Ribbon-algorithm characterization for pretzel knots

Let $P(p_1,\dots,p_n)$ be a pretzel knot with $|p_i|>1$ for every $i$. The paper's algorithm, given by Proposition~, detects certain orderings of the parameters by exhibiting a ribbon presentation. Ribbon-algorithm conjecture. The pretzel knots with $|p_i|>1$ for all $i$ that are ribbon are precisely those detected by the algorithm in Proposition~. The algorithm supplies a sufficient condition in the examples considered, but its necessity and sufficiency in full generality remain open.