The topological sliceness of 3-strand pretzel knots

TL;DR

This paper characterizes when odd 3-strand pretzel knots are topologically slice, showing they are so if and only if they are ribbon or have trivial Alexander polynomial, using Casson-Gordon invariants.

Contribution

It provides a complete topological sliceness classification for odd 3-strand pretzel knots and extends understanding of their ribbon and slice properties.

Findings

Odd 3-strand pretzel knots are topologically slice iff they are ribbon or have trivial Alexander polynomial.

Topologically slice even 3-strand pretzel knots are mostly ribbon, except possibly in Lecuona's exceptional family.

Casson-Gordon invariants are used to determine the topological sliceness of these knots.

Abstract

We give a complete characterization of the topological slice status of odd 3-strand pretzel knots, proving that an odd 3-strand pretzel knot is topologically slice if and only if either it is ribbon or has trivial Alexander polynomial. (By work of [FS85], a nontrivial odd 3-strand pretzel knot $K$ cannot both be ribbon and have $\Delta_K(t)=1$.) We also show that topologically slice even 3-strand pretzel knots (except perhaps for members of Lecuona's exceptional family of [Lec13]) must be ribbon. These results follow from computations of the Casson-Gordon 3-manifold signature invariants associated to the double branched covers of these knots.