Locally flat proper surfaces and concordance collars

Let $X$ be a four-dimensional manifold with boundary and let $W$ be a compact surface with boundary. An embedding $e:(W,\mathrm{\partial }W)\to (X,\mathrm{\partial }X)$ is proper when $e^{-1}(\mathrm{\partial }X)=\mathrm{\partial }W$. It is locally flat at an interior point of $W$ when there are neighborhoods of the point and its image and a homeomorphism of pairs from those neighborhoods to $({\mathbb{R}}^4,{\mathbb{R}}^2)$, with the standard linear plane ${\mathbb{R}}^2\subset {\mathbb{R}}^4$. It is locally flat at a boundary point when there are such neighborhoods and a homeomorphism of pairs to $({\mathbb{R}}_{+}^4,{\mathbb{R}}_{+}^2)$, where ${\mathbb{R}}_{+}^4=\{x_4\ge 0\}$ and ${\mathbb{R}}_{+}^2=\{x_2=x_3=0,x_4\ge 0\}$; the line $\mathrm{\partial }{\mathbb{R}}_{+}^2$ lies in $\mathrm{\partial }{\mathbb{R}}_{+}^4$. A collar of a boundary circle of $W$ is an embedding of that circle times $[0,\epsilon )$ into $W$ that is the identity on the circle at parameter $0$; a concordance collar is required to agree with the corresponding product collar of the ambient end. A smooth proper surface has the analogous smooth charts and collars, which are also topological locally flat charts.

Oriented knots and the standard product ambient space