Tangle decompositions of doubled knots (Q1267146)

Let \(K\) be a doubled knot of a knot \(K_1\). \(K_1\) is called the companion knot of \(K\). If a decomposing 2-sphere divides \(K\) in two essential tangles with \(m\) strings, then it is proved that \(m\) is even and the same 2-sphere divides the companion \(K_1\) in two essential tangles with \(m/2\) strings. As a corollary, if \(K_1\) is \(n\)-string prime for every \(n\geq 1\) then so is \(K\).