Displaying similar documents to “A self-linking invariant of virtual knots”

An operator invariant for handlebody-knots

Kai Ishihara, Atsushi Ishii (2012)

Fundamenta Mathematicae

Similarity:

A handlebody-knot is a handlebody embedded in the 3-sphere. We improve Luo's result about markings on a surface, and show that an IH-move is sufficient to investigate handlebody-knots with spatial trivalent graphs without cut-edges. We also give fundamental moves with a height function for handlebody-tangles, which helps us to define operator invariants for handlebody-knots. By using the fundamental moves, we give an operator invariant.

Virtual knot invariants arising from parities

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

Banach Center Publications

Similarity:

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...

The equation [B,(A-1)(A,B)] = 0 and virtual knots and links

Stephen Budden, Roger Fenn (2004)

Fundamenta Mathematicae

Similarity:

Let A, B be invertible, non-commuting elements of a ring R. Suppose that A-1 is also invertible and that the equation [B,(A-1)(A,B)] = 0 called the fundamental equation is satisfied. Then this defines a representation of the algebra ℱ = A, B | [B,(A-1)(A,B)] = 0. An invariant R-module can then be defined for any diagram of a (virtual) knot or link. This halves the number of previously known relations and allows us to give a complete solution in the case when R is the quaternions. ...

Virtual biquandles

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

Fundamenta Mathematicae

Similarity:

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...

Edge number results for piecewise-Linear knots

Monica Meissen (1998)

Banach Center Publications

Similarity:

The minimal number of edges required to form a knot or link of type K is the edge number of K, and is denoted e(K). When knots are drawn with edges, they are appropriately called piecewise-linear or PL knots. This paper presents some edge number results for PL knots. Included are illustrations of and integer coordinates for the vertices of several prime PL knots.

Virtual knot theory-unsolved problems

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

Fundamenta Mathematicae

Similarity:

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.

Vassiliev invariants as polynomials

Simon Willerton (1998)

Banach Center Publications

Similarity:

Three results are shown which demonstrate how Vassiliev invariants behave like polynomials.

Invariants of piecewise-linear knots

Richard Randell (1998)

Banach Center Publications

Similarity:

We study numerical and polynomial invariants of piecewise-linear knots, with the goal of better understanding the space of all knots and links. For knots with small numbers of edges we are able to find limits on polynomial or Vassiliev invariants sufficient to determine an exact list of realizable knots. We thus obtain the minimal edge number for all knots with six or fewer crossings. For example, the only knot requiring exactly seven edges is the figure-8 knot.

A topological model of site-specific recombination that predicts the knot and link type of DNA products

Karin Valencia (2014)

Banach Center Publications

Similarity:

This is a short summary of a topological model of site-specific recombination, a cellular reaction that creates knots and links out of circular double stranded DNA molecules. The model is used to predict and characterise the topology of the products of a reaction on double stranded DNA twist knots. It is shown that all such products fall into a small family of Montesinos knots and links, meaning that the knot and link type of possible products is significantly reduced, thus aiding their...

Applications of topology to DNA

Isabel Darcy, De Sumners (1998)

Banach Center Publications

Similarity:

The following is an expository article meant to give a simplified introduction to applications of topology to DNA.

Positive knots, closed braids and the Jones polynomial

Alexander Stoimenow (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Similarity:

Using the recent Gauß diagram formulas for Vassiliev invariants of Polyak-Viro-Fiedler and combining these formulas with the Bennequin inequality, we prove several inequalities for positive knots relating their Vassiliev invariants, genus and degrees of the Jones polynomial. As a consequence, we prove that for any of the polynomials of Alexander/Conway, Jones, HOMFLY, Brandt-Lickorish-Millett-Ho and Kauffman there are only finitely many positive knots with the same polynomial and no...

Every knot is a billiard knot

P. V. Koseleff, D. Pecker (2014)

Banach Center Publications

Similarity:

We show that every knot can be realized as a billiard trajectory in a convex prism. This proves a conjecture of Jones and Przytycki.

The writhes of a virtual knot

Shin Satoh, Kenta Taniguchi (2014)

Fundamenta Mathematicae

Similarity:

Kauffman introduced a fundamental invariant of a virtual knot called the odd writhe. There are several generalizations of the odd writhe, such as the index polynomial and the odd writhe polynomial. In this paper, we define the n-writhe for each non-zero integer n, which unifies these invariants, and study various properties of the n-writhe.

Nielsen number is a knot invariant

Alexander Fel'shtyn (2007)

Banach Center Publications

Similarity:

We show that the Nielsen number is a knot invariant via representation variety.