Infinitely many virtual knots with any given sequence of n-writhes

Satoh and Taniguchi introduced the [Formula: see text]-writhe [Formula: see text] for each non-zero integer [Formula: see text], which is an integer invariant for virtual knots. It is obvious that the virtualization of a real crossing is an unknotting operation for virtual knots. In our previous paper, we showed that if [Formula: see text] is a sequence of integers with [Formula: see text], then there exists a virtual knot [Formula: see text] whose virtual unknotting number equals one and [Formula: see text] for any [Formula: see text]. In this paper, we show that there exist infinitely many virtual knots having such properties by using the vertex connected sum on Gauss diagrams.