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1
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A self-linking invariant of virtual knots

100%
Kauffman L.
Fundamenta Mathematicae
|
2004
|
tom 184
|
nr 1
135-158
EN
This paper introduces a self-linking invariant for virtual knots and links, and relates this invariant to a state model called the binary bracket, and to a class of coloring problems for knots and links that include classical coloring problems for cubic graphs.
2
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Virtual biquandles

64%
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
Content available remote

Local moves on knots and products of knots

64%
Kauffman L., Ogasa E.
Banach Center Publications
|
2014
|
tom 103
|
nr 1
159-209
EN
We show a relation between products of knots, which are generalized from the theory of isolated singularities of complex hypersurfaces, and local moves on knots in all dimensions. We discuss the following problem. Let K be a 1-knot which is obtained from another 1-knot J by a single crossing change (resp. pass-move). For a given knot A, what kind of relation do the products of knots, K ⊗ A and J ⊗ A, have? We characterize these kinds of relation between K ⊗ A and J ⊗ A by using local moves on high dimensional knots. We also discuss a connection between local moves and knot invariants. We show a new form of identities for knot polynomials associated with a local move.
4
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Virtual braids

64%
Kauffman L., Lambropoulou S.
Fundamenta Mathematicae
|
2004
|
tom 184
|
nr 1
159-186
EN
This paper gives a new method for converting virtual knots and links to virtual braids. Indeed, the braiding method given here is quite general and applies to all the categories in which braiding can be accomplished. This includes the braiding of classical, virtual, flat, welded, unrestricted, and singular knots and links. We also give reduced presentations for the virtual braid group and for the flat virtual braid group (as well as for other categories). These reduced presentations are based on the fact that these virtual braid groups for n strands are generated by a single braiding element plus the generators of the symmetric group on n letters.
5
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Parity biquandle

64%
Kaestner A., Kauffman L.
Banach Center Publications
|
2014
|
tom 100
|
nr 1
131-151
EN
We use crossing parity to construct a generalization of biquandles for virtual knots which we call parity biquandles. These structures include all biquandles as a standard example referred to as the even parity biquandle. We find all parity biquandles arising from the Alexander biquandle and quaternionic biquandles. For a particular construction named the z-parity Alexander biquandle we show that the associated polynomial yields a lower bound on the number of odd crossings as well as the total number of real crossings and virtual crossings for the virtual knot. Moreover we extend this construction to links to obtain a lower bound on the number of crossings between components of a virtual link.
6
Content available remote

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.
7
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Equivalence classes of colorings

51%
Ge J., Jablan S., Kauffman L., Lopes P.
Banach Center Publications
|
2014
|
tom 103
|
nr 1
63-76
EN
For any link and for any modulus m we introduce an equivalence relation on the set of non-trivial m-colorings of the link (an m-coloring has values in Z/mZ). Given a diagram of the link, the equivalence class of a non-trivial m-coloring is formed by each assignment of colors to the arcs of the diagram that is obtained from the former coloring by a permutation of the colors in the arcs which preserves the coloring condition at each crossing. This requirement implies topological invariance of the equivalence classes. We show that for a prime modulus the number of equivalence classes depends on the modulus and on the rank of the coloring matrix (with respect to this modulus).
8
Content available remote

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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