Polynomial splittings of correction terms and doubly slice knots

TL;DR

This paper investigates the relationship between polynomial splittings, correction terms, and doubly slice knots, showing that certain obstructions vanish for knots with coprime Alexander polynomials, and providing new examples of topologically but not smoothly doubly slice knots.

Contribution

It establishes a link between polynomial splittings and correction terms as obstructions to smooth double sliceness, and introduces new examples of knots with specific sliceness properties.

Findings

Correction terms vanish for connected sums of knots with coprime Alexander polynomials if the sum is doubly slice.

New examples of topologically doubly slice but not smoothly doubly slice knots are constructed.

The results differentiate between smooth and topological sliceness properties of knots.

Abstract

We show that if the connected sum of two knots with coprime Alexander polynomials is doubly slice, then the Ozsv\'ath-Szab\'o correction terms as smooth double sliceness obstructions vanish for both knots. Recently, Jeffrey Meier gave smoothly slice knots that are topologically doubly slice, but not smoothly doubly slice. As an application, we give a new example of such knots that is distinct from Meier's knots modulo doubly slice knots.