Slice implies mutant ribbon for odd 5–stranded pretzel knots

Abstract

A pretzel knot K is called odd if all its twist parameters are odd, and mutant ribbon if it is mutant to a simple ribbon knot. We prove that the family of odd, 5-stranded pretzel knots satisfies a weaker version of the Slice-Ribbon Conjecture: All slice, odd, 5stranded pretzel knots are mutant ribbon. We do this in stages by first showing that 5stranded pretzel knots having twist parameters with all the same sign or with exactly one parameter of a different sign have infinite order in the topological knot concordance group, and thus in the smooth knot concordance group as well. Next, we show that any odd, 5stranded pretzel knot with zero pairs or with exactly one pair of canceling twist parameters is not slice.

Surgeries on torus knots, rational balls, and cabling

“…Since b 1 = 2, we know that c 1 > 2 which implies in turn that c 1 = 5, so that b 1 + c 1 = 7. From here an easy induction shows that the string L we are looking at is either [7,2,2,2], or [7 [s+1] , 2, 2, (3, 2 [4] ) [s] , 2]. Since this corresponds to b = [7 [s+1] , 5], we obtain the fractions…”

“…Montesinos family bounding surfaces of Euler characteristic 1 by several authors (see for example [28,29,18,7,25,26,31,11]).…”

“…Two examples are representative of this issue. First, the manifold S 3 −1 (T 2,3 ), which is the Brieskorn sphere Σ (2,3,7), is known to bound a rational homology ball [15], but the construction of the rational homology ball is highly nontrivial. Indeed, since the Rokhlin invariant of Σ(2, 3, 7) is non-trivial, any handle decomposition of such a rational homology ball must contain a 3-handle.…”

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]).…”

Pretzel links, mutation, and the slice-ribbon conjecture

“…The papers of Greene-Jabuka, Lecuona, and Choe-Park mentioned above sidestep mutant knots since there is no non-trivial mutation within the families they consider. However, the problem remains for more general families of knots and links if using double branched covers, as seen in the work of [Lon14,Bry17,Ace18].…”