A Short Survey of the Work of Cochran-Orr-Teichner on Knot Concordance

Definition 1.1. [8] An oriented knotK is a topologically slice knot if there is an oriented embedded locally flat disk D2 ⊆ D4 whose boundary ∂D2 ⊂ ∂D4 = S3 is the knot K. Here locally flat means locally homeomorphic to a standardly embedded R2 ⊆ R4. Two knots K1,K2 are concordant if there is an embedded locally flat annulus S 1 × I ⊂ S3 × I such that ∂(S1×I) ⊆ S3×I is the disjoint union of the knots K1⊔−K2, where the knot −K arises from K by reversing the orientation of the knot and of the ambient space S3: on diagrams this latter means switching under crossings to over crossings and vice versa. The set of concordance classes of knots form a group C under the operation of connected sum with the identity element given by the class of slice knots.