On knots with no 3-state

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 sn(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 s3(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 s3(D) among all possible diagrams D for K. The second aim of this paper is to study several properties of the virtual knots with s3(K)=0.