Exceptional pretzel knots are not algebraically slice

Abstract

We consider the three-strand pretzel knot $P_a=P(a,-a-2,-(a+1{)}^2/2)$ and prove that, when $a\equiv 1(\mathrm{mod}4)$, its Alexander polynomial fails the Fox--Milnor condition. The same remains true after adding any number of opposite odd parameter pairs $q,-q$ and reordering the parameters. Combined with work of Kim--Lee--Song, this gives the same conclusion for every odd $a\ge 3$. In particular, no knot in Lecuona's exceptional family is algebraically or topologically slice. Together with earlier work of Lecuona, Miller, and Kim--Lee--Song, this completes the slice--ribbon and topological-sliceness classifications for three-strand pretzel knots and removes the nonexceptional hypothesis from the Lecuona--Wand classification of prime fibered ribbon pretzel knots.

arXiv comment8 pages, 2 figures. Revision: minor changes and a remark