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 ≠ 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.