Displaying 601 – 620 of 643

Showing per page

Virtual biquandles

Louis H. Kauffman, Vassily O. Manturov (2005)

Fundamenta Mathematicae

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...

Virtual braids

Louis H. Kauffman, Sofia Lambropoulou (2004)

Fundamenta Mathematicae

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...

Virtual knot invariants arising from parities

Denis Petrovich Ilyutko, Vassily Olegovich Manturov, Igor Mikhailovich Nikonov (2014)

Banach Center Publications

In [12, 15] it was shown that in some knot theories the crucial role is played by parity, i.e. a function on crossings valued in {0,1} and behaving nicely with respect to Reidemeister moves. Any parity allows one to construct functorial mappings from knots to knots, to refine many invariants and to prove minimality theorems for knots. In the present paper, we generalise the notion of parity and construct parities with coefficients from an abelian group rather than ℤ₂ and investigate them...

Virtual knot theory-unsolved problems

Roger Fenn, Louis H. Kauffman, Vassily O. Manturov (2005)

Fundamenta Mathematicae

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.

Virtual strings

Vladimir Turaev (2004)

Annales de l'Institut Fourier

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 cobordism. We show that the homotopy invariants of strings form an infinite dimensional Lie group. We also discuss connections between virtual strings and virtual knots.

Wirtinger presentations for higher dimensional manifold knots obtained from diagrams

Seiichi Kamada (2001)

Fundamenta Mathematicae

A Wirtinger presentation of a knot group is obtained from a diagram of the knot. T. Yajima showed that for a 2-knot or a closed oriented surface embedded in the Euclidean 4-space, a Wirtinger presentation of the knot group is obtained from a diagram in an analogous way. J. S. Carter and M. Saito generalized the method to non-orientable surfaces in 4-space by cutting non-orientable sheets of their diagrams by some arcs. We give a modification to their method so that one does not need to find and...

Yamada polynomial and crossing number of spatial graphs.

Tomoe Motohashi, Yoshiyuki Ohyama, Kouki Taniyama (1994)

Revista Matemática de la Universidad Complutense de Madrid

In this paper we estimate the crossing number of a flat vertex graph in 3-space in terms of the reduced degree of its Yamada polynomial.

θ -curves inducing two different knots with the same 2 -fold branched covering spaces

Soo Hwan Kim, Yangkok Kim (2003)

Bollettino dell'Unione Matematica Italiana

For a knot K with a strong inversion i induced by an unknotting tunnel, we have a double covering projection Π : S 3 S 3 / i branched over a trivial knot Π fix i , where fix i is the axis of i . Then a set Π fix i K is called a θ -curve. We construct θ -curves and the Z 2 Z 2 cyclic branched coverings over θ -curves, having two non-isotopic Heegaard decompositions which are one stable equivalent.

Currently displaying 601 – 620 of 643