Pretzel links, mutation, and the slice-ribbon conjecture

Abstract

Let p and q be distinct integers greater than one. We show that the 2-component pretzel link P (p, q, −p, −q) is not slice, even though it has a ribbon mutant, by using 3-fold branched covers and an obstruction based on Donaldson's diagonalization theorem. Combining this result with previous work of the first author we prove the slice-ribbon conjecture for 4stranded 2-component pretzel links.

Non-slice 3-stranded pretzel knots

“…Lecuona [Lec15] confirmed the slice-ribbon conjecture for all 3-stranded pretzel knots P (p, q, r) except for infinitely many 3-stranded pretzel knots of the form P (a, −a − 2, − (a+1) 2 2 ) for a ≥ 3 odd. There are further interesting results on the slice-ribbon conjecture for general pretzel knots and links (for example, see [Lon14,Bry17,AKPR18,KST20]).…”