Virtual Knots with Properties of Kishino's Knot

Abstract

Satoh and Taniguchi defined an invariant of virtual knots $J_n$ for a non-zero integer $n$. It is called the $n$-writhe. 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 unknotting number by the virtualization is called the virtual unknotting number. Kishino's knot is a virtual unknotting number one knot which has the trivial $n$-writhe and the trivial Jones polynomial. In this paper, we construct infinitely many virtual knots which have the same properties as Kishino's knot.