The purpose of this paper is to give an introduction to virtual knot theory and to record a collection of research problems that the authors have found fascinating. The second section of the paper introduces the theory and discusses problems in that context. The third section is a list of specific problems.

The purpose of this paper is to give an introduction to virtual knot theory and to record a collection of research problems that the authors have found fascinating. The second section of the paper introduces the theory and discusses problems in that context. The third section is a list of specific problems.

The purpose of this paper is to give an introduction to virtual knot theory and to record a collection of research problems that the authors have found fascinating. The second section of the paper introduces the theory and discusses problems in that context. The third section is a list of specific problems.

The purpose of this paper is to give an introduction to virtual knot theory and to record a collection of research problems that the authors have found fascinating. The second section of the paper introduces the theory and discusses problems in that context. The third section is a list of specific problems.

This paper extends the construction of invariants for virtual knots to virtual long knots and introduces two new invariant modules of virtual long knots. Several interesting features are described that distinguish virtual long knots from their classical counterparts with respect to their symmetries and the concatenation product. 1

Abstract. We introduce the concept of a relative Tutte polynomial of colored graphs. We show that this relative Tutte polynomial can be computed in a way similar to the classical spanning tree expansion used by Tutte in his original paper on this subject. We then apply the relative Tutte polynomial to virtual knot theory. More specifically, we show that the Kauffman bracket polynomial (hence the Jones polynomial) of a virtual knot can be computed from the relative Tutte polynomial of its face (Tait) graph with some suitable variable substitutions. Our method offers an alternative to the ribbon graph approach, using the face graph obtained from the virtual link diagram directly. 1.

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