Algebraic & Geometric Topology

Slice implies mutant ribbon for odd $5$–stranded pretzel knots

Kathryn Bryant

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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 5–stranded pretzel knots are mutant ribbon, meaning they are mutant to a ribbon knot. We do this in stages by first showing that 5–stranded 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 5–stranded pretzel knot with zero pairs or with exactly one pair of canceling twist parameters is not slice.

Article information

Algebr. Geom. Topol., Volume 17, Number 6 (2017), 3621-3664.

Received: 21 September 2016
Revised: 7 February 2017
Accepted: 26 February 2017
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32S55: Milnor fibration; relations with knot theory [See also 57M25, 57Q45] 57-XX: MANIFOLDS AND CELL COMPLEXES {For complex manifolds, see 32Qxx}

slice ribbon pretzel knot Donaldson's theorem d-invariant


Bryant, Kathryn. Slice implies mutant ribbon for odd $5$–stranded pretzel knots. Algebr. Geom. Topol. 17 (2017), no. 6, 3621--3664. doi:10.2140/agt.2017.17.3621.

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