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

Abstract

Satoh and Taniguchi introduced the $n$-writhe $J_{n}$ for each non-zero integer $n$, which is an integer invariant for virtual knots. The sequence of $n$-writhes $\{J_{n}\}_{n \neq 0}$ of a virtual knot $K$ satisfies $\sum_{n \neq 0} nJ_{n}(K) = 0$. They showed that for any sequence of integers $\{c_{n}\}_{n \neq 0}$ with $\sum_{n \neq 0} nc_{n} = 0$, there exists a virtual knot $K$ with $J_n(K) = c_{n}$ for any $n \neq 0$. 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 $\{c_{n}\}_{n \neq 0}$ is a sequence of integers with $\sum_{n \neq 0} nc_{n} = 0$, then there exists a virtual knot $K$ such that $u^{v}(K) = 1$ and $J_{n}(K) = c_{n}$ for any $n \neq 0$.