What is a virtual link

@article{Kuperberg2003WhatIA,
  title={What is a virtual link},
  author={Greg Kuperberg},
  journal={Algebraic \& Geometric Topology},
  year={2003},
  volume={3},
  pages={587-591}
}
  • G. Kuperberg
  • Published 5 August 2002
  • Philosophy
  • Algebraic & Geometric Topology
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 thickened, compact, oriented surface S such that the link complement (SI)nL has no essential vertical cylinder. AMS Classication 57M25; 57M27 57M15 
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225 Citations
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225 Citations

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References

SHOWING 1-10 OF 16 REFERENCES
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.
Virtual Knot Theory
This paper is an introduction to the theory of virtual knots. It is dedicated to the memory of Francois Jaeger.
The rack space
The main result of this paper is a new classification theorem for links (smooth embeddings in codimension 2). The classifying space is the rack space and the classifying bundle is the first James…
Seifert fibered spaces in 3-manifolds
Publisher Summary This chapter describes Seifert Fibered Spaces in 3-Manifolds. There exist finitely many disjoint, non-contractible, pairwise non-parallel, embedded 2-spheres in M, whose homotopy…
Homotopy Equivalences of 3-Manifolds with Boundaries
General theory.- Essential singular surfaces in some special 3-manifolds.- Characteristic submanifolds.- Singular surfaces and characteristic submanifolds.- Singular submanifolds and characteristic…
INVOLUTORY HOPF ALGEBRAS AND 3-MANIFOLD INVARIANTS
We establish a 3-manifold invariant for each finite-dimensional, involutory Hopf algebra. If the Hopf algebra is the group algebra of a group $G$, the invariant counts homomorphisms from the…
A Unique Decomposition Theorem for 3-Manifolds
Die eindeutige Zerlegbarkeit eines Knotens in Primknoten
Shalen, Seifert- bered spaces in 3-manifolds
  • Mem. Amer. Math. Soc
  • 1979
Algebraic & Geometric Topology
  • Algebraic & Geometric Topology
  • 2003
...
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