Year: 2026
Type: preprint
Abstract: We consider the three-strand pretzel knot $P_a=P(a,-a-2,-(a+1)^2/2)$ and prove that, when $a\equiv1\pmod4$, 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 $a\ge3$. 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.
Source: arXiv (Cornell University)
Author: Saurabh, Suman
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214785908