Narrow Results

We introduce an algebra ℤ[X, S] associated to a pair X, S of a virtual birack X and X-shadow S. We use modules over ℤ[X, S] to define enhancements of the virtual birack shadow counting invariant, extending the birack shadow module invariants to virtual case. We repeat this construction for the twisted virtual case. As applications, we show that the new invariants can detect orientation reversal and are not determined by the knot group, the Arrow polynomial and the Miyazawa polynomial, and that the twisted version is not determined by the twisted Jones polynomial.

This paper presents a combined framework of Multi-Objective Generic Algorithm (MOGA) and Monte Carlo Simulation (MCS) in order to improve backbone topology by leveraging the Virtual Link (VL) system in an hierarchical Link-State (LS) routing domain. Given that the sound backbone topology structure has a great impact on the overall routing performance in a hierarchical LS domain, the importance of this research is evident. The proposed decision model is to find an optimal configuration of VLs that properly meets two-pronged engineering goals in installing and maintaining VLs: i.e., operational costs and network reliability. The experiment results clearly indicates that it is essential to the effective operations of hierarchical LS routing domain to consider not only engineering aspects but also specific benefits from systematical layout of VLs, thereby presenting the validity of the decision model and MOGA with MCS.

In 2002, Hrencecin and Kauffman defined a filamentation invariant on oriented chord diagrams that may determine whether the corresponding flat virtual knot diagrams are non-trivial. A virtual knot diagram is non-classical if its related flat virtual knot diagram is non-trivial. Hence filamentations can be used to detect non-classical virtual knots. We extend these filamentation techniques to virtual links with more than one component. We also give examples of virtual links that they can detect as non-classical.

The Jones polynomial is a well-defined invariant of virtual links. We observe the effect of a generalised mutation M of a link on the Jones polynomial. Using this, we describe a method for obtaining invariants of links which are also invariant under M.

The Jones polynomial of welded links is not well-defined in ℤ[q^1/4, q^-1/4]. Taking M = F_o allows us to pass to a quotient of ℤ[q^1/4, q^-1/4] in which the Jones polynomial is well-defined. We get the same result for M = F_u, so in fact, the Jones polynomial in this ring defines a fused isotopy invariant. We show it is non-trivial and compute it for links with one or two components.

We study the quandle counting invariant for a certain family of finite quandles with trivial orbit subquandles. We show how these invariants determine the linking number of classical two-component links up to sign.

We introduce an additional arrow structure on ribbon graphs. We extend the dichromatic polynomial to ribbon graphs with this structure. This extended polynomial satisfies the contraction–deletion relations and behaves naturally with respect to the partial duality of ribbon graphs. From a virtual link, we construct an arrow ribbon graph whose extended dichromatic polynomial specializes to the arrow polynomial of the virtual link recently introduced by H. Dye and L. Kauffman. This result generalizes the classical Thistlethwaite theorem to the arrow polynomial of virtual links.

We introduce two kinds of structures, called v-structures and t-structures, on biquandles. These structures are used for colorings of diagrams of virtual links and twisted links such that the numbers of colorings are invariants. Given a biquandle or a quandle, we give a method of constructing a biquandle with these structures. Using the numbers of colorings, we show that Bourgoin's twofoil and non-orientable virtual m-foils do not represent virtual links.

This paper is an introduction to virtual knot theory and an exposition of new ideas and constructions, including the parity bracket polynomial, the arrow polynomial, the parity arrow polynomial and categorifications of the arrow polynomial.

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.

Generalized Reidemeister moves provide an extended set of moves to work with virtual knots and links. We introduce virtual tangle moves, generalization of classical rational tangle moves and show that such generalizations are essential to develop new invariants of virtual knots and links. We show that every 2-algebraic virtual link is a virtual 4-move equivalent to a trivial link or Hopf link. The properties of virtual tangle move are analyzed on few existing invariants associated with virtual knots and links.

This paper studies rotational virtual knot theory and its relationship with quantum link invariants. Every quantum link invariant for classical knots and links extends to an invariant of rotational virtual knots and links. We give examples of non-trivial rotational virtuals that are undectable by quantum invariants.

We introduce the even index polynomial of virtual link diagrams and some even state sum polynomials which are the even Jones–Kauffman, even Miyazawa and even arrow polynomials. By simple computation, we can distinguish virtual links with these even polynomial invariants. Also, we give examples so that these invariants are different from the original ones.

The Wirtinger number of a virtual link is the minimum number of generators of the link group over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. We prove that the Wirtinger number of a virtual link equals its virtual bridge number. Since the Wirtinger number is algorithmically computable, it gives a more effective way to calculate an upper bound for the virtual bridge number from a virtual link diagram. As an application, we compute upper bounds for the virtual bridge numbers and the quandle counting invariants of virtual knots with 6 or fewer crossings. In particular, we found new examples of nontrivial virtual bridge number one knots, and by applying Satoh’s Tube map to these knots we can obtain nontrivial weakly superslice links.

For any virtual link $L=S\cup T$ that may be decomposed into a pair of oriented $n$-tangles $S$ and $T$, an oriented local move of type $T\mapsto \varphi (T)$ is a replacement of $T$ with the $n$-tangle $\varphi (T)$ in a way that preserves the orientation of $L$. After developing a general decomposition for the Jones polynomial of the virtual link $L=S\cup T$ in terms of various (modified) closures of $T$, we analyze the Jones polynomials of virtual links $L_1,L_2$ that differ via a local move of type $T\mapsto \varphi (T)$. Succinct divisibility conditions on $V(L_1)-V(L_2)$ are derived for broad classes of local moves that include the $\mathrm{\Delta }$-move and the double-$\mathrm{\Delta }$-move as special cases. As a consequence of our divisibility result for the double-$\mathrm{\Delta }$-move, we introduce a necessary condition for any pair of classical knots to be $S$-equivalent.

It is known that the Kauffman–Murasugi–Thislethwaite type inequality becomes an equality for any (possibly virtual) adequate link diagram. We refine this condition. As an application we obtain a criterion for virtual link diagram with exactly one virtual crossing to represent a properly virtual link.