Journal of the
Korean Mathematical Society JKMS

A symmetric union is a diagram of a knot, obtained from diagrams of a knot in the 3-space and its mirror image, which are symmetric with respect to an axis in the 2-plane, by connecting them with 2-tangles with twists along the axis and 2-tangles with no twists. This paper presents an invariant of knots with symmetric union presentations, which is called the minimal twisting number, and the minimal twisting number of $10_{42}$ is shown to be two. This paper also presents a sufficient condition for non-amphicheirality of a knot with a certain symmetric union presentation.

Keywords: symmetric union, Jones polynomial, ribbon knot

MSC numbers: Primary 57M25; Secondary 57M27