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

Abstract

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}n̸=0 of a virtual knot K satisfies ∑ n̸=0 nJn(K) = 0. They showed that for any sequence of integers {cn}n̸=0 with ∑ n̸=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 uv . In this paper, we show that if {cn}n̸=0 is a sequence of integers with ∑ n̸=0 ncn = 0, then there exists a virtual knot K such that uv(K) = 1 and Jn(K) = cn for any n ̸= 0.