Concordance group of virtual knots

@article{Boden2016ConcordanceGO,
  title={Concordance group of virtual knots},
  author={Hans U. Boden and Matthias Nagel},
  journal={arXiv: Geometric Topology},
  year={2016}
}
We study concordance of virtual knots. Our main result is that a classical knot K is virtually slice if and only if it is classically slice. From this we deduce that the concordance group of classical knots embeds into the concordance group of long virtual knots. 

Figures from this paper

Virtual Knot Theory and Virtual Knot Cobordism
  • L. Kauffman
  • Mathematics
    Knots, Low-Dimensional Topology and Applications
  • 2019
This paper is an introduction to virtual knot theory and virtual knot cobordism [37, 39]. Non-trivial examples of virtual slice knots are given and determinations of the four-ball genus of positive
Virtual Knot Cobordism and Bounding the Slice Genus
TLDR
The graded genus of Turaev's graded genus is remarkably effective as a slice obstruction, and an algorithm is developed that applies virtual unknotting operations to determine the slice genus of many virtual knots with six or fewer crossings.
Signature and concordance of virtual knots
We introduce signature invariants of virtual knots and use them to investigate virtual knot concordance. The signatures, which depend on a choice of Seifert surface, are defined first for almost
Algebraic concordance and almost classical knots
A virtual knot is called almost classical if it can be a represented by a homologically trivial knot K in a thickened surface Σ× [0, 1], where Σ is closed and oriented. Then K bounds a Seifert
Smooth and topological almost concordance
We investigate the disparity between smooth and topological almost concordance of knots in general 3-manifolds Y. Almost concordance is defined by considering knots in Y modulo concordance in Yx[0,1]
Ascent concordance
We show that there exist concordant links in thickened surfaces between which a concordance can only be realised by passing through thickenings of higher genus surfaces. We exhibit an infinite family
Virtual knot homology and concordance
We construct and investigate the properties of a new extension of Khovanov homology to virtual links, known as doubled Khovanov homology. We describe a perturbation of doubled Khovanov homology,
Cobordisms of graphs: A sliceness criterion for stably odd free knots and related results on cobordisms
In [1], the authors proved a sliceness criterion for odd free knots: free knots with odd chords. In the present paper, we give a similar criterion for stably odd free knots. In essence, free knots
Computations of the slice genus of virtual knots
From chord parity to chord index
We give a brief survey of virtual knot invariants derived from chord parity or chord index. These invariants have grown into an area in its own right due to rapid developing in the last decade. Sev...
...
...

References

SHOWING 1-10 OF 17 REFERENCES
Virtual Knot Cobordism
This paper defines a theory of cobordism for virtual knots and studies this theory for standard and rotational virtual knots and links. Non-trivial examples of virtual slice knots are given.
Cobordism of knots on surfaces
We introduce a relation of cobordism for knots in thickened surfaces and study cobordism invariants of such knots.
STABLE EQUIVALENCE OF KNOTS ON SURFACES AND VIRTUAL KNOT COBORDISMS
TLDR
An equivalence relation, called stable equivalence, is introduced on knot diagrams and closed generically immersed curves on surfaces and it is shown that Kauffman's example of a virtual knot diagram is not equivalent to a classical knot diagram.
What is a virtual link
Several authors have recently studied virtual knots and links because they admit invariants arising from R-matrices. We prove that every virtual link is uniquely represented by a link L SI in a
ABSTRACT LINK DIAGRAMS AND VIRTUAL KNOTS
TLDR
It is prove that there is a bijection from the equivalence classes of virtual link diagrams to those of abstract link diagrams, and a generalization to higher dimensional cases is introduced, and the state-sum invariants are treated.
Virtual Knot Theory
This paper is an introduction to the theory of virtual knots. It is dedicated to the memory of Francois Jaeger.
Virtual Covers of Links II
A fibered concordance of knots, introduced by Harer, is a concordance between fibered knots that is well-behaved with respect to the fibrations. We consider semi-fibered concordance of two component
Compact and long virtual knots
The theory of virtual knots is a generalization of the theory of classical knots proposed by Kauffman in 1996. In this paper solutions of certain problems of the theory of virtual knots are given.
Band-Passes and Long Virtual Knot Concordance
Every classical knot is band-pass equivalent to the unknot or the trefoil. The band-pass class of a knot is a concordance invariant. Every ribbon knot, for example, is band-pass equivalent to the
...
...