On the slice-ribbon conjecture for pretzel knots

TL;DR

This paper investigates the sliceness of pretzel knots with one even parameter, providing conditions for sliceness and analyzing obstructions using classical and modern invariants.

Contribution

It establishes necessary and sometimes sufficient conditions for pretzel knot sliceness with one even parameter, and compares different obstructions including Alexander polynomial, Heegaard-Floer, and Casson-Gordon invariants.

Findings

Certain pretzel knots' non-sliceness is detected by Alexander polynomial.

Heegaard-Floer correction terms do not obstruct rational homology ball bounding.

Casson-Gordon invariants show the double branched covers do not bound rational homology balls.

Abstract

We give a necessary, and in some cases sufficient, condition for sliceness inside the family of pretzel knots $P (p_1,...,p_n)$ with one $p_i$ even. The three stranded case yields two interesting families of examples: the first consists of knots for which the non-sliceness is detected by the Alexander polynomial while several modern obstructions to sliceness vanish. The second family has the property that the correction terms from Heegaard-Floer homology of the double branched covers of these knots do not obstruct the existence of a rational homology ball; however, the Casson-Gordon invariants show that the double branched covers do not bound rational homology balls.