Cobordism of knots on surfaces

@article{Turaev2008CobordismOK,
  title={Cobordism of knots on surfaces},
  author={Vladimir Turaev},
  journal={Journal of Topology},
  year={2008},
  volume={1}
}
  • V. Turaev
  • Published 2 March 2007
  • Mathematics
  • Journal of Topology
We introduce a relation of cobordism for knots in thickened surfaces and study cobordism invariants of such knots. 
A Lower Bound for the Crossing Number of Links in Thickened Surfaces
  • V. Tarkaev
  • Mathematics
    Siberian Mathematical Journal
  • 2018
We introduce the notion of homological multiplicity for an oriented link in a thickened orientable closed surface. Using the notion, we establish some lower bounds for the crossing number of a link
Chord index for knots in thickened surfaces
In this note, we construct a chord index homomorphism from a subgroup of H1(Σ, Z) to the group of chord indices of a knot K in Σ × I. Some knot invariants derived from this homomorphism are discussed.
Invariants in Low-Dimensional Topology and Knot Theory
We explain our general program to prove the existence of a cycle of curves on minimal class VII surfaces with b_2>0 and the recent progress obtained by the author in this direction.
Concordance group of virtual knots
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
Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory
This paper is a concise introduction to virtual knot theory, coupled with a list of research problems in this field.
Crossing tribes of tangles in a thickened surface
Crossings of knot diagrams can be divided into classes (tribes) compatible with Reidemeister moves. Tribes can be considered as localization of the notion of weak chord index introduced by M. Xu. In
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
Index polynomials for virtual tangles
  • Nicolas Petit
  • Mathematics
    Journal of Knot Theory and Its Ramifications
  • 2018
We generalize the index polynomial invariant, originally introduced by Turaev [Cobordism of knots on surfaces, J. Topol. 1(2) (2008) 285–305] and Henrich [A sequence of degree one vassiliev
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.
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 24 REFERENCES
Virtual Knot Theory
This paper is an introduction to the theory of virtual knots. It is dedicated to the memory of Francois Jaeger.
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.
Gauss Diagram Invariants for Knots and Links
Preface. Introduction and announcement. 1. The space of diagrams. 2. Invariants of knots and links by Gauss sums. 3. Applications. 4. Global knot theory in F2xR. 5. Isotopies with restricted cusp
Virtual knots and links
This paper is an introduction to the subject of virtual knot theory and presents a discussion of some new specific theorems about virtual knots. The new results are as follows: Using a 3-dimensional
A generalization of several classical invariants of links
We extend several classical invariants of links in the 3-sphere to links in so-called quasi-cylinders. These invariants include the linking number, the Seifert form, the Alexander module, the
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
VIRTUAL STRINGS
A virtual string is a scheme of self-intersections of a closed curve on a surface. We study algebraic invariants of strings as well as two equivalence relations on the set of strings: homotopy and
CLOSED CURVES THAT NEVER EXTEND TO PROPER MAPS OF DISKS
If a closed curve in an orientable surface bounds a disk in a handle- body, then the double points on the boundary admit certain pairings that are called filamentations. Intersection numbers are
Categorification of the Kauffman bracket skein module of I-bundles over surfaces
Khovanov defined graded homology groups for links LR 3 and showed that their polynomial Euler characteristic is the Jones polyno- mial of L. Khovanov's construction does not extend in a
A Survey of Classical Knot Concordance
...
...