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Parity biquandle

Aaron Kaestner, Louis H. Kauffman (2014)

Banach Center Publications

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...

Plane curve singularities and carousels

Lê Dung Tráng (2003)

Annales de l’institut Fourier

In this paper we give a direct and explicit description of the local topological embedding of a plane curve singularity using the Puiseux expansions of its branches in a given set of coordinates.

Polynomial invariants of 2-component links.

Kunio Murasugi (1985)

Revista Matemática Iberoamericana

Let L = X U Y be an oriented 2-component link in S3. In this paper we will define two different types of polynomials which are ambient isotopic invariants of L. One is associated with a cyclic cover branched along one of their components, an the other is associated with a metabelian cover of L. This invariants are defined for any link unless the linking number lk(X,Y), is ±1.

Positive knots, closed braids and the Jones polynomial

Alexander Stoimenow (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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 positive knot...

Presentations of surface braid groups by graphs

Paolo Bellingeri, Vladimir Vershinin (2005)

Fundamenta Mathematicae

We extend and generalise Sergiescu's results on planar graphs and presentations for the braid group Bₙ to other topological generalisations of Bₙ.

Principal congruence link complements

Mark D. Baker, Alan W. Reid (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

In this paper we study principal congruence link complements in S 3 . It is known that there are only finitely many such link complements, and we make a start on enumerating them using a combination of theoretical methods and computer calculations with MAGMA.

Pure virtual braids homotopic to the identity braid

H. A. Dye (2009)

Fundamenta Mathematicae

Two virtual link diagrams are homotopic if one may be transformed into the other by a sequence of virtual Reidemeister moves, classical Reidemeister moves, and self crossing changes. We recall the pure virtual braid group. We then describe the set of pure virtual braids that are homotopic to the identity braid.

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