Yasuyuki Miyazawa

MAGNETIC GRAPHS AND AN INVARIANT FOR VIRTUAL LINKS

Introducing a type of graph named a virtual magnetic graph diagram, we define a virtual link invariant, which generalizes the Jones-Kauffman polynomial and the 2-variable polynomial invariant defined by N. Kamada and the author. We show that this invariant can evaluate the number of virtual crossings of a virtual link.

A MULTI-VARIABLE POLYNOMIAL INVARIANT FOR VIRTUAL KNOTS AND LINKS

We construct a multi-variable polynomial invariant for virtual knots and links via the concept of a decorated virtual magnetic graph diagram. The invariant is a generalization of the Jones–Kauffman polynomial for virtual knots and links. We show some features of the invariant including an evaluation of the virtual crossing number of a virtual knot or link.

VASSILIEV LINK INVARIANTS OF ORDER THREE

We express a basis for the space of all Vassiliev invariants of order less than or equal to three for an ordered oriented link in terms of the Conway polynomials, the linking numbers, and the HOMFLY polynomials. An applications, we give a formula relating to the Casson-Walker invariant of 3-manifolds from a result of Kirk and Livingston, and give a new proof for a formula of Hoste concerning the coefficient of the Conway polynomial of a link.

GORDIAN DISTANCE AND POLYNOMIAL INVARIANTS

Some evaluations of the Gordian distance from a knot to another knot are given by using three polynomial invariants called the HOMFLY polynomial, the Jones polynomial and the Q-polynomial. Furthermore, Gordian distances between a lot of pairs of knots are determined.

A MULTI-VARIABLE POLYNOMIAL INVARIANT FOR UNORIENTED VIRTUAL KNOTS AND LINKS

We construct a multi-variable polynomial invariant Y for unoriented virtual links as a certain weighted sum of polynomials, which are derived from virtual magnetic graphs with oriented vertices, on oriented virtual links associated with a given virtual link. We show some features of the Y-polynomial including an evaluation of the virtual crossing number of a virtual link.

The Third Derivative of the Jones Polynomial

Let L be an oriented n-component link and let VL (t) be the Jones polynomial of L. A relationship between (1) and the coefficients of the Conway polynomials of L is given.

A VIRTUAL LINK POLYNOMIAL AND THE VIRTUAL CROSSING NUMBER

We construct a multi-variable polynomial invariant for virtual knots and links by using the concepts of a decorated virtual magnetic graph diagram and a weight map. We show that the invariant is a variety of virtual link polynomial with multiple variables introduced in [16] by the author and it gives a sharpened evaluation of the virtual crossing number of a virtual knot or link.

Knots with a Trivial Coefficient Polynomial

By using a tangle decomposition of a knot, we give a method for the construc- tion of a knot with the lowest trivial HOMFLY coefficient polynomial. Applying this, we show that there exist infinitely many 2-bridge knots with such a coefficient polynomial.

A LINK INVARIANT DOMINATING THE HOMFLY AND THE KAUFFMAN POLYNOMIALS

By using a graph diagram named a magnetic graph diagram, we construct a polynomial invariant for knots and links. We show that it is a generalization of both the HOMFLY and the Kauffman polynomials.