The topological sliceness of 3-strand pretzel 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.

17 pages, 3 figures

References in corpus (2)

• The signature of a splice
• On 2-bridge knots with differing smooth and topological slice genera

Cited by in corpus (3)

• Surgeries on torus knots, rational balls, and cabling
• Non-slice 3-stranded pretzel knots
• Explicit Formulas for the Alexander Polynomial of Pretzel Knots