STABLE EQUIVALENCE OF KNOTS ON SURFACES AND VIRTUAL KNOT COBORDISMS

By J. Scott Carter, S. Kamada, Masahico Saito. 2000

241 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics, Neurology

View Paper: https://www.semanticscholar.org/paper/f4353fc68d06b46c29f3fb1bfa4147d3a8eed43f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Louis H. Kauffman
2. V. Manturov
3. Sofia Lambropoulou
4. Andrew J. Nicas
5. Robin Gaudreau
6. Hans U Boden
7. Lindsay White
8. Eric Harper

In Mathematical Physics

1. Louis H. Kauffman
2. V. Manturov
3. Andrew J. Nicas
4. Robin Gaudreau
5. Hans U Boden
6. Lindsay White
7. Eric Harper
8. H. Dye

In Neurology

1. Louis H. Kauffman

Citations in the corpus

• VIRTUAL BRAIDS AND THE L-MOVE (2005, background): “For reference to previous work on virtual links and braids the reader should consult [2, 6, 7, 9, 11, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 27, 28, 32, 36, 37].”
• Cobordism of knots on surfaces (2007, background): “) In the language of [3], these facts may be reformulated by saying that stably equivalent knots are cobordant.”
• INTRODUCTION TO VIRTUAL KNOT THEORY (2011, background): “The same figure shows a virtual diagram on the left and an “abstract knot diagram” [3, 45] on the right.”
• Virtual knots and links (2005, background): “the addition and subtraction of thickened 1-handles from the surface that do not have any part of the knot or link embedded in them) [KaV2, KaV4, Ma1, Ma8, Ma10, CKS , KUP].”
• Virtual Braids (2004, background): “For reference to previous work on virtual braids the reader should consult [5, 6, 9, 10, 11, 17, 18, 19, 12, 13, 14, 20, 21, 23, 25, 26, 27, 28].”
• Virtual knot groups and almost classical knots (2015, background): “Every virtual knot K can be realized as a knot on a thickened surface, and by [7] there is a one-to-one correspondence between virtual knots and stable equivalence classes of knots on thickened surfaces.”
• Minimal surface representations of virtual knots and links (2004, background): “ilized by adding further handles which do not meet the diagram. This diagram can then be regarded as a genuine link in the surface thickened by crossing with a unit interval. Carter, Kamada and Saito [3] have shown that desingularization induces a bijection between equivalence classes of virtual lin”