Virtualization and $n$-writhes for virtual knots

Satoh and Taniguchi introduced the $n$-writhe $J_n$ for each non-zero integer $n$, which is an invariant for virtual knots. The $n$-writhes give the coefficients of some polynomial invariants for virtual knots including the index polynomial, the odd writhe polynomial and the affine index polynomial. It is obvious that the virtualization of a real crossing is an unknotting operation for virtual knots. The values of $n$-writhes changed by some local moves are calculated. However for the virtualization, it is unknown. In this paper, we show that for any given non-zero integer $n$ and any given integer $N$, there exists a virtual knot whose unknotting number by the virtualization is one and the value of the $n$-writhe equals $N$. Namely, the virtualization of a real crossing changes the value of $n$-writhe by any given integer $N$.