#### Volume 18, issue 5 (2014)

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A knot characterization and $1$–connected nonnegatively curved $4$–manifolds with circle symmetry

### Karsten Grove and Burkhard Wilking

Geometry & Topology 18 (2014) 3091–3110
##### Abstract

We classify nonnegatively curved simply connected $4$–manifolds with circle symmetry up to equivariant diffeomorphisms. The main problem is to rule out knotted curves in the singular set of the orbit space. As an extension of this work we classify all knots in ${\mathbb{S}}^{3}$ that can be realized as an extremal set with respect to an inner metric on ${\mathbb{S}}^{3}$ that has nonnegative curvature in the Alexandrov sense.

##### Keywords
nonnegative curvature, circle actions, knots, Alexandrov geometry
##### Mathematical Subject Classification 2010
Primary: 53C23
Secondary: 57M25, 57M60