On satellite knots with symmetric union presentations

Abstract

A symmetric union in the 3-space $\mathbb{R}^3$ is a knot, obtained from a knot in $\mathbb{R}^3$ and its mirror image, which are symmetric with respect to a 2-plane in $\mathbb{R}^3$, by taking the connected sum of them and moreover by connecting them with some vertical twists along the plane, which is a generalized operation of the connected sum of a knot and its mirror image. In this paper, we show that a satellite symmetric union with minimal twisting number one such that the order of the pattern is an odd number $\ge 3$ has at least two disjoint non-parallel essential tori in the complement.