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

摘要

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≠0) of a virtual knot K satisfies ∑_(n≠0) nJ_n(K) = 0. They showed that for any sequence of integers {c_n}_(n≠0) with ∑_(n≠0) nC_n = 0, there exists a virtual knot K with J_n(K) = c_n 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 {c_n}_(n≠0) is a sequence of integers with ∑_(n≠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 ≠ 0.