Locally flat PL knot concordance

Use the standard PL three-sphere given by the boundary of a fixed PL four-simplex and the product PL structure on $S^3\times I$. A PL map is a map that is affine on every simplex after finite subdivisions of triangulated domain and target. An oriented PL knot is the image of a PL embedding of an oriented triangulated circle into this PL three-sphere. Two such knots are PL concordant when a PL embedding of a triangulated annulus into $S^3\times I$ has them as its lower and upper boundary circles with boundary orientation $-K_0\times \{0\}\bigsqcup K_1\times \{1\}$, and the embedding is proper, locally flat in the PL category, and has PL product collars agreeing with the ambient end products: for some $\epsilon >0$, its image near the lower end is $K_0\times [0,\epsilon )$ and near the upper end is $K_1\times (1-\epsilon ,1]$. PL local-flatness means that each interior pair chart is PL-homeomorphic to $({\mathbb{R}}^4,{\mathbb{R}}^2)$ and each boundary pair chart is PL-homeomorphic to $({\mathbb{R}}_{+}^4,{\mathbb{R}}_{+}^2)$. Annuli with cone singularities or other non-locally-flat points are excluded. This definition makes no assertion identifying PL concordance with smooth or locally flat topological concordance.

Properness and the standard pair-chart convention

The standard three-sphere and oriented smooth knot model