On composite knots with symmetric union presentations

Abstract

A symmetric union in the 3-space [Formula: see text] is obtained from a knot in [Formula: see text] and its mirror image, which is symmetric with respect to a 2-plane in [Formula: see text], by connecting them with some vertical 2-string tangles with twists and a horizontal 2-string tangle with no twists along the plane. We introduce the minimal twisting number of a symmetric union and show that if a knot [Formula: see text] is a composite symmetric union with minimal twisting number one, then [Formula: see text] has a non-trivial knot and its mirror image as connected summands. As an application, we study the minimal twisting number of a symmetric union and show that there exist infinitely many symmetric unions with minimal twisting number two.