Abstract
We exhibit a knot $P$ in the solid torus, representing a generator of first homology, such that for any knot $K$ in the 3-sphere, the satellite knot with pattern $P$ and companion $K$ is not smoothly slice in any homology 4-ball. As a consequence, we obtain a knot in a homology 3-sphere that does not bound a piecewise-linear disk in any homology 4-ball.
Highlights
A knot K in the boundary of a smooth 4-manifold X is called smoothly slice (in X ) if it bounds a smoothly embedded disk in X , and topologically slice if it merely bounds a locally flatly embedded disk (that is, a continuous embedding with a topological normal bundle)
A knot K in the boundary of a smooth 4-manifold X is called smoothly slice if it bounds a smoothly embedded disk in X, and topologically slice if it merely bounds a locally flatly embedded disk
The classical study of knot concordance aims to classify which knots in the 3-sphere S3 are slice in the 4-ball D4
Summary
A knot K in the boundary of a smooth 4-manifold X is called smoothly slice (in X ) if it bounds a smoothly embedded disk in X , and topologically slice if it merely bounds a locally flatly embedded disk (that is, a continuous embedding with a topological normal bundle). 2k}, let Vect(Y, s) and Spinc(Y, s) denote the set of homotopy classes and homology classes, respectively, of nonvanishing vector fields on Y restricting to vs Elements of Spinc(Y ) are called relative spinc structures (relative to vs); note that Spinc(Y, s) is an affine set for H 2(Y, ∂Y ; Z).
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