Exceptional pretzel knots are not algebraically slice

Abstract: We prove that the Alexander polynomial of every knot in Lecuona's exceptional family of pretzel knots fails the Fox--Milnor condition. Consequently, none of these knots is algebraically or topologically slice. Together with work of Lecuona, Miller, and Kim--Lee--Song, this completes the slice--ribbon and topological-sliceness classifications for three-strand pretzel knots. It also removes the nonexceptional hypothesis from the Lecuona--Wand classification of prime fibered ribbon pretzel knots up to reordering of parameters.