Non-slice 3-stranded pretzel knots

TL;DR

This paper investigates the sliceness of certain 3-stranded pretzel knots, showing that most in a specific family are not slice, thus contributing to the understanding of the slice-ribbon conjecture for these knots.

Contribution

It extends previous work by proving that four-fifths of the remaining family of pretzel knots are not slice, advancing the classification of these knots.

Findings

Most of the remaining family are not slice

Confirmed non-sliceness for a large subset of the family

Progress towards the slice-ribbon conjecture for pretzel knots

Abstract

Greene-Jabuka and Lecuona confirmed the slice-ribbon conjecture for 3-stranded pretzel knots except for an infinite family $P(a,-a-2,-\frac{(a+1)^2}{2})$ where $a$ is an odd integer greater than $1$. Lecuona and Miller showed that $P(a,-a-2,-\frac{(a+1)^2}{2})$ are not slice unless $a\equiv 1, 11, 37, 47, 59 \pmod{60}$. In this note, we show that four-fifths of the remaining knots in the family are not slice.