Periodic ends and knot concordance
Two knots \(K_ 0\) and \(K_ 1\) in \(S^ 3\) are smoothly concordant if there is a smooth annulus embedded in \(S^ 3\times I\) with boundary \(K_ 0\times \{0\}\cup K_ 1\times \{1\}\). The set \({\mathcal K}_{DIFF}\) of smooth concordance classes of knots is an abelian group under connected sum. Elements of the identity class are called smoothly slice. Similarly, the topological concordance group \({\mathcal K}_{TOP}\) is obtained by using flat topological annuli in place of smooth ones. Inclusion gives a canonical epimorphism i: \({\mathcal K}_{DIFF}\to {\mathcal K}_{TOP}\). The main result of the paper is that the kernel of i is larger than Z, even modulo torsion. The proof uses the Taubes theorem on nonexistence of certain definite 4-manifolds with periodic ends. A corollary of the proof is that there are uncountably many diffeomorphism types of Casson handles. A specific collection of doubled knots is exhibited, no two members of which are smoothly concordant.
uncountably many diffeomorphism types of Casson handles
scientific article; zbMATH DE number 4126320