Planar extensions in o-minimal structures

Let $X\subset {\mathbb{R}}^2$ be a closed definable set of dimension at most $1$, and let $h:X\to {\mathbb{R}}^2$ be a definable continuous injective map. In this paper, we establish necessary and sufficient combinatorial conditions, formulated in terms of cyclic orders at topological singular points and orientations of Jordan curves, for $h$ to admit a definable homeomorphic extension to the whole plane.