Narrow Results

We introduce the pole diagram, which helps to retrieve information from a knot diagram when we smooth crossings. By using the notion, we define a bracket polynomial for the Miyazawa polynomial. The bracket polynomial gives a simple definition and evaluation for the Miyazawa polynomial. Then we show that the virtual crossing number of a virtualized alternating link is determined by its diagram. Furthermore, we construct infinitely many virtual link diagrams which attain the minimal real and virtual crossing numbers together.

We introduce a new infinite family of enhancements of the biquandle homset invariant called biquandle arrow weights. These invariants assign weights in an abelian group to intersections of arrows in a Gauss diagram representing a classical or virtual knot depending on the biquandle colors associated to the arrows. We provide examples to show that the enhancements are nontrivial and proper, i.e. not determined by the homset cardinality.

Following the suggestion of Alexei Morozov, Andrey Morozov and Anton Morozov, Phys. Lett. B737, 48 (2014), arXiv:1407.6319, to lift the knot polynomials for virtual knots and links from Jones to HOMFLY, we apply the evolution method to calculate them for an infinite series of twist-like virtual knots and antiparallel two-strand links. Within this family one can check topological invariance and understand how differential hierarchy is modified in virtual case. This opens a way towards a definition of colored (not only cabled) knot polynomials, though problems still persist beyond the first symmetric representation.

Virtual knots, defined by Kauffman, provide a natural generalization of classical knots. Most invariants of knots extend in a natural way to give invariants of virtual knots. In this paper we study the fundamental groups of virtual knots and observe several new and unexpected phenomena.

In the classical setting, if the longitude of a knot is trivial in the knot group then the group is infinite cyclic. We show that for any classical knot group there is a virtual knot with that group and trivial longitude. It is well known that the second homology of a classical knot group is trivial. We provide counterexamples of this for virtual knots.

For an arbitrary group G, we give necessary and sufficient conditions for the existence of a virtual knot group that maps onto G with specified behavior on the peripheral subgroup. These conditions simplify those that arise in the classical setting.

We introduce an equivalence relation, called stable equivalence, on knot diagrams and closed generically immersed curves on surfaces. We give bijections between the set of abstract knots, the set of virtual knots, and the set of the stable equivalence classes of knot diagrams on surfaces. Using these bijections, we define concordance and link homology for virtual links. As an application, it is shown that Kauffman's example of a virtual knot diagram is not equivalent to a classical knot diagram.

It is an open question whether there are Vassiliev invariants that can distinguish an oriented knot from its inverse, i.e., the knot with the opposite orientation. In this article, an example is given for a first order Vassiliev invariant that takes different values on a virtual knot and its inverse. The Vassiliev invariant is derived from the Conway polynomial for virtual knots. Furthermore, it is shown that the zeroth order Vassiliev invariant coming from the Conway polynomial cannot distinguish a virtual link from its inverse and that it vanishes for virtual knots.

Virtual links are generalizations of classical links that can be represented by links embedded in a "thickened" surface Σ×I, product of a Riemann surface of genus h with an interval. In this paper, we show that virtual alternating links and tangles are naturally associated with the 1/N² expansion of an integral over N×N complex matrices. We suggest that it is sufficient to count the equivalence classes of these diagrams modulo ordinary (planar) flypes. To test this hypothesis, we use an algorithm coding the corresponding Feynman diagrams by means of permutations that generates virtual diagrams up to 6 crossings and computes various invariants. Under this hypothesis, we use known results on matrix integrals to get the generating functions of virtual alternating tangles of genus 1 to 5 up to order 10 (i.e. 10 real crossings). The asymptotic behavior for n large of the numbers of links and tangles of genus h and with n crossings is also computed for h=1,2,3 and conjectured for general h.

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.

The Witten–Reshetikhin–Turaev invariant of classical link diagrams is generalized to virtual link diagrams. This invariant is unchanged by the framed Reidemeister moves and the Kirby calculus. As a result, it is also an invariant of the 3-manifolds represented by the classical link diagrams. This generalization is used to demonstrate that there are virtual knot diagrams with a non-trivial Witten–Reshetikhin–Turaev invariant and trivial 3-manifold fundamental group.

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.

In this paper, we define and give examples of a family of polynomial invariants of virtual knots and links. They arise by considering certain 2 × 2 matrices with entries in a possibly non-commutative ring, for example, the quaternions. These polynomials are sufficiently powerful to distinguish the Kishino knot from any classical knot, including the unknot.

In this paper, we show how generalized quaternions including some 2 × 2 matrices, can be used to find solutions of the equation

Let A, B be invertible, non-commuting elements of a ring R. Suppose that A - 1 is also invertible and that the equation

In this paper, based on a talk given at the Oberwolfach research centre in May 2008 I will describe how biquandles and their big brother, biracks, can be used to differentiate isotopy classes of virtual (and welded) knots and links.

A group-theoretical method, via Wada's representations, is presented to distinguish Kishino's virtual knot from the unknot. Biquandles are constructed for any group using Wada's braid group representations. Cocycle invariants for these biquandles are studied. These invariants are applied to show the non-existence of Alexander numberings and checkerboard colorings.

For ordinary knots in R³, there are no degree one Vassiliev invariants. For virtual knots, however, the space of degree one Vassiliev invariants is infinite-dimensional. We introduce a sequence of three degree one Vassiliev invariants of virtual knots of increasing strength. We demonstrate that the strongest invariant is a universal Vassiliev invariant of degree one for virtual knots in the sense that any other degree one Vassiliev invariant can be recovered from it by a certain natural construction. To prove these results, we extend the based matrix invariant introduced by Turaev for virtual strings to the class of singular flat virtual knots with one double-point.

In this paper, we investigate twist sequences for Kauffman finite-type invariants and Goussarov–Polyak–Viro finite-type invariants. It is shown that one obtains a Kauffman or GPV type of degree ≤ n if and only if an invariant is a polynomial of degree ≤ n on every twist lattice of the right form. The main result of this paper is an application of this technique to the coefficients of the Jones–Kauffman polynomial. It is shown that the Kauffman finite-type invariants obtained from these coefficients are not GPV finite-type invariants of any degree by explicitly showing they can never be polynomials. This generalizes a result of Kauffman [8], where it is known for degree k = 2.

Two welded (respectively virtual) link diagrams are homotopic if one may be transformed into the other by a sequence of extended Reidemeister moves, classical Reidemeister moves, and self crossing changes. In this paper, we extend Milnor's μ and invariants to welded and virtual links. We conclude this paper with several examples, and compute the μ invariants using the Magnus expansion and Polyak's skein relation for the μ invariants.

In this paper, it is shown that there are no nonconstant Goussarov–Polyak–Viro finite-type invariants that are invariant under the virtualization move. As an immediate corollary, we obtain the theorem of [1] which states none of the Birman coefficients of the Jones–Kauffman polynomial are of GPV finite type.

A virtual string can be defined as an equivalence class of planar diagrams under certain kinds of diagrammatic moves. Virtual strings are related to virtual knots in that a simple operation on a virtual knot diagram produces a diagram for a virtual string. In this paper we consider three operations on a virtual string or virtual strings which produce another virtual string, namely covering, composition and cabling. In particular we study virtual strings unchanged by the covering operation. We also show how the based matrix of a composite virtual string is related to the based matrices of its components, correcting a result by Turaev. Finally we investigate what happens under cabling to some invariants defined by Turaev.