A virtual knot whose virtual unknotting number equals one and a sequence of n-writhes

Satoh and Taniguchi introduced the n-writhe Jn for each non-zero integer n, which is an integer invariant for virtual knots. The sequence of n-writhes {Jn}_n≠₀ of a virtual knot K satisfies ^∑_n≠₀ nJn(K) = 0. They showed that for any sequence of integers {cn}_n≠₀ with ^∑_n≠₀ ncn = 0, there exists a virtual knot K with Jn(K) = cn for any n ≠ 0. It is obvious that the virtualization of a real crossing is an unknotting operation for virtual knots. The unknotting number by the virtualization is called the virtual unknotting number and is denoted by u^v. In this paper, we show that if {cn}_n≠0 is a sequence of integers with ^∑_n≠₀ ncn = 0, then there exists a virtual knot K such that u^v(K) = 1 and Jn(K) = cn for any n ≠ 0.