redrawing of boundaries allows us to revisit traditional categories of distinction such as the inside and the outside, and is evidence of the way colonial models of subjugation boomerang to the present.

Biquandle virtual brackets

We introduce an infinite family of quantum enhancements of the biquandle counting invariant which we call biquandle virtual brackets. Defined in terms of skein invariants of biquandle colored oriented knot and link diagrams with values in a commutative ring [Formula: see text] using virtual crossings as smoothings, these invariants take the form of multisets of elements of [Formula: see text] and can be written in a “polynomial” form for convenience. The family of invariants defined herein includes as special cases all quandle and biquandle 2-cocycle invariants, all classical skein invariants (Alexander–Conway, Jones, HOMFLYPT and Kauffman polynomials) and all biquandle bracket invariants defined in [S. Nelson, M. E. Orrison and V. Rivera, Quantum enhancements and biquandle brackets, J. Knot Theory Ramifications 26(5) (2017) 1750034] as well as new invariants defined using virtual crossings in a fundamental way, without an obvious purely classical definition.

The generalized Yamada polynomials of virtual spatial graphs

Knot theory can be generalized to virtual knot theory and spatial graph theory. In 2007, Fleming and Mellor combined and generalized them to virtual spatial graph theory in a combinatorial way and extended the Yamada polynomial from spatial graphs to virtual spatial graphs by ignoring virtual crossings. In this paper, we introduce a topological definition of the virtual spatial graph which is similar to that of a virtual link. Our main goal is to generalize the Yamada polynomial from spatial graphs to virtual spatial graphs such that it could capture the virtuality. It is realized via adding a new variable and integrating the topological Tutte polynomial instead of the Tutte polynomial into our polynomial. We define the generalized Yamada polynomial for virtual spatial graphs via their diagrams and prove that it can be normalized to be a rigid vertex isotopic invariant of virtual spatial graphs and to be a pliable vertex isotopic invariant for virtual spatial graphs with maximum degree at most 3. We consider the connection and difference between our generalized Yamada polynomial and the Dubrovnik polynomial of a classical link, and proved the generalized Yamada polynomial specializes to a version of the Dubrovnik polynomial for classical links which can be used to sometimes detect the virtuality of virtual links. We also prove the generalized Yamada polynomial specializes to the bracket polynomial of the associated diagram obtained from a virtual spatial graph diagram with the Jones–Wenzl projector P2 acting on it, which can be used to write a program for calculating this special parametrization of the generalized Yamada polynomial based on Mathematica Code.

Alexander invariants of ribbon tangles and planar algebras

Ribbon tangles are proper embeddings of tori and cylinders in the $4$-ball~$B^4$, bounding $3$-manifolds with only ribbon disks as singularities. We construct an Alexander invariant $\mathsf{A}$ of ribbon tangles equipped with a representation of the fundamental group of their exterior in a free abelian group $G$. This invariant induces a functor in a certain category $\mathsf{R}ib_G$ of tangles, which restricts to the exterior powers of Burau-Gassner representation for ribbon braids, that are analogous to usual braids in this context. We define a circuit algebra $\mathsf{C}ob_G$ over the operad of smooth cobordisms, inspired by diagrammatic planar algebras introduced by Jones, and prove that the invariant $\mathsf{A}$ commutes with the compositions in this algebra. On the other hand, ribbon tangles admit diagrammatic representations, throught welded diagrams. We give a simple combinatorial description of $\mathsf{A}$ and of the algebra $\mathsf{C}ob_G$, and observe that our construction is a topological incarnation of the Alexander invariant of Archibald. When restricted to diagrams without virtual crossings, $\mathsf{A}$ provides a purely local description of the usual Alexander poynomial of links, and extends the construction by Bigelow, Cattabriga and the second author.

Virtual unknotting numbers of certain virtual torus knots

The virtual unknotting number of a virtual knot is the minimal number of crossing changes that makes the virtual knot to be the unknot, which is defined only for virtual knots virtually homotopic to the unknot. We focus on the virtual knot obtained from the standard [Formula: see text]-torus knot diagram by replacing all crossings on one overstrand into virtual crossings and prove that its virtual unknotting number is equal to the unknotting number of the [Formula: see text]-torus knot, that is, [Formula: see text].

Linking invariants of even virtual links

A virtual link diagram is even if the virtual crossings divide each component into an even number of arcs. The set of even virtual link diagrams is closed under classical and virtual Reidemeister moves, and it contains the set of classical link diagrams. For an even virtual link diagram, we define a certain linking invariant which is similar to the linking number. In contrast to the usual linking number, our linking invariant is not preserved under the forbidden moves. In particular, for two fused isotopic even virtual link diagrams, the difference between the linking invariants of them gives a lower bound of the minimal number of forbidden moves needed to deform one into the other. Moreover, we give an example which shows that the lower bound is best possible.

Virtual crossings and a filtration of the triply graded link homology of a link diagram

A filtration of Soergel bimodules by virtual crossing bimodules extends to Rouquier’s complexes associated with braid words. We show that these complexes are invariant up to filtered homotopy with respect to the second Reidemeister move, and the filtration of the triply graded link diagram homology, constructed by Khovanov through the application of the Hochschild homology, is invariant under Markov moves. We also prove that the homotopy equivalence of the complexes of braid words related by the third Reidemeister move violates filtration by at most two units.

Using smartphone technology to deliver a virtual pedestrian environment: usability and validation.

Various programs effectively teach children to cross streets more safely, but all are labor- and cost-intensive. Recent developments in mobile phone technology offer opportunity to deliver virtual reality pedestrian environments to mobile smartphone platforms. Such an environment may offer a cost- and labor-effective strategy to teach children to cross streets safely. This study evaluated usability, feasibility, and validity of a smartphone-based virtual pedestrian environment. A total of 68 adults completed 12 virtual crossings within each of two virtual pedestrian environments, one delivered by smartphone and the other a semi-immersive kiosk virtual environment. Participants completed self-report measures of perceived realism and simulator sickness experienced in each virtual environment, plus self-reported demographic and personality characteristics. All participants followed system instructions and used the smartphone-based virtual environment without difficulty. No significant simulator sickness was reported or observed. Users rated the smartphone virtual environment as highly realistic. Convergent validity was detected, with many aspects of pedestrian behavior in the smartphone-based virtual environment matching behavior in the kiosk virtual environment. Anticipated correlations between personality and kiosk virtual reality pedestrian behavior emerged for the smartphone-based system. A smartphone-based virtual environment can be usable and valid. Future research should develop and evaluate such a training system.

Bikei invariants and gauss diagrams for virtual knotted surfaces

Marked vertex diagrams provide a combinatorial way to represent knotted surfaces in [Formula: see text]; including virtual crossings allows for a theory of virtual knotted surfaces and virtual cobordisms. Biquandle counting invariants are defined only for marked vertex diagrams representing knotted orientable surfaces; we extend these invariants to all virtual marked vertex diagrams by considering colorings by involutory biquandles, also known as bikei. We introduce a way of representing marked vertex diagrams with Gauss diagrams and use these to characterize orientability.

Simulation of Van Allen Probes plasmapause encounters

AbstractWe use anE×B‐driven plasmapause test particle (PTP) simulation to provide global contextual information for in situ measurements by the Van Allen Probes (Radiation Belt Storm Probes (RBSP)) during 15–20 January 2013. During 120 h of simulation time beginning on 15 January, geomagnetic activity produced three plumes. The third and largest simulated plume formed during enhanced convection on 17 January, and survived as a rotating, wrapped, residual plume for tens of hours. To validate the simulation, we compare its output with RBSP data. Virtual RBSP satellites recorded 28 virtual plasmapause encounters during 15–19 January. For 26 of 28 (92%) virtual crossings, there were corresponding actual RBSP encounters with plasmapause density gradients. The mean difference in encounter time between model and data is 36 min. The mean model‐data difference in radial location is 0.40 ± 0.05 RE. The model‐data agreement is better for strong convection than for quiet or weakly disturbed conditions. On 18 January, both RBSP spacecraft crossed a tenuous, detached plasma feature at approximately the same time and nightside location as a wrapped residual plume, predicted by the model to have formed 32 h earlier on 17 January. The agreement between simulation and data indicates that the model‐provided global information is adequate to correctly interpret the RBSP density observations.

A METHOD OF ENCODING GENERALIZED LINK DIAGRAMS

We describe a method of encoding various types of link diagrams, including those with classical, flat, rigid, welded, and virtual crossings. We show that this method may be used to encode link diagrams, up to equivalence, in a notation whose length is a cubic function of the number of "riser marks". For classical knots, the minimal number of such marks is twice the bridge index, and a classical knot diagram in minimal bridge form with bridge index b may be encoded in space [Formula: see text]. A set of moves on the notation is defined. As a demonstration of the utility of the notation we give another proof that the Kishino virtual knot is non-classical.

VIRTUAL CROSSING NUMBERS FOR VIRTUAL KNOTS

The aim of the present paper is to prove that the minimal number of virtual crossings for some families of virtual knots grows quadratically with respect to the minimal number of classical crossings. All previously known estimates for virtual crossing number ([D. M. Afanasiev, Refining the invariants of virtual knots by using parity, Sb. Math.201(6) (2010) 3–18; H. A. Dye and L. H. Kauffman, Virtual crossing numbers and the arrow polynomial, in The Mathematics of Knots. Theory and Applications, eds. M. Banagl and D. Vogel (Springer-Verlag, 2010); S. Satoh and Y. Tomiyama, On the crossing number of a virtual knot, Proc. Amer. Math. Soc.140 (2012) 367–376] and so on) were principally no more than linear in the number of classical crossings (or, what is the same, in the number of edges of a virtual knot diagram) and no virtual knot was found with virtual crossing number greater than the classical crossing number.

A Categorical Structure for the Virtual Braid Group

This article gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. This point of view makes many relationships between the virtual braid group and the pure virtual braid group apparent, and makes representations of the virtual braid groups and pure virtual braid groups via solutions to the algebraic Yang–Baxter Equation equally transparent. In this categorical framework, the virtual braid group has nothing to do with the plane and nothing to do with virtual crossings. It is a natural group associated with the structure of algebraic braiding.

ON A GENERALIZATION OF THE ALEXANDER POLYNOMIAL FOR LONG VIRTUAL KNOTS

We construct a new invariant polynomial for long virtual knots. It is a generalization of the Alexander polynomial. We designate it as ζ by meaning an analogy with ζ-polynomial for virtual links. The degree of ζ-polynomial estimates the virtual crossing number. We describe some application of ζ-polynomial for the study of minimal long virtual diagrams with respect to the number of virtual crossings.

Gauss paragraphs of classical links and a characterization of virtual link groups

AbstractA classical link in 3-space can be represented by a Gauss paragraph encoding a link diagram in a combinatorial way. A Gauss paragraph may code not a classical link diagram, but a diagram with virtual crossings. We present a criterion and a linear algorithm detecting whether a Gauss paragraph encodes a classical link. We describe Wirtinger presentations realizable by virtual link groups.

BIQUANDLES FOR VIRTUAL KNOTS

This paper studies the biquandle as an invariant of virtual knots and links, analyzing its structure via composition of biquandle morphisms where elementary morphisms are associated with the crossings and virtual crossings in the diagram.

MAGNETIC GRAPHS AND AN INVARIANT FOR VIRTUAL LINKS

Introducing a type of graph named a virtual magnetic graph diagram, we define a virtual link invariant, which generalizes the Jones-Kauffman polynomial and the 2-variable polynomial invariant defined by N. Kamada and the author. We show that this invariant can evaluate the number of virtual crossings of a virtual link.

VIRTUAL CROSSING REALIZATION

We study virtual isotopy sequences with classical initial and final diagrams, asking when such a sequence can be changed into a classical isotopy sequence by replacing virtual crossings with classical crossings. An example of a sequence for which no such virtual crossing realization exists is given. A conjecture on conditions for realizability of virtual isotopy sequences is proposed, and a sufficient condition for realizability is found. The conjecture is reformulated in terms of 2-knots and knots in thickened surfaces.