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Flat hierarchy

100%
Manturov V.
Fundamenta Mathematicae
|
2005
|
tom 188
|
nr 1
147-154
EN
We consider the hierarchy flats, a combinatorial generalization of flat virtual links proposed by Louis Kauffman. An approach to constructing invariants for hierarchy flats is presented; several examples are given.
2
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Virtual biquandles

63%
Kauffman L., Manturov V.
Fundamenta Mathematicae
|
2005
|
tom 188
|
nr 1
103-146
EN
We describe new approaches for constructing virtual knot invariants. The main background of this paper comes from formulating and bringing together the ideas of biquandle [KR], [FJK], the virtual quandle [Ma2], the ideas of quaternion biquandles by Roger Fenn and Andrew Bartholomew [BF], the concepts and properties of long virtual knots [Ma10], and other ideas in the interface between classical and virtual knot theory. In the present paper we present a new algebraic construction of virtual knot invariants, give various presentations of it, and study several examples. Several conjectures and unsolved problems are presented throughout the paper.
3
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Virtual knot theory-unsolved problems

51%
Fenn R., Kauffman L., Manturov V.
Fundamenta Mathematicae
|
2005
|
tom 188
|
nr 1
293-323
EN
The present paper gives a quick survey of virtual and classical knot theory and presents a list of unsolved problems about virtual knots and links. These are all problems in low-dimensional topology with a special emphasis on virtual knots. In particular, we touch new approaches to knot invariants such as biquandles and Khovanov homology theory. Connections to other geometrical and combinatorial aspects are also discussed.
4
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Unsolved problems in virtual knot theory and combinatorial knot theory

51%
Fenn R., Ilyutko D., Kauffman L., Manturov V.
Banach Center Publications
|
2014
|
tom 103
|
nr 1
9-61
EN
This paper is a concise introduction to virtual knot theory, coupled with a list of research problems in this field.
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