Alexander invariants for virtual knots

Giving a presentation of the group of a 2-braid virtual knot or link, we consider the groups of three families of 2-braid virtual knots. Each of them has a certain feature; for example, we can show: for any positive integer N, there exists a virtual knot group with an element of order N. It is known that the collection of the virtual knot groups is the same as that of the ribbon T2-knot groups. Using our examples we discuss the relationship among the virtual knot groups and other knot groups such as ribbon S2-knot groups, S2-knot groups, T2-knot groups, and S3-knot groups.

Milnor's $\bar{\mu}$-invariants of links in the $3$-sphere $S^3$ vanish on any link concordant to a boundary link. In particular, they are trivial on any knot in $S^3$. Here we consider knots in thickened surfaces $\Sigma \times [0,1]$, where $\Sigma$ is closed and oriented. We construct new concordance invariants by adapting the Chen-Milnor theory of links in $S^3$ to an extension of the group of a virtual knot. A key ingredient is the Bar-Natan $\textit{Zh}$ map, which allows for a geometric interpretation of the group extension. The group extension itself was originally defined by Silver-Williams. Our extended $\bar{\mu}$-invariants obstruct concordance to homologically trivial knots in thickened surfaces. We use them to give new examples of non-slice virtual knots having trivial Rasmussen invariant, graded genus, affine index (or writhe) polynomial, and generalized Alexander polynomial. Furthermore, we complete the slice status classification of all virtual knots up to five classical crossings and reduce to four (out of 92800) the number of virtual knots up to six classical crossings having unknown slice status. Our main application is to Turaev's concordance group $\mathscr{VC}$ of long knots on surfaces. Boden and Nagel proved that the concordance group $\mathscr{C}$ of classical knots in $S^3$ embeds into the center of $\mathscr{VC}$. In contrast to the classical knot concordance group, we show $\mathscr{VC}$ is not abelian; answering a question posed by Turaev.

Virtual links can be regarded as links embedded in thickened surfaces. Virtual link groups correspond to the fundamental groups of the complement spaces of the virtual links in thickened surfaces. In this paper, we demonstrate that for a virtual link [Formula: see text] with supporting genus [Formula: see text], the complement space of [Formula: see text] is irreducible and forms a [Formula: see text]-space. Furthermore, we establish that under certain hypotheses, there is an isomorphism between the intersecting subgroup and the symmetric commutator subgroup of all normal closures of meridians in the virtual link group.

Study of stable isotopy classes of a finite collection of immersed circles without triple or higher intersections on closed oriented surfaces is considered as a planar analogue of virtual knot theory, a far reaching generalisation of classical knot theory. Recent works have established Alexander and Markov theorems in the planar setting. In the classical case, the role of groups is played by twin groups, a class of right-angled Coxeter groups. A new class of groups called virtual twin groups, that extends twin groups in a natural way, plays the role of groups in the virtual case. The virtual twin group V Tn contains the pure virtual twin group P V Tn, a planar analogue of the pure Artin braid group. In this paper, we prove that the pure virtual twin group P V Tn is an irreducible right-angled Artin group with trivial center and give it's precise presentation. We show that P V Tn has a decomposition as an iterated semi-direct product of infinite rank free groups. We give a complete description of the automorphism group of P V Tn and establish splitting of natural exact sequences of automorphism groups. As applications, we show that V Tn is residually finite and P V Tn has the R∞-property.

In this paper, we define the virtual Alexander polynomial following the works of Boden et al. (2016) [Alexander invariants for virtual knots, J. Knot Theory Ramications 24(3) (2015) 1550009] and Kaestner and Kauffman [Parity biquandles, in Knots in Poland. III. Part 1, Banach Center Publications, Vol. 100 (Polish Academy of Science Mathematical Institute, Warsaw, 2014), pp. 131–151]. The properties of this invariant are explored and some examples are computed. In particular, the invariant demonstrates that many virtual knots cannot be unknotted by crossing changes on only odd crossings.

foreword by Ronald Rice The vast, international web of computer networks that is the Internet offers millions of users the opportunity to exchange electronic mail, photographs, and sound clips; to search databases for books, CDs, cars, and term papers; to participate in real-time audio- and video-conferencing; and to shop for products both virtual and physical. This huge conglomerate of links, hyperlinks, and virtual links is not just a technology for linking computers--it is a medium for communication.The convergence of computer and communication technologies creates a social convergence as well. People meet in chat rooms and discussion groups to converse on everything from auto mechanics to postmodern art. Networked groups form virtually and on-the-fly, as common interests dictate. Like interpersonal communication, the networks are participatory, their content made up by their audience. Like mass-mediated communication, they involve large audiences. But the networks are neither purely interpersonal nor purely mass--they are a new phenomenon.Network and Netplay addresses the mutual influences between information technology and group formation and development, to assess the impact of computer-mediated communications on both work and play. Areas discussed include the growth and features of the Internet, network norms and experiences, and the essential nature of network communication.Contributors : Michael Berthold, Lee Li-Jen Chen, Richard Coyne, Brenda Danet, Patrick Doyle, Brian R. Gaines, Barbara Hayes-Roth, Steve Jones, Sandra Katzman, Edward Mabry, Richard MacKinnon, Margaret McLaughlin, Sid Newton, Kerry Osborne, Sheizaf Rafaeli, Yehudit Rosenbaum-Tamari, Lucia Ruedenberg, Christine Smith, Fay Sudweeks, Alexander Voiskounsky, Diane Witmer.

Non-abelian tensor product and circular orderability of groups

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.

In this paper, we construct quantum invariants for knotoid diagrams in $\mathbb{R}^2$. The diagrams are arranged with respect to a given direction in the plane ({\it Morse knotoids}). A Morse knotoid diagram can be decomposed into basic elementary diagrams each of which is associated to a matrix that yields solutions of the quantum Yang-Baxter equation. We recover the bracket polynomial, and define the rotational bracket polynomial, the binary bracket polynomial, the Alexander polynomial, the generalized Alexander polynomial and an infinity of specializations of the Homflypt polynomial for Morse knotoids via quantum state sum models.

Let [Formula: see text] be the algebra of Laurent polynomials in the variable [Formula: see text] and let [Formula: see text] be the algebra of Laurent polynomials in the variable [Formula: see text] and standard polynomials in the variables [Formula: see text] For [Formula: see text] we denote by [Formula: see text] the virtual braid group on [Formula: see text] strands. We define two towers of algebras [Formula: see text] and [Formula: see text] in terms of diagrams. For each [Formula: see text] we determine presentations for both, [Formula: see text] and [Formula: see text]. We determine sequences of homomorphisms [Formula: see text] and [Formula: see text], we determine Markov traces [Formula: see text] and [Formula: see text], and we show that the invariants for virtual links obtained from these Markov traces are the [Formula: see text]-polynomial for the first trace and the arrow polynomial for the second trace. We show that, for each [Formula: see text] the standard Temperley–Lieb algebra [Formula: see text] embeds into both, [Formula: see text] and [Formula: see text], and that the restrictions to [Formula: see text] of the two Markov traces coincide.

Two virtual link diagrams are homotopic if one may be transformed into the other by a sequence of virtual Reidemeister moves, classical Reidemeister moves, and self crossing changes. We recall the pure virtual braid group. We then describe the set of pure virtual braids that are homotopic to the identity braid.