## Algebraic & Geometric Topology

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

Kathryn Bryant

#### 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 $5$–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

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

Dates
Revised: 7 February 2017
Accepted: 26 February 2017
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841516

Digital Object Identifier
doi:10.2140/agt.2017.17.3621

Mathematical Reviews number (MathSciNet)
MR3709656

Zentralblatt MATH identifier
1376.32035

#### Citation

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. https://projecteuclid.org/euclid.agt/1510841516

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