Double branched covers and pretzel knots (Q794027)

Let \(K(p_ 1,p_ 2,...,p_ n)\) be a pretzel knot, and \(\phi\) be any permutation of the numbers 1,2,...,n. Then the author shows that the double branched covers of \(S^ 3\) along \(K(p_ 1,...,p_ n)\) and \(K(p_{\phi(1)},...,p_{\phi(n)})\) are homeomorphic. However \(K(p_ 1,...,p_ n)\) and \(K(p_{\phi(1)},...,p_{\phi(n)})\) are of the same knot type if and only if \(\phi\) is a cyclic permutation, an order reversing permutation or a combination of these.