Displaying similar documents to “Synopsis Palmariorum Matheseos”

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...

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...

On the AJ conjecture for cables of twist knots

Anh T. Tran (2015)

Fundamenta Mathematicae

Similarity:

We study the AJ conjecture that relates the A-polynomial and the colored Jones polynomial of a knot in S³. We confirm the AJ conjecture for (r,2)-cables of the m-twist knot, for all odd integers r satisfying ⎧ (r+8)(r−8m) > 0 if m > 0, ⎨ ⎩ r(r+8m−4) > 0 if m < 0.

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.