Non-slice 3-stranded 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 $a\equiv 1,11,37,47,59(\mathrm{mod}60)$. In this note, we show that four-fifths of the remaining knots in the family are not slice.