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Although the connected sum of trivial knots is trivial in classical knot theory, there is a nontrivial connected sum of trivial knots in virtual knot theory. In this paper, we give connected sum formulae of the intersection polynomials of virtual knots introduced in the preceding paper [6]. As an application, we show that there are infinitely many connected sums of any pair of virtual knots. [ABSTRACT FROM AUTHOR]

We go along a knot diagram, and get a sequence of over- and under- crossing points. We will study which kinds of sequences are realized by diagrams of the trefoil knot. As an application, we will characterize the Shimizu warping polynomials for diagrams of the trefoil knot. [ABSTRACT FROM AUTHOR]

Every virtual knot has a pair of groups called the upper and lower groups of the knot. In this paper we treat two topics on those groups: We first give a sufficient condition for a pair of groups which are realized by a certain virtual knot as the upper and lower groups. Secondly we give a necessary condition for a virtual knot to be almost classical in terms of the first elementary ideals of the groups. [ABSTRACT FROM AUTHOR]

The first aim of this paper is to give an example of a virtual knot which vanishes all the known invariants and to show that it is a non-classical (in particular, non-trivial) knot. The second is to study some properties of the Z-polynomial of a virtual knot with virtual crossing number one and to show that there are infinitely many virtual knots with virtual crossing number two. [ABSTRACT FROM AUTHOR]

It is known that the pass move is not an unknotting operation in classical knot theory. In this paper, we prove that the pass move is an unknotting operation in welded knot theory. [ABSTRACT FROM AUTHOR]

A state of a virtual knot diagram D is a disjoint union of circles obtained from D by splicing all real crossings. For each positive integer n , we denote by s n ( D ) the number of states of D consisting of n circles. The first aim of this paper is to characterize the virtual knot diagrams with s 3 ( D ) = 0 in terms of their Gauss diagrams. The 3-state number of a virtual knot K is defined to be the minimal number of s 3 ( D ) among all possible diagrams D for K . The second aim of this paper is to study several properties of the virtual knots with s 3 ( K ) = 0 . [ABSTRACT FROM AUTHOR]

The writhe polynomial is a fundamental invariant of an oriented virtual knot. We introduce a set of local moves for oriented virtual knots called shell moves. The first aim of this paper is to prove that two oriented virtual knots have the same writhe polynomial if and only if they are related by a finite sequence of shell moves. The second aim of this paper is to classify oriented 2-component virtual links up to shell moves by using several invariants of virtual links. [ABSTRACT FROM AUTHOR]