On composite knots with symmetric union presentations.

A symmetric union in the 3-space ℝ 3 is obtained from a knot in ℝ 3 and its mirror image, which is symmetric with respect to a 2-plane in ℝ 3 , 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 K is a composite symmetric union with minimal twisting number one, then K 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. [ABSTRACT FROM AUTHOR]